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You tell me about it. I was just minding my own business in room 654, and all of a sudden the receptionist told me to pack up and climb the stairs all the way up to some new room with a 197 digit number. Not even sure what the point is, no one else moved into my old room, it was just left vacant.
To be honest, they could solve their issues by simply making sure that they always have infinitely many rooms available at any given time. Have everybody move once, and you're done. And if they have a problem with it, there are plenty of ways to make the proportion of empty rooms arbitrarily small the higher their numbers.
I hate staying at Hilbert's Hotel. I had this great room right by the pool, and right across the hall from the where they serve the complementary breakfast, then this uncountably infinite group of guys shows up and they make _me_ move rooms. Now I have to wake up crazy early if I want to get to the breakfast before the heat death of the universe.
@@Alex-mh4lt They very well could be just a countably infinite group of guys, for all I know, because I haven't actually tried to count them to figure that out.
@@Alex-mh4lt Yeah... keeping uncountably many visitors in countably many rooms means uncountably many guests have been stuffed into a single room somewhere! Such practices by the hotel administration are outrageous.
If an infinite amount of guest had to come to the hotel everyone could just go to the room double of theirs leaving infinite spaces so I hope you are in a room below 10k
when you've been walking down the hall in Hilbert's Hotel for the past 10^6875 years to reach your room and an announcement plays saying "please move to the room number that doubles yours"
But you could easily fit all of Hilbert's hotel on the real interval [0,1]. So you could take the length of the hall to be finite, the problem is trying to stop at a particular room since you can pass infinitely many of them on a finite length.
The paradox of the Gabriel's horn is that you need an infinite amount of dye to paint it inside with a brush, but only a finite amount of dye to fill it (and paint it anyway).
The biggest reason why these paradoxes exist is because they assume and treat infinity like it's a finite number. It's the same kind of logic as when people say 0 divided by 0 is 1, because of the observed rule that every other number divided by itself results in 1, which while true, doesn't mean that it also necessarily applies to 0. It's an interesting thought experiment though.
Yeah, like mathematicians think 0! = 1. Just makes life easier for them when in actual fact 0! must equal 0. All numbers are values, but with the exception of 0 and infinity. That's why they don't play nice with others.
@@brettr7970 isn't it 1 by mere definition of factorial tho idk it works out everytime i have to do maths involucring factorial and 0 , better than it does saying that 0/0 = 1 lol
I always thought of it as sets of a thing. If you have nothing and divide it into nothing sets you have nothing. 1 divide into 1 set is 1. 5 divided into 5 sets is 1. 10 divided into 10 sets is 1. And so on. You could could say a set of nothing nothings is 1 set of that…. But then 1 set divided into 1 set is 1 set of that… which is just stupid to say.
For Gabriel's horn you were supposed to calculate the surface area of the area of revolution, not the area beneath the curve of 1/x. The surface area integral is the integral from 1 to infinity of ((1/x)sqrt(1+(1/x^4)))dx which you can bound below by the integral you did, which diverges.
My long-time statistics hot take is that the "expected value" operation is given undue significance. Its name is very suggestive of it being the inherently correct way to judge risk and reward, but the fact of the matter is that it is fairly uncommon for it to actually yield a value which you can, in any meaningful way, expect. The St Petersburg paradox shows this well. There is less than a 7% chance of getting $32 or more from the game. A person simply cannot, in the ordinary sense of the word, expect more than $32, regardless of the value of the operation which has been named "expected value". Pascal's mugging is another "paradox" that just comes down to the expectation operator being treated as gospel. A mugger says if you give them your money, they will give you x amount back later. No matter how improbable you believe it to be that they will uphold the deal, they can give an x for which the "expected value" of taking the deal is positive, giving muggers a surefire strategy against people who treat the "expected value" operation as the ultimate arbiter. "Expected value" is only genuinely accurate to its name when the risk is taken infinitely many times. Humans aren't immortal. The risks also often have an ante of some kind, and if the risk failed enough times, you can't afford the ante anymore and can't try again. Even with large corporations, which can absorb many more and much larger failed risks than people can, risk assessment goes beyond expected value -- they qualify or quantify the level of risk itself and compare that to a risk tolerance
I was a professional poker player for a long time, so EV is a concept I'm vary familiar with. It can be incredibly beneficial to think in terms of EV in countless real life scenarios. However, real life is just as you said, not infinite. So you also have to factor in utility, otherwise you get into the absurdities you mentioned and the concept loses the value and importance it has.
Expected value is great for getting an idea of how things shpuld trend. But you have to temper it with analysis into short term risks. The combination works really well for evaluating an overall idea by seperating short and long term benefits
What do you think about taking the median instead of the mean? (Me using statistics terms instead of probability terms.) It sounds like the median gives you a much better idea of the short-term reward: you have 50% chance to get above the median and 50% chance to get below.
My thinking on the Ross-Littlewood paradox is that if the balls are removed in order, whenever any ball of number ‘n’ is removed, the vase will still contain ball n+1. So even though for all values of n, ‘ball n’ will be removed, it is impossible for the vase to be empty.
While I think your point is valid, I think the question is asking about the state of the vase after all actions have been taken. The paradox is saying that every ball will have been removed at some action before the end, thus one cannot claim to have a ball still left in the vase.
@@n484l3iehugtil And I understand that. My thinking is along the line of Cantor's Diagonal Argument for proving the real's are uncountable. Simplifying for brevity, it says that if you list "all" reals in a numbered list, a new real can be constructed from the nth digit of every number n, adding 1 (without carrying, so 9s roll back to 0). Since this number differs from all others in the list by at least one digit, it must not be in the set, and thus the set does not contain all reals. So even when starting from the premise that the set contains all real numbers, it can be shown that it doesn't. Similarly, even with the premise that "all" balls will be removed, the last ball removed will always have its successor still within the vase, and thus the vase can never be empty. Another more mechanical proof is that with each step, you add 10 balls and remove 1, so the number of balls at each step is 10x - x, and the limit as x approaches infinity for that is infinity.
@@n484l3iehugtil I get what you’re saying but, (and I may be wrong) at least in their example there is always 9 times as many balls as have been taken out so no matter what number of steps you take even if it’s an infinite number of times there will always be at least 9 times as many balls in the vase as have been taken out. The mathematical format of what you appear to be stating is: “The limit as n approaches infinity of (10n-1n)” Which would normally create a indeterminate form of (infinity-infinity) but, it can always be factored which then just gives n(10-1) or n(9) with the limit as n approaches infinity therefore = infinity
The problem with the paradox is in the idea of a super-task. It doesn't matter if the steps take 0 time, an infinite number of steps are never finished. Imagine you're driving around a roundabout, what direction are you facing when you get to the "end" of the roundabout? It doesn't matter how fast you drive, or if you're doing extra stuff, it's a circle. What does this program do if it's run on a machine that's infinitely fast? 10 GOTO 10 20 PRINT "Hello" It just hangs. Putting balls in a vase doesn't change anything: 10 balls = 0 20 balls = balls + 10 30 balls = balls - 1 40 GOTO 20 50 PRINT balls You never get to line 50. It's just an infinite loop with busywork.
@@nisonaticthe point of the thought experiment is to assume it happens and explore the outcome, not to explore whether or not its possible. thats like when someone asks u "what would u do in a zombie apocalypse" and then u respond "zombies dont exist".
The spot you choose on the dartboard may be infinitely small. But the dart itself is not! Therefore, the dart does not actually hit any one spot. Instead, it hits a small area. And you can totally calculate the probability of your chosen spot lying inside that area.
@@herrhartmann3036 i did that in another comment thread. Using a dart point area of a hundredth of a square millimeter and a regulation 13.25 in diameter dartboard, there are roughly 9 million potential dart hits in that don't overlap. Now, if you are counting overlapping points as different, we have the issue of atomic width. After all, a dart cannot go through a foam atom. It can only go between two, and so this too is calculable and finite (though I'm not doing it).
If we take the fact that the sum of all probabilities of all points should be equal to 1, and therefore the probability of a single point is infinitesimally small but not zero. What am I missing?
@@iseslc By definition, a "point" has no area. Therefore, the fraction of the whole dart board which is represented by any given point on it, is equal to 0. And the probability of hitting a size 0 point is also 0. However, as stated above, the dart actually hits a small area, rather than a point. This area has a size greater than 0, and therefore it represents a fraction of the whole dart board greater than 0. And the probability that our arealess point is located within this area can be calculated.
I find the St. Petersburg paradox very interesting, it highlights (as many other paradoxes here do) how "right" results are wrong when we ignore other variables applicable to that context. For example looking at the variance. While the EV is theoretically infinite we can plot every price of the game to the odds of turning a profit, and how much total capital we would need to ensure a profit in the long-run.
You could calculate with a logarithmic function how much it would do to you where you to win or lose and that would then be based on the money you own and you would get a more accurate result, this assumes that doubling your money has the same absolute impact as halfing your money
the probability of hitting any individual point on a dartboard changes quite a lot when you remember that the point of the dart itself ALSO has an area to it, and thus its own infinite number of points
Infinity is infinite, even if you subtract infinity from infinity there will still be infinty left over. The amount of points on the dart changes nothing.
@@Lisa-dd1uq it's not a matter of subtracting, but dividing. and in analyzing this particular problem physically there is actually no need to resort to infinites - both the relevant areas (dart and board) can be measured in real units with physical tools
@@Lisa-dd1uq ok, think of it this way: if there are an infinite number of points on the dartboard, and an infinite number of points at the tip of the dart, then you say that the chance of ever hitting one specific point on the dartboard is zero. but if the numbers are both infinite, then the chance of hitting any individual point on the tip of the dart must ALSO be zero - even though geometrically we assume that ALL points on the tip of the dart are hit when it enters the dartboard.
★☆☆☆☆ · Would NOT recommend! I was at Hilbert's Hotel for a bit, I stayed in room 92e+12, then they said to go to double your room number, it's been 30 octillion years, I'm nowhere near that room.
Thomson's lamp reminds me of calculating average values - in average, every person, who has two arms, has more arms than the average person, because there are (for sure) more persons missing limbs than having extra one's.
An important thing to note about Cantors Diagonalisation proof is that is is a proof by contradiction; it is assumed that there is a mapping between all the real numbers between 0 and 1 and the naturals then using logical steps you show that there is a number between 0 and 1 not in the mapping. From there a contradiction is raised (the value both being in the mapped values and not) and the initial premise is rejected. It’s a bit pedantic but distinct to being able to add it to the mappings and do it again.
Perhaps the real bullshit is holding hard onto the notion that the Universe in general gives two shits about what human epistemology considers a paradox, and will adjust itself accordingly to satisfy our limited capcities of observation, comprehension, and rationality.
Yet sometimes it's good to recognise what kind of paradox you're dealing with and recognise the limits of your models to make a useful decision. Way too often we use our models without considering their limitations and tell people to shut up when they point out the limitations and how they could lead to problems if ignored.
You talk about the dartboard as if the tip of the dart was infinitely small as well, but the dart does take up space. The dart can hit multiple infinitely small points on the board at the same time. If you measure the surface area of the dart (D = Dart = N1) and the surface area of the board (B = Board = N2) then you will have a N1 in N2 chance of hitting any given spot. It’s an interesting theory though. If the dart tip was infinitely small and the very tip was the only part that would ever hit the board, it would be very cool to think about.
Regarding the St. Petersburg Paradox, I think the reason for why people would not pay an infinite amount of money to play this game is that if the probability of tails is even slightly less than 50%, then the expected value immedeatly becomes finite. For example if the probability of tails was 47.9%, then paying 25 Dollars would already put you at a disadvantage as a player.
Although this is mathematically true, I disagree that this is the primary psychological reasoning. Borrowing from economics, a person might intuitively aim to maximize expected “utility”. Bc of diminishing returns, utility is a sub-liner function of money and thus under-weights rare, high-value outcomes. This generalizes to other behaviors as well
@@LoganPost-c6pthank you for saying this. money usually doesn’t add never-ending value/happiness to one’s life. therefore, there’s no paradox in not investing in this game
I surmise that it is because the chance of losing is much bigger than the chance of winning. If I had to pay $1000, my chance of net winning money would be approx 0.1% which feels negligible (unless I made an off-by-one-error, but it would still be very small). The fact that I would win a lot if I win does not outweight that (you could insert utility here, but I suspect people are simply sensitive to probability of winning in addition to expected payout).
An interesting variant (or at least a related) of the Ross-Littlewood (Vase and Balls) paradox: 1] Alice starts building a stack of dollars bill by adding 1000 bills at a time. This continues for an infinite number of turns. 2] Bob’s task is to ensure that there are no bills left in the stack at “the end”. 3] Bob must choose between the following two strategies: Option 1: Each time Alice places her 1000 bills, Bob can remove 999 bills from the top of the stack. Or Option 2: Each time Alice places her 1000 bills, Bob can remove 1 bill from the bottom of the stack. So, in short, which is the winning strategy?
@@benjaminhill6171 : Right. I’ve always thought that the correct answer (that is, removing 1 bill each time results in 0 bills left in the stack, while removing 999 bills each times results in an infinite number of bills) is pretty much the definition of a counter-intuitive answer. ;-)
@@simohayha6031 My gut says the expected value would be $2 - that is, if you flip the coin infinitely many times, you would always hover around $2 - but I'm feeling too lazy to try to prove it, haha.
I mean correct me if I’m wrong mathematically but the Ross-Littlewood paradox is just the infinite sum of 10-1 (or just 9) with n=any natural number because no matter what on each step, the net number of balls being added in = 9. At any given point in time there is at least 9 times as many balls as have been taken out. Of course graphically it just goes to infinity which can be proven with a limit, I don’t quite get the paradox of “Well at some point every ball has to have been taken out of the jar” as mathematically that’s just impossible. Even if you tried to make it a limit formed as “the limit as n approaches infinity of 10n-1n” (as to of make it indeterminate by (infinity-infinity)) you can just factor it out as n(10-1) which after taking the limit still approaches infinity
The problem is that you can't necessarily factor infinite numbers like that. The algebra concept you're using is factoring for finite numbers. Infinite numbers are not guaranteed to be so well-behaved, so it's not a valid operation. This is a paradox because the solution is completely indeterminate - if you assume that any number is the answer, it leads to a contradiction. If you assume there are balls left, then they must have numbers, and once the supertask is finished all of those numbers must have been removed, so there can't be any balls left. But if you assume that there are none left, that's also a contradiction because there must be at least nine left because that's the net number added in the "last" step. Anything you assume leads to a contradiction.
@@benjaminhill6171 agreeing with benjamin, since the limit of the function '10n - n' can be said to go towards infinity, we plug in infinity to get '10*inf - inf', however any number * infinity is still infinity, so we get 'infinity - infinity', which is indeterminant.
the reason why it breaks is because infinity doesnt have an end. the very idea of even finishing the task is ridiculous, because finishing requires there to be an end to infinity. thus, the entire thing is illogical, but even then, since there is no end, there is no point where ALL balls were removed, because you cant have an end! you can confidently say, even though the answer to the question is null, that even if there were an answer it would be infinite.
@@aidenaune7008 I was actually corrected since my last comment here, and at least according to common mathematical axioms, the "correct" answer to this paradox is zero. Because of the weirdness of infinity, if you assume the supertask is completed, there must be no balls left. I found a whole Wikipedia article explaining the various purported solutions, and the accepted mathematical solution is zero. So, at least mathematically, there is no direct contradiction here, so this isn't a "paradox" paradox. Regarding there being an "end to infinity" as you called it, if you've ever learned about geometric series you know that it's a perfectly sound concept. This has nothing to do with actually being able to observe each step in the infinite process; you just assume the process is completed, which, again, is logically sound. In math you can just assume that anything you want happens, and then observe the consequences. That's what we're doing here, and the consequence is that there are no balls left after the task is done.
@@benjaminhill6171 your defense of there being an end to a definitionally endless number is that the geometric series assumes that there is one? if I were to start with the number 0 and add one to it an infinite number of times over a finite amount of time, what number would I end with? and no, its not infinity, infinity is not a number, its a concept. if you give ANY definable number that can be placed concretely and discretely somewhere on a number line, there is a finite number of numbers before it. if you dont, then what you gave me is not a number, its a concept, and once again cannot be the result of a series of math problems, as they will always give a number. infinity fundamentally is not a number, and cannot be treated like a number. it cannot be combined with any number or mathematical operation because the very concept is undefinable in the mathematical sphere. the concept of infinity definitionally does not have an end. if it did, then there would be a finite number of numbers before it, and thus it would be finite. and I am not talking about an end to its string of digits either. if you were to make any assertion about its smallest or largest digit then it would still fail to be placeable on the number line due to its infinite digits in between, which is proven by the exact argument I made above. you CANNOT make assertions about what the end of an infinite order of operations looks like, because infinity does not end. there is no final result, there is no outcome that you can discretely point to, there is nothing. the answer is null. I dont care what mathematicians say, they are wrong. just because a group of self proclaimed experts agree on something, that doesnt make it true. if radiologists were to wake up tomorrow and all proclaim the sky to be purple, it would not be purple!
It’s only countable infinity, so if an infinite number of buses filled with an infinite number of people arrive at the hotel looking for a room, it’s uncountable infinity because the hotel will always need to provide +1 room than the number of tenants to maintain vacancy, and there are two layers of infinity stacked against the hotel’s singular plane of infinity.
@@Xhu666I don't think that answers their question. @fulltimeslackerii8229 The paradox isn't whether or not it's physically possible to have an infinite hotel, or for it to be completely filled. The paradox is about what happens if you could. This is a purely mathematical paradox because it's just a way to visualize how you can fit more numbers into an infinite set without changing its size. Also, this is the kind of paradox that is only called a paradox because it's unintuitive, not because there's actually a logical contradiction - because there's no contradiction here.
@@josephoyek6574 Great question! All the rooms are already full at the start, so there actually isn't an empty room at any "n+1" place. That's why it's unintuitive: there's no room at the start, but just reordering the people opens up a room. It's an analogy to show that if you add things to an infinite set, that set doesn't get any bigger.
If we take the lamp on/off one literally while considering the constant speed of light, then having the light half on would be a measurable result if the light is turning on and off at as the rate at which it is alternating between on and off is at the speed of light, at which point, it can no longer be turned on or off any faster. So we'd see a dimmed light, as though it were half on. Phenomena we can observe and measure at least seems to have some finite limitations like the plank length, and the speed of light. Gravity was a constant according to Newton's observations, until we could model it as a curvature in space and time caused by mass thanks to relativity. We don't even have a fundamental explanation as to "why" mass curves space and time. Gravity still isn't fully understood. Numbers, and models and math are all abstract. The reality may be indefinite; a bunch of larger or smaller infinities, and so I'm inclined to think that math and physics are always going to be shy of total consistency.
I think your explanation of Cantor's diagonal argument doesn't hold up. Why can we map the reals to the naturals in the first place (what if we run out of them)? Also how did you map them? And why shouldn't we be able to do the Hilbert's hotel thing and move all the reals one space down to make room for more? These things can be explained/worked around, but you left out the parts of the proof that do that. The parts you did keep don't really follow a logical flow either. Why are we even constructing a new number? You explain it, but only after making it...
I'm not sure which bit of it you're questioning, but the point is that you can't map the naturals to the reals, no matter what way you try to link the numbers together. The reason for not explaining the way they are mapped is because it has to be arbitrary, or the proof has to work no matter what way they're mapped. if you said "well 1 will align with 0.0000001, 2 with 0.0000002" and so on, obviously that won't work for listing all real numbers, but you've only proved it for that specific arrangement, not any arbitrary arrangement. The point of the diagonal is that you *always* can add one more real number, even an infinitely long one to an infinitely long list. if your list is infinitely long, you've listed all the natural numbers already, and theoretically that should also mean you've listed all the reals. but if you can add another, or even infinitely more, that means you don't have all of them, so the mapping doesn't work. In regards to the hilbert hotel thing, yes, you can keep adding numbers to the beginning of the list! But, since you're doing the exact same checking and number creation every time you add a number, you're not actually going to make any progress in listing all of the real numbers, and you're not going to hit a point where you have all of them and the diagonal strategy doesn't work, even after an infinite number of reals added. Hopefully that makes a little more sense, I feel like I said number and infinity a few too many times :P
The whole point of the proof by contradiction is that it started with the assumption that we could map the reals to the naturals, and then proved that assumption wrong, because we created a real number that had no mapping. If it were possible to do the map in any way, we could not create that number. The mapping method doesn't matter because the process of generating the new real number works for any mapping method.
@@elderfrost9892 I think you're just mapping the numbers wrong. If you match the naturals to reals like: 1 = 0.1 2 = 0.2 3 = 0.3 ... 9 = 0.9 10 = 0.01 11 = 0.11 12 = 0.21 ... 99 = 0.99 100 = 0.001 That'll work just fine. If you want to pick a random infinite string, you must do it for both sides: ...36492749 = 0.887592738... ...47494693 = 0.474926957... ...39572074 = 0.164959378... Naturals (like 3) can be thought of as having an infinite series of leading zeros (...000003). This is exactly how we think about reals, just reversed. Reals (like 0.5) have an infinite series of tailing zeros (0.500000). If you think about it like this you'll realize that the size (cardinality) of the set of naturals and the set of reals are exactly the same. There are absolutely not more reals between 0 and 1 than natural numbers. Both are uncountable.
The problem with the Hilbert's hotel paradox and many other infinity paradoxes is that they violate a fundamental law of numbers. We have been told over and over again that infinity is NOT a number, but rather a concept. But when we say that Hilbert's hotel has an infinite number of rooms, we treating infinity as a number! It only is a paradox because we are violating the rules.
@CosmicButterfly2 no. Most of these paradoxes are nonsense they make and then break their own rules. They're just dumb word games. If a set contains infinite people, you couldn't add infinite people to it because they would already be in it.
Ross-Littlewood paradox is just Grandhi's series 2.0: instead of series alternating between 1 and -1, this one alternates between 10 and -1 in both cases the sum cannot be defined properly, Ross-Littlewood just needs an extra step to see that
Fun counterexampke to ross little wood, add 10 balls, numbered, remove the smallest number divisible by 3, repeat. After the super task is complete there will be infinite balls, every ball not a multiple of 3
The point of the ross little wood formulation is that you are removing *the entire set of natural numbers* from the jar,, the same set you are adding to it. If you only remove a subset of it then there's no paradox.
@@unadulterated i know, its a cardinality bijection thing. Im just giving an example of a sanity check for the sequence, how this kind of operation is incredibly malleable to leave you with any amount leftover
For the dartboard paradox, with infinitely small points, when split into a quadrants, how does the premise of a dart being able to hit the precise center or even just the precise divide between two quadrants affect it, wouldn’t that mean there’s a non zero chance of it landing outside just one quadrant, meaning it wouldn’t be precisely 1/4, but infinitely close to 1/4?
The probability of the dart hitting the dividing line is 0, so the probability of the dart hitting the interior of the quadrant is the same as the probability of the dart hitting the interior or the border. 1/4+0=1/4
Hilbert's Hotel's Murphy's Law: Even though they can accomodate an infinite amount of guests they never seem to have enough money to fix the ice machine
Good video, BUT click bait title!!! Knowing infinity, it obvious there are infinitely many. Banach-Tarski paradox, or Pythagoras theorem which is also link with infinity (infinitely small)
Some more infinity paradoxes Consider "the set of everything." By the diagonal argument (on subsets rather than on digits) one can prove that it doesn't coontain the power set of "the set of everything", hence "the set of everything" doesn't contain everything. Solution: modern set theory gives axioms that give a restriction to what can be a set, denying "the class of everything" as a set. So it has no power set. According to (modern) set theory, cardinal numbers and ordinal numbers are both sets. A cardinal number is defined to be an ordinal number alpha that is equal to the lowest ordinal number beta such that alpha can be mapped injectively into beta. Then the smalles transfinite ordinal number often is denoted w (in fact we call it omega.) It is also a cardinal number, and as a cardinal number it will often be denoted Aleph_0. But, w^w is countable while Aleph_0^Aleph_0 is uncountable. Explanation/solution: when doing ordinal numbers, alpha^beta is an ordered set. But when doing cardinal numbers, alpha^beta is just a set (and we forget about order.) So w^w differs from Aleph_0^Aleph_0. Banach - Tarski Paradox. It is possible to 'cut' a solid ball into a finite amount of pieces and reassemble those pieces such that you assemble two balls, each being a perfect copy of the original ball. However, each of those 'pieces' is in fact an infinite scattering of points, rather than a solid piece of the ball. Achilles Paradox. Consider a race between Achilles (who was able to run very fast) and a turtoise (who was quite slow), with the turtoise starting ahead. Each time Achilles covers the distance between him and the turtoise, the turtoise also moved a little bit. Hence Achilles has to cover another distance to catch up the turtoise. This repeats infinitely many times. "So Achilles can't win the race." It took quite the time before this paradox was finally solved. First we had to define a notion of "converging limits" before finally reason well why Achilles is able to come ahead of the turtoise. The point where Achilles will pass the turtoise is the limit point of the set of points where either is at any of the mentioned moments. These are the moment the race started, the moment Achilles has reached the starting point of the turtoise, the moment Achilles has reached the point where the turtoise was at the moment Achilles reached the starting point of the turtoise, etc.
Proper video on this topic should be infinitely long. It is impossible to explain every infinity paradoxes in finite amount of time. Or it create new paradox.
The word is probably one of the most overused, long with the words "unsolved" (generally, not in maths) and "mystery". Like saying the Bermuda triangle is mysterious. It isn't.
2:50 this one doesn’t make sense to me because even if the new real number is not found anywhere in the current real numbers index there will always be another natural number to reference that index. In this wise I can see it being paradoxical but not in the way the problem is stated. In my interpretation of the paradox it’s more like how can a set that can be infinitely broken down also itself be considered infinite logic would dictate the real numbers must be a larger infinity but because there will always be a natural number index that can increase with every real number in that index both must be the same magnitude, or maybe I’m misunderstanding something. 7:15 I also am having a problem with the gabriel’s horn not because the proof isn’t understandable but because despite pi being represented as a solid number solution it has infinite places after the decimal meaning that itself must represent an infinite volume as it’s the same a saying 3 + 1/10 + 4/100 + 1/1000 etc, with every decimal point being the equivalent of adding a fraction of volume equivalent to number over place if that makes sense, so though it would appear to be finite and is indeed useful in finite calculations it is still infinite. Or once again I may be misunderstanding something, I’m really not any sort of mathematician. 8:28 this one doesn’t seem like a paradox to me since every iteration is the equivalent of saying “balls in vase=balls in vase+9” because every iteration, even if you pull out the index value ball, you are still leaving 9 balls in its place with every index. So the real problem would look like “balls in vase=balls in vase + 10 - 1” which is just “balls in vase=balls in vase + 9” because no matter what index we are at we are still adding 10 and removing 1 so it will never and could never empty completely.
For pi, yes you are misunderstanding decimals. No matter how many numbers come after the decimal point in pi, it will NEVER be greater than 3. This is obvious when you consider that 1/3 is 0.333 repeating infinity, but we know for a fact that 3/3 is 1. So no matter how many 3’s there are after the decimal point, they will not make 1/3 infinitely large. As for the jar, you’re misunderstanding infinity. Infinity is not a number. You cannot add 9 to infinity and make it bigger, it doesn’t work like that. The reason it’s a paradox is because if you remove the ball labeled 1 the first time you add 10 balls, and remove the 10 the 10th time, infinite times, then you have removed every single number. It doesn’t matter that 10 times infinity is added, because 10 times infinity is still infinity. You can’t do algebra with infinity because it’s not a number. Though paradoxically, if when you add the 10 balls you remove a ball that is even, like 2, then once you’ve removed the infinite amount of balls, there’s still the infinite amount of numbers in the jar that are odd.
@@UnluckyLilly still having trouble understanding. If Pi will never be larger than 3 (which it is but I understand your point) then the volume as well could never be greater than x amount meaning as pi can’t represent infinite volume in this circumstance then the volume itself can’t be infinite. The jar explanation still doesn’t make sense. Infinity isn’t a number but a representation of uncountable numbers, possibly unending. So yes adding numbers to the concept of infinity doesn’t change the concept, just as subtracting numbers from it wouldn’t change it, but the uncountable number it’s added to would change by 1. That’s not entirely the argument I want to make anyways but it seems to me like a more grey argument than anything else. I think that it goes along with the larger and lesser infinite concepts you are subtracting a lesser infinity from a larger infinity which would still be infinity. But I’m not all caught up on infinity theory so again I could be mistaken.
The bit he jumped over in the Cantor's theorem part of his video is that the "set" of natural numbers, which is infinite, is a function. The set of real numbers is a function too. Functions are how we get or return those numbers out of infinity. The natural numbers look like f(n) = n, so that every number is in a room and if you want to get that number you go to that room; f(1) returns 1, f(2340) = 2340. You can perform operations on real numbers function (uncountable set) that produces a number not contained in its set -- that's the diagonal bit -- but you can't perform those operations on countable sets like the set of natural numbers. Again, its important to stress that it's not the number that is limited it is the function or, if you rather, the set. Yes, you can make a larger natural number, by adding or multiplying, but you can do that to your real numbers as well so those parts of these sets are "one-to-one." The difference is that in the function of the real numbers you can derive a new number and thus that set is, cardinally, larger than a set which is limited to the elements it already contains. You could swap any amount of digits by any amount of numbers in the natural number function but you'll only return another number already contained in the set--the function can't produce a new number because of the function/set. Intuitively, it makes sense that a two dimensional infinity is smaller than a three dimensional one, and that the three dimensional one is infinitely larger. Despite an unlimited amount of numbers available to us in two dimensions, we can't map any numbers to all the ones in the third dimension that aren't already mapped to the other two dimensions.
@@anhi399 that actually does make alot more sense than just saying one infinity is larger than another. I can understand how one function can’t get a return from another set, that makes complete sense. Thank you for explaining it!
It is very well known in maths that there are far more numbers than integers, because there are many more ways of being specific over whatever range you care to name
Those who are truly smart understand there is always something more to learn. Those who arent smart will think theyve got an understanding of everything already.
Hilbert’s Hotel paradox - The concept calls for an unreal premise to begin with with an incorrect set up. An Infinity Hotel can never be fully occupied so the question is invalid.
Math is just counting an infinite amount of zeros. All negative and positive numbers cancel each other out. Zero flips the number line from negative to positive (-0+). Infinity flips it from positive to negative (+0-). Which is how we can fit infinity into a finite space.
@@magnuslunzer2335the countable rooms are full that's the closest to it if not an infinite room can't be full if its full then it can't accommodate any new person
@@yungdkay1008 But it‘s a countable infinite amount because it‘s N_0. All numbers from N_0 never fit in N_0 is technically what this „paradox“ is saying.
This video is look more entertaining then your other videos. I liked the sarcasm in this video and also the animation you put in which make video easy to understand. Kudos to you 👏
Can't you disprove Cantor's diagonal argument by putting natural numbers on the left side of the table with an infinite number of zeros in front of them and then do the same thing proving their are more natural numbers than natural numbers?
@@unadulterated no i mean starting with a list like 0001 - 1 0002 - 2 0003 - 3 0004 - 4 And doing the same thing along a diagonal from the top right, generating the number 1112, and like said in the video, add it it to the list, then repeat to infinitely generate another unpaired number?
The reason Cantor's arguement can't be applied to the natural numbers is due to how the natural numbers are defined. Natural numbers are inherently finite. There are an infinite number of them of course, but any specific natural number has a finite amount of digits (excluding the trailing 0s because left side trailing 0s don't actually change anything). Attempting to do the diagonalization method on the full in-order list you mentioned (assuming it is infinitely long) will result in an 2 followed by an infinite string of 1s (i.e. ...111111111112, where the 1s go on forever). This number is NOT a natural number. By definition, it cannot be, as it does not have a finite number of non-left trailing 0s, but instead an infinite amount of 1s. This applies even if you are smarter about this and randomize the order of natural numbers. You will have an infinite string of random digits, which is also not a natural number.
The total volume increases infinitely in the cone, its just increasing by an unimaginably low amount at smaller and smaller decimal places so that it essentially is pi, but it never stops increasing, just as each new digit of pi is an increase from the previous. I think this should be considered infinite, just because we humans dont view the number as large doesnt mean its finite, because it will never stop increasing.
Pi has an infinite number of digits when you write it down but that doesn't mean it's "increasing" lol, it has a definite, unchanging value, the fact that you cannot write every digit down is due to it not being rational but that's a notation problem and not a magical property of pi.
Exactly this. The answer is that you consider all the moments up to the two minute mark, but you don't define what happens at the two minute mark. It's like saying "Walk one mile north Monday, three miles West on Tuesday, eight miles north on Wednesday, where are you on Thursday?"
I doubt this is actually as similar to Zeno's Paradox as you might think. I disagree that the moment of Achilles finally arriving is not defined, because that paradox is only unintuitive, and there's no direct contradiction there. He definitely reaches the destination. But in the example of the lamp, there *is* a direct contradiction in trying to find the solution. There certainly was a last step - we know, because after two minutes it's definitely over - but if we assume the lamp is on or off then our assumption can be disproven by the Archimedean Principle, that there's always another counting number. The lamp paradox goes one step farther than Zeno's Paradox.
@@benjaminhill6171 There is no last step of the lamp, because there's always a step afterwards, right? The paradox is "let's do an infinite number of things in a finite length of time." That's the paradox. It's no more paradoxical than saying "1+1-1+1-1....." and then asking what the answer is. It's *similar* to Zeno's paradox in that in both cases, the problem tells you to look closer and closer to the end of the process without looking at the end of the process. In the case of Zeno's, we say "use the calculus of limits and *define* the answer to be the limit of the progression." But the light bulb has no limit because it doesn't converge to an answer. Yes, after two minutes, it's definitely over. *But* the problem doesn't specify what happens after you've done the infinite number of changes. It simply says "you're getting closer and closer to being done, working more and more, now what happens when you *are* done?" I could give exactly the same "paradox" and add "and at two minutes, turn it off." And then there wouldn't be a paradox, nor would it contradict the process leading up to it. There is no "last step" because you do an infinite number of steps.
@@benjaminhill6171 There is no last step of the lamp, because there's always a step afterwards, right? The paradox is "let's do an infinite number of things in a finite length of time." That's the paradox. It's no more paradoxical than saying "1+1-1+1-1....." and then asking what the answer is. It's *similar* to Zeno's paradox in that in both cases, the problem tells you to look closer and closer to the end of the process without looking at the end of the process. In the case of Zeno's, we say "use the calculus of limits and *define* the answer to be the limit of the progression." But the light bulb has no limit because it doesn't converge to an answer. Yes, after two minutes, it's definitely over. *But* the problem doesn't specify what happens after you've done the infinite number of changes. It simply says "you're getting closer and closer to being done, working more and more, now what happens when you *are* done?" I could give exactly the same "paradox" and add "and at two minutes, turn it off." And then there wouldn't be a paradox, nor would it contradict the process leading up to it. There is no "last step" because you do an infinite number of steps.
I don't play darts often. One time showed up at cousins house and they were playing darts. Take a turn and first shot hit a bullseye. Shocked as I don't recall if I had ever done that before. Threw a second dart and it stuck into the back of the first dart. It was an "I am in the matrix" moment. My cousins were in disbelief.
Quantum Mechanics has completely nullified everything represented in the video. There are not infinite divisions, infinite steps or infinite amounts of time. Those are Classical descriptions, which don't exist in the real world.
I feel like hilbert's paradox works because we say that it works. How can a countable set of things be infinite? If you go backwards and empty the hotel using n-1 do we eventually get to 1 person occuping the 1 room in the infinite hotel making infinity =1?
Well that wouldn't work because removing 1 person at each step will leave out infinitely many still in their rooms. Unless we do that infinitely many times but I don't think that makes sense.
@@PhilipHaseldine To see why this is true, we need to make an observation: Rearranging a collection of objects is to give me them a new order. Thus, we can associate a tuple to each rearrangement. Example: Consider the set S = {a,b,c} The arrangements of S would be represented by 3-tuples Here are all of them: (a,b,c) (a,c,b) (b,c,a) (b,a,c) (c,a,b) (c,b,a) This definition can work for any set of n elements. Now, let's apply the definition to the empty set {}. The arrangements of 0 elements correspond to the 0-tuples. There is only 1 of them which is the empty tuple () . It is empty so it's a valid arrangement of {} Thus, the amount of rearrangements of 0 things is 1.
You know what? We can actually empty the hotel, but not in the way you suggest. 1st, in order for us to do infinitely many steps, we will use the same trick as in the Ross-Littlewood paradox (the vase paradox) . We do step 1, 1 minute before midnight, step 2, 30 seconds before midnight, step 3, 15 seconds before midnight and so on, each time leaving out half the remaining time. This way we can do infinitely many steps in 1 minute. If in each step, 1 person leaves and then everyone shifts 1 room back, then the hotel will never be empty even after doing infinitely many steps. But if they don't do the shift then *each room* will get emptied eventually at some point, and thus in th end, the hotel is empty. *Thinking about the process proposed* I have just realized! In these two scenarios, we have asked in each step the same person to leave, so they should lead to the same result, that everyone will leave. I think I have just created a new paradox. *Thinking again* I think I was just wrongly concluding that in the scenarion where everyone shifts, that the hotel won't be emptied because in each step every room stays occupied due to the shifts and thus it seemed to me that they will stay like that after the infinitely many steps. I was wrongly assuming that lim (n→∞) f(n) = f(∞). f(n) prsents the state of a room after n steps. In the end, the answer to your question is: Yes, we can empty the hotel, but no, this won't lead us to the scenario where 1 person stays in room 1, we transition directly from infinitely many occupied room to 0 occupied rooms.
@@PhilipHaseldine I have actually. I've argued with professors about it. I disagree with the phrasing of the premise behind the experiment. And I have rarely had professors agree. This one, and the balls into the bin are the only ones I take issue with.
The interesting thing about paradoxes is, they all seem to contain infinity. Because infinity is incomprehensible. It's sucha thing that it could not physically exist, but the concept can. It's freaky.
@@Y_tho yeah, also they try to say that “infinity” is synonymous with “all” and “never ending”, yet this dumb thought experiment (that’s all it is) requires you to treat “infinity” as a finite number that you can place into an equation. And for any geeks who are sure I’m being obtuse, I’ve got a cute hotel room where each of the four corners of the room are smaller than 90 degrees.
@0:35 If all the rooms up to infinity are “filled”, how can N be moved to N+1…? That is only possible if you switch to different, a larger infinite-hotel……therefore it’s Not the same hotel…so it is just word-trickery…
@@michaelmicek Because by definition, all the rooms are occupied ( or listed)….. that is what the first type of infinity means….ie to be “counted or listed” If it’s all listed, then how can you have a unlisted room…? In order to move to N+1, it’s switching to a different hotel , ie a different set..
@dumblr In order for N to move to N+1, you must have a empty room (or another decimal place; ie 2.11. Vs 2.111) In that case, it’s just word play. (Room)-occupied doesn’t mean it’s “All” occupied, it just means “only the listed rooms” are occupied! If I say there are no numbers between 2.1 and 2.2… You say: 2.12 exists (the extra room). I say: 2.12 was never listed. If every decimal point was not listed, then the hotel was never “occupied” to start with…
In the thought experiment, it doesn't even have to be simultaneous per se. The guest in room N just knocks on the door of room N+1 and tells the occupant that everyone is being moved to the next door down. Or to the room with twice the current number, in the case where the bus with an infinite number of guests arrives. The point is that that's the nature of infinity. Unlike a finite hotel, where moving everyone down one requires a space at the end, the infinite hotel doesn't have an end, so there's no problem.
I really don't understand the criteria mathematicians use to call something a paradox, like the first one just sounds like a logical error, how can an infinite something be full? This is not a paradox is a logical contradiction, the same thing with other two "paradoxes" that can be basically reduced to you can't complete something doing only half the work left. It just sounds that they made up a formula and because it doesn't fit something, instead of the formula being wrong or something, no it's a paradox
Some people need to realize that paradoxes aren't necessarily formulated to 'prove' anything, but rather to show a gap in our understanding of the subject on hand.
For instance, people like to clown on Zeno's paradox for being 'stupid' since Achilles obviously would outrun the tortoise. However, with what was known to the Greeks at the time, one could reasonably land at the conclusion that Achilles would never outrun the tortoise. The paradox shows us that there was a gap in our knowledge of infinity.
Paradox only exist in language, never in reality. Languages are only medical to the extent they are descriptive of reality, so things that cannot occur in reality but are still phrased grammatically correctly can be a paradox. Also statements that only reference themselves, like "this sentence is a lie". No external validity makes any internal logic meaningless.
@@vincentb5431 dude i think the greeks understood Zeno's paradox just fine lol, it's not like a bunch of ancient greek imbeciles stumbled upon it by accident and went "well no point in trying to catch up with a tortoise ever again"
The problem with infinity is: It does not exist. At least not that we know of. And paradoxes involving things that don't exist do not exist themselves. You can make up all kind of crazy sh*t but in the end all you did is making things up.
@dumblr We invented math to describe our reality. And it turns out that you can calculate something new to make a prediction, and then when you test the prediction in real life, the results actually match the theoretical prediction. So you can predict reality using only math, but that doesn't mean that everything you can calculate actually matches reality. It's like language. I can use language to describe reality, and I can make logical predictions with it (if Peter is the only child of Mary and George, and Peter has no children, I predict that Mary and George do not have grandchildren), but I can also use it to make up a fairy tale that has nothing to do with reality.
Indeed. It seems to me that applied mathematics is always in the business of mapping things from reality and remapping things, extracting properties and methods that were not yet there in the open from the outset. "Infinity" does simply not exist, whereas actual members of mapping do exist, i.e., all members of groups that are not infinite. There is no practical use in mapping "infinity", unless it is meant to indicate a direction. It will never be reached, because it does not exist, and vice versa. It is useless to try to play with it in analogous ways as we play with non-infinite mapping scenarios. Encountering paradoxes is therefore to be expected. The mere fact that twice infinity equals once infinity already says it all. I'm convinced that it can support some approaches in mathematics (mainly infinitesimal algebra), but not as a philosophical topic.
There is an issue. You say infinity doesn’t exist, but how long would it take until my rubix cube eats my grandma and burps louder than any volcanos that ever existed?
@@skellious You are touching on fractals, not infinity as such. Fractals depict issues of scale and repeatability. That is not the same as infinity being something real.
Looking at the comments I'm chuckling at the number of people simply dismissing Hilbert's Hotel out of hand when it's literally one of the most famous thought experiments in maths, like one day a mathematician will wake up and go: "Oh yeah! How silly of us!!" and it will be be all over the news that it's dead easy to solve (Just showing they really don't understand it all).
@@skellious Philosophy is WAY more easier than maths. All you need to understand is - "lead by example" or "Practice what you preach", and watch as 90% of "philosophers" jump out the window head first.
@@SuperSky9 No. I mean if you plan to work with infinity. You must accept certain properties that are simply unintuitve and entirely different than our typical number system
The Hilbert Hotel is so trash in my opinion. It basically says "If you carefully build 2 infinite sets of numbers so that they have exactly the same size then one can be bigger than the other one.". Sounds like complete nonsense.
Not ".. then one can be bigger". However, you can construct two infinite sets of equal cardinality such that one is a proper subset of the other. That is what the Hilbert hotel construction does.
Literally the opposite of what it's trying to show, which might suggest that it doesn't do it's job very well! If you think of the initial full hotel as containing all positive integers, and then having all negative integers turn up and the hotel being able to make space for them, we see that the set of positive integers and the set of all integers are actually the same size
for the dartboard: if we go to point like targets and pint like darts, the dart tip has no volume so it does not exist, hence it cannot hit the board at that state. if we assume any actual size, the probability is not zero.
The dart one is a prime example of "snarky maths nerds hiding obvious mcguffins in the verbage used to describe the 'problem' and working against finding a realistic solution" Instead of stating the obvious "hur dur its amazing" trash 'it could land infinitely anywhere on the board (completely ignoring the grain structure of the cork the board is made of) you could easily state it in the much more realistic, not masturbating over your ti-82, way of 'the dart tip diameter is 3mm diameter. What is the probability it will land within the 3mm area you believe it will.' My purposefully introducing infinity the answer trends towards infinity, therefore it being locked to an infiniti is not that special. It's like saying I have an infinite amount of eggs how many eggs do I have.... It would be amazing if I had two eggs, it would not be amazing if I had an infinite amount of eggs. Other than actually possessing an infinite amount of eggs....
My favourite 'paradox' is the continuum hypothesis which is likely the simplest to express unaswerable question in ZFC. Though explaining exactly how it works and why it is the case is a lot of fairly heafty maths
Not unanswerable but rather, independent. The thing about it is, there exist systems for which it is true and there exist systems for which it is false. Both options are acceptable and neither leads to contradictions.
The issue with the dartboard paradox is the fact that the dart will have an area in which the head takes up. Therefore the probability is related to the size of the dartboard in relation to the size of the dart.
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you get only 7 days not 30...
Note to self never stay at Halbert's hotel. I'm not tryna keep switching rooms everytime a new schmuck shows up
No worries, just ask your neighbour to move instead. And they can ask their neighbour, and they can ask theirs…
@@seanpierce9386 double it and give it to the next person
You tell me about it.
I was just minding my own business in room 654, and all of a sudden the receptionist told me to pack up and climb the stairs all the way up to some new room with a 197 digit number.
Not even sure what the point is, no one else moved into my old room, it was just left vacant.
To be honest, they could solve their issues by simply making sure that they always have infinitely many rooms available at any given time. Have everybody move once, and you're done.
And if they have a problem with it, there are plenty of ways to make the proportion of empty rooms arbitrarily small the higher their numbers.
An infinite amount of people can refuse to do that, and you'd still have an infinite amount of people who accept!
I hate staying at Hilbert's Hotel. I had this great room right by the pool, and right across the hall from the where they serve the complementary breakfast, then this uncountably infinite group of guys shows up and they make _me_ move rooms. Now I have to wake up crazy early if I want to get to the breakfast before the heat death of the universe.
Wait I don't think an uncountable infinite number of people fit in hilberts hotel since theres a countable infinite amount of rooms
@@Alex-mh4lt They very well could be just a countably infinite group of guys, for all I know, because I haven't actually tried to count them to figure that out.
Its a hotel made for an infinite number of guests
@@Alex-mh4lt Yeah... keeping uncountably many visitors in countably many rooms means uncountably many guests have been stuffed into a single room somewhere! Such practices by the hotel administration are outrageous.
If an infinite amount of guest had to come to the hotel everyone could just go to the room double of theirs leaving infinite spaces so I hope you are in a room below 10k
when you've been walking down the hall in Hilbert's Hotel for the past 10^6875 years to reach your room and an announcement plays saying "please move to the room number that doubles yours"
But you could easily fit all of Hilbert's hotel on the real interval [0,1]. So you could take the length of the hall to be finite, the problem is trying to stop at a particular room since you can pass infinitely many of them on a finite length.
The lamp is gonna be off by the end of the 2 minutes, because if it's not, Dad's gonna yell at me for wasting electricity.
this is the solution!!!!
Well a lightbulb's lifespan is usually calculated by how often it is turned on, so the lamp would be neither on nor off, but broken
My mom said the lamp is just broken.
The paradox of the Gabriel's horn is that you need an infinite amount of dye to paint it inside with a brush, but only a finite amount of dye to fill it (and paint it anyway).
Arguably you need 0 paint to paint it since its infinitely thin
@@piercexlr878 On my horn the paint is r/3 thin.
@@PrzemyslawSliwinski I feel like that has to converge, but I'm gonna guess it doesn't
@PrzemyslawSliwinski Oh right you can convert that into an area which gives a finite amount of paint
@@piercexlr878 My r tends to zero as x tends to infinity.
6:38 change yo smoke detector batteries
Bro how would you notice
@@Alex-mh4lt i got that special mix of brain damage
@@Halo56782💀
The ceiling birdie is chirping again
how to find if the narrator is black
I LOVE that you just hit the road running and started the vid w/o intro. Thanks for respecting our time.
How does this affect the local trout population
negatively
I asked some trout that very question and the answer they gave me was, disturbing.
As a trout, yes
Infinitely. 🐟
The biggest reason why these paradoxes exist is because they assume and treat infinity like it's a finite number. It's the same kind of logic as when people say 0 divided by 0 is 1, because of the observed rule that every other number divided by itself results in 1, which while true, doesn't mean that it also necessarily applies to 0. It's an interesting thought experiment though.
Yeah, like mathematicians think 0! = 1. Just makes life easier for them when in actual fact 0! must equal 0.
All numbers are values, but with the exception of 0 and infinity. That's why they don't play nice with others.
Yep, infinity is a concept not a number
@@brettr7970 isn't it 1 by mere definition of factorial tho
idk it works out everytime i have to do maths involucring factorial and 0 , better than it does saying that 0/0 = 1 lol
I always thought of it as sets of a thing.
If you have nothing and divide it into nothing sets you have nothing.
1 divide into 1 set is 1.
5 divided into 5 sets is 1.
10 divided into 10 sets is 1.
And so on.
You could could say a set of nothing nothings is 1 set of that…. But then 1 set divided into 1 set is 1 set of that… which is just stupid to say.
Infinity is a process, not a number. It should never be put on the number scale.
I feel that we created infinity and then began complaining about infinity lore.
Infinity is barely real
@@mastermonke2413 ?
6:39 please fix that smoke alarm
For Gabriel's horn you were supposed to calculate the surface area of the area of revolution, not the area beneath the curve of 1/x. The surface area integral is the integral from 1 to infinity of ((1/x)sqrt(1+(1/x^4)))dx which you can bound below by the integral you did, which diverges.
I was just about to point this out.
bro what
@@jimmydavis8172 He’s just saying what we all knew anyway.
@@seyerus cap no normal person knows what this means
@@jimmydavis8172 In the video, he calculated 1/x crosswise initially. That’s why it didn’t diverge.
My long-time statistics hot take is that the "expected value" operation is given undue significance. Its name is very suggestive of it being the inherently correct way to judge risk and reward, but the fact of the matter is that it is fairly uncommon for it to actually yield a value which you can, in any meaningful way, expect. The St Petersburg paradox shows this well. There is less than a 7% chance of getting $32 or more from the game. A person simply cannot, in the ordinary sense of the word, expect more than $32, regardless of the value of the operation which has been named "expected value".
Pascal's mugging is another "paradox" that just comes down to the expectation operator being treated as gospel. A mugger says if you give them your money, they will give you x amount back later. No matter how improbable you believe it to be that they will uphold the deal, they can give an x for which the "expected value" of taking the deal is positive, giving muggers a surefire strategy against people who treat the "expected value" operation as the ultimate arbiter.
"Expected value" is only genuinely accurate to its name when the risk is taken infinitely many times. Humans aren't immortal. The risks also often have an ante of some kind, and if the risk failed enough times, you can't afford the ante anymore and can't try again. Even with large corporations, which can absorb many more and much larger failed risks than people can, risk assessment goes beyond expected value -- they qualify or quantify the level of risk itself and compare that to a risk tolerance
I was a professional poker player for a long time, so EV is a concept I'm vary familiar with. It can be incredibly beneficial to think in terms of EV in countless real life scenarios. However, real life is just as you said, not infinite. So you also have to factor in utility, otherwise you get into the absurdities you mentioned and the concept loses the value and importance it has.
Expected value is very useful in science. Ignorant comment.
This, very much, makes sense 👍🏻.
Expected value is great for getting an idea of how things shpuld trend. But you have to temper it with analysis into short term risks. The combination works really well for evaluating an overall idea by seperating short and long term benefits
What do you think about taking the median instead of the mean? (Me using statistics terms instead of probability terms.) It sounds like the median gives you a much better idea of the short-term reward: you have 50% chance to get above the median and 50% chance to get below.
My thinking on the Ross-Littlewood paradox is that if the balls are removed in order, whenever any ball of number ‘n’ is removed, the vase will still contain ball n+1. So even though for all values of n, ‘ball n’ will be removed, it is impossible for the vase to be empty.
While I think your point is valid, I think the question is asking about the state of the vase after all actions have been taken. The paradox is saying that every ball will have been removed at some action before the end, thus one cannot claim to have a ball still left in the vase.
@@n484l3iehugtil And I understand that. My thinking is along the line of Cantor's Diagonal Argument for proving the real's are uncountable. Simplifying for brevity, it says that if you list "all" reals in a numbered list, a new real can be constructed from the nth digit of every number n, adding 1 (without carrying, so 9s roll back to 0). Since this number differs from all others in the list by at least one digit, it must not be in the set, and thus the set does not contain all reals. So even when starting from the premise that the set contains all real numbers, it can be shown that it doesn't. Similarly, even with the premise that "all" balls will be removed, the last ball removed will always have its successor still within the vase, and thus the vase can never be empty.
Another more mechanical proof is that with each step, you add 10 balls and remove 1, so the number of balls at each step is 10x - x, and the limit as x approaches infinity for that is infinity.
@@n484l3iehugtil I get what you’re saying but, (and I may be wrong) at least in their example there is always 9 times as many balls as have been taken out so no matter what number of steps you take even if it’s an infinite number of times there will always be at least 9 times as many balls in the vase as have been taken out.
The mathematical format of what you appear to be stating is: “The limit as n approaches infinity of (10n-1n)” Which would normally create a indeterminate form of (infinity-infinity) but, it can always be factored which then just gives n(10-1) or n(9) with the limit as n approaches infinity therefore = infinity
The problem with the paradox is in the idea of a super-task. It doesn't matter if the steps take 0 time, an infinite number of steps are never finished.
Imagine you're driving around a roundabout, what direction are you facing when you get to the "end" of the roundabout?
It doesn't matter how fast you drive, or if you're doing extra stuff, it's a circle.
What does this program do if it's run on a machine that's infinitely fast?
10 GOTO 10
20 PRINT "Hello"
It just hangs. Putting balls in a vase doesn't change anything:
10 balls = 0
20 balls = balls + 10
30 balls = balls - 1
40 GOTO 20
50 PRINT balls
You never get to line 50. It's just an infinite loop with busywork.
@@nisonaticthe point of the thought experiment is to assume it happens and explore the outcome, not to explore whether or not its possible. thats like when someone asks u "what would u do in a zombie apocalypse" and then u respond "zombies dont exist".
Your production quality is getting better and better! keep up good work :)
The spot you choose on the dartboard may be infinitely small.
But the dart itself is not!
Therefore, the dart does not actually hit any one spot. Instead, it hits a small area.
And you can totally calculate the probability of your chosen spot lying inside that area.
@@herrhartmann3036 i did that in another comment thread. Using a dart point area of a hundredth of a square millimeter and a regulation 13.25 in diameter dartboard, there are roughly 9 million potential dart hits in that don't overlap.
Now, if you are counting overlapping points as different, we have the issue of atomic width. After all, a dart cannot go through a foam atom. It can only go between two, and so this too is calculable and finite (though I'm not doing it).
If we take the fact that the sum of all probabilities of all points should be equal to 1, and therefore the probability of a single point is infinitesimally small but not zero. What am I missing?
@@iseslc By definition, a "point" has no area.
Therefore, the fraction of the whole dart board which is represented by any given point on it, is equal to 0.
And the probability of hitting a size 0 point is also 0.
However, as stated above, the dart actually hits a small area, rather than a point.
This area has a size greater than 0, and therefore it represents a fraction of the whole dart board greater than 0.
And the probability that our arealess point is located within this area can be calculated.
I find the St. Petersburg paradox very interesting, it highlights (as many other paradoxes here do) how "right" results are wrong when we ignore other variables applicable to that context.
For example looking at the variance.
While the EV is theoretically infinite we can plot every price of the game to the odds of turning a profit, and how much total capital we would need to ensure a profit in the long-run.
You could calculate with a logarithmic function how much it would do to you where you to win or lose and that would then be based on the money you own and you would get a more accurate result, this assumes that doubling your money has the same absolute impact as halfing your money
6:35 *BEEP* change your smoke detector battery dude
I'm waiting for one of these theories to be named after someone who isn't the guy who thought it up or supposedly uses it
That wouldn't be fair on the person who came up with it
thomson's lamp: [∞/0.5]
ross Littlewood: [∞/10 -∞/1 = ∞/9]
dartboard: [-1/∞]
The furniture and laundry budget to keep Hilbert's hotel running must be a nightmare.
Unsurprisingly, “Littlewood” spent his immense leisure time pondering sums that don’t exist
the probability of hitting any individual point on a dartboard changes quite a lot when you remember that the point of the dart itself ALSO has an area to it, and thus its own infinite number of points
☝🤓 Well for the sake of theoretical argument, assume the dart's point is infinitely thin.........
Infinity is infinite, even if you subtract infinity from infinity there will still be infinty left over. The amount of points on the dart changes nothing.
@@Lisa-dd1uq it's not a matter of subtracting, but dividing. and in analyzing this particular problem physically there is actually no need to resort to infinites - both the relevant areas (dart and board) can be measured in real units with physical tools
@@giddycadet it doesn't matter if your subtracting or dividing, there is an infinite number of infinitys in infinity
@@Lisa-dd1uq ok, think of it this way: if there are an infinite number of points on the dartboard, and an infinite number of points at the tip of the dart, then you say that the chance of ever hitting one specific point on the dartboard is zero. but if the numbers are both infinite, then the chance of hitting any individual point on the tip of the dart must ALSO be zero - even though geometrically we assume that ALL points on the tip of the dart are hit when it enters the dartboard.
★☆☆☆☆ · Would NOT recommend!
I was at Hilbert's Hotel for a bit, I stayed in room 92e+12, then they said to go to double your room number, it's been 30 octillion years, I'm nowhere near that room.
Thomson's lamp reminds me of calculating average values - in average, every person, who has two arms, has more arms than the average person, because there are (for sure) more persons missing limbs than having extra one's.
Vishnu makes up for it 🤣
@@PhilipHaseldinebruh 😂
An important thing to note about Cantors Diagonalisation proof is that is is a proof by contradiction; it is assumed that there is a mapping between all the real numbers between 0 and 1 and the naturals then using logical steps you show that there is a number between 0 and 1 not in the mapping. From there a contradiction is raised (the value both being in the mapped values and not) and the initial premise is rejected. It’s a bit pedantic but distinct to being able to add it to the mappings and do it again.
The key to resolving any apparent paradox is one's ability to spot the bullshit. Once the bullshit is removed the paradox ceases to exist.
Exactly. "Infinite number" .... no such thing.... the strawman of a veridical paradox.
lol
😅
Perhaps the real bullshit is holding hard onto the notion that the Universe in general gives two shits about what human epistemology considers a paradox, and will adjust itself accordingly to satisfy our limited capcities of observation, comprehension, and rationality.
Yet sometimes it's good to recognise what kind of paradox you're dealing with and recognise the limits of your models to make a useful decision. Way too often we use our models without considering their limitations and tell people to shut up when they point out the limitations and how they could lead to problems if ignored.
I'm so high right now that this actually makes sense!
Good one!!
A true nerd, high and still watching maths videos
That's meth for ya
YOO SAMEEE
The real question about Gabriel's horn is "does it go doot?"
Lets pray it doesn't go doot anytime soon! It's many times more spooky them Mr. Skeltal's...
The infinite amount of surface area implies an infinite amount of friction, so unfortunately not
@@brandonm1708 tragic.
You talk about the dartboard as if the tip of the dart was infinitely small as well, but the dart does take up space. The dart can hit multiple infinitely small points on the board at the same time. If you measure the surface area of the dart (D = Dart = N1) and the surface area of the board (B = Board = N2) then you will have a N1 in N2 chance of hitting any given spot. It’s an interesting theory though. If the dart tip was infinitely small and the very tip was the only part that would ever hit the board, it would be very cool to think about.
Regarding the St. Petersburg Paradox, I think the reason for why people would not pay an infinite amount of money to play this game is that if the probability of tails is even slightly less than 50%, then the expected value immedeatly becomes finite. For example if the probability of tails was 47.9%, then paying 25 Dollars would already put you at a disadvantage as a player.
Although this is mathematically true, I disagree that this is the primary psychological reasoning. Borrowing from economics, a person might intuitively aim to maximize expected “utility”. Bc of diminishing returns, utility is a sub-liner function of money and thus under-weights rare, high-value outcomes. This generalizes to other behaviors as well
it's actually because the increase of payout is proportional to the chance of the payout
@@LoganPost-c6pthank you for saying this. money usually doesn’t add never-ending value/happiness to one’s life. therefore, there’s no paradox in not investing in this game
Pretty sure it's because most people don't have infinite money laying around.
I surmise that it is because the chance of losing is much bigger than the chance of winning.
If I had to pay $1000, my chance of net winning money would be approx 0.1% which feels negligible (unless I made an off-by-one-error, but it would still be very small). The fact that I would win a lot if I win does not outweight that (you could insert utility here, but I suspect people are simply sensitive to probability of winning in addition to expected payout).
An interesting variant (or at least a related) of the Ross-Littlewood (Vase and Balls) paradox:
1] Alice starts building a stack of dollars bill by adding 1000 bills at a time. This continues for an infinite number of turns.
2] Bob’s task is to ensure that there are no bills left in the stack at “the end”.
3] Bob must choose between the following two strategies:
Option 1: Each time Alice places her 1000 bills, Bob can remove 999 bills from the top of the stack.
Or
Option 2: Each time Alice places her 1000 bills, Bob can remove 1 bill from the bottom of the stack.
So, in short, which is the winning strategy?
That's really interesting.
@@benjaminhill6171 : Right. I’ve always thought that the correct answer (that is, removing 1 bill each time results in 0 bills left in the stack, while removing 999 bills each times results in an infinite number of bills) is pretty much the definition of a counter-intuitive answer. ;-)
Shooting Alice and running away with the infinite money
You have 2 dollars, if you flip H, it doubles, if T it halves. Is this game gonna earn or lose you money on average or neutral?
@@simohayha6031 My gut says the expected value would be $2 - that is, if you flip the coin infinitely many times, you would always hover around $2 - but I'm feeling too lazy to try to prove it, haha.
I would play the Saint Petersburg game if someone paid me 1/12 dollars.
* Negative 1/12
@@udaysingh-wr2kw seller paying buyer x dollars is the same as buyer paying seller negative x dollars
I love that the video just starts without any of the common distractions! 👍
3:57 the lamp will be off because you broke it with infinite toggling
Yah but we have to take an assumption which is technically impossible.
@@blackwarrior823so don’t take it. lol
Exactly my point
Mathemathician trying to make a paradox without the use the concept of infinity:
Level Impossible
I mean correct me if I’m wrong mathematically but the Ross-Littlewood paradox is just the infinite sum of 10-1 (or just 9) with n=any natural number because no matter what on each step, the net number of balls being added in = 9. At any given point in time there is at least 9 times as many balls as have been taken out. Of course graphically it just goes to infinity which can be proven with a limit, I don’t quite get the paradox of “Well at some point every ball has to have been taken out of the jar” as mathematically that’s just impossible.
Even if you tried to make it a limit formed as “the limit as n approaches infinity of 10n-1n” (as to of make it indeterminate by (infinity-infinity)) you can just factor it out as n(10-1) which after taking the limit still approaches infinity
The problem is that you can't necessarily factor infinite numbers like that. The algebra concept you're using is factoring for finite numbers. Infinite numbers are not guaranteed to be so well-behaved, so it's not a valid operation.
This is a paradox because the solution is completely indeterminate - if you assume that any number is the answer, it leads to a contradiction. If you assume there are balls left, then they must have numbers, and once the supertask is finished all of those numbers must have been removed, so there can't be any balls left. But if you assume that there are none left, that's also a contradiction because there must be at least nine left because that's the net number added in the "last" step. Anything you assume leads to a contradiction.
@@benjaminhill6171 agreeing with benjamin, since the limit of the function '10n - n' can be said to go towards infinity, we plug in infinity to get '10*inf - inf', however any number * infinity is still infinity, so we get 'infinity - infinity', which is indeterminant.
the reason why it breaks is because infinity doesnt have an end. the very idea of even finishing the task is ridiculous, because finishing requires there to be an end to infinity.
thus, the entire thing is illogical, but even then, since there is no end, there is no point where ALL balls were removed, because you cant have an end! you can confidently say, even though the answer to the question is null, that even if there were an answer it would be infinite.
@@aidenaune7008 I was actually corrected since my last comment here, and at least according to common mathematical axioms, the "correct" answer to this paradox is zero. Because of the weirdness of infinity, if you assume the supertask is completed, there must be no balls left. I found a whole Wikipedia article explaining the various purported solutions, and the accepted mathematical solution is zero. So, at least mathematically, there is no direct contradiction here, so this isn't a "paradox" paradox.
Regarding there being an "end to infinity" as you called it, if you've ever learned about geometric series you know that it's a perfectly sound concept. This has nothing to do with actually being able to observe each step in the infinite process; you just assume the process is completed, which, again, is logically sound. In math you can just assume that anything you want happens, and then observe the consequences. That's what we're doing here, and the consequence is that there are no balls left after the task is done.
@@benjaminhill6171 your defense of there being an end to a definitionally endless number is that the geometric series assumes that there is one?
if I were to start with the number 0 and add one to it an infinite number of times over a finite amount of time, what number would I end with?
and no, its not infinity, infinity is not a number, its a concept.
if you give ANY definable number that can be placed concretely and discretely somewhere on a number line, there is a finite number of numbers before it. if you dont, then what you gave me is not a number, its a concept, and once again cannot be the result of a series of math problems, as they will always give a number.
infinity fundamentally is not a number, and cannot be treated like a number. it cannot be combined with any number or mathematical operation because the very concept is undefinable in the mathematical sphere.
the concept of infinity definitionally does not have an end. if it did, then there would be a finite number of numbers before it, and thus it would be finite. and I am not talking about an end to its string of digits either. if you were to make any assertion about its smallest or largest digit then it would still fail to be placeable on the number line due to its infinite digits in between, which is proven by the exact argument I made above.
you CANNOT make assertions about what the end of an infinite order of operations looks like, because infinity does not end. there is no final result, there is no outcome that you can discretely point to, there is nothing. the answer is null.
I dont care what mathematicians say, they are wrong. just because a group of self proclaimed experts agree on something, that doesnt make it true. if radiologists were to wake up tomorrow and all proclaim the sky to be purple, it would not be purple!
"Some infinities are bigger than others" must be one of the most misunderstood phrases ever (not here of course)
The hotel one isn’t even a paradox because how could it ever be full in the first place?
It’s only countable infinity, so if an infinite number of buses filled with an infinite number of people arrive at the hotel looking for a room, it’s uncountable infinity because the hotel will always need to provide +1 room than the number of tenants to maintain vacancy, and there are two layers of infinity stacked against the hotel’s singular plane of infinity.
@@Xhu666I don't think that answers their question.
@fulltimeslackerii8229 The paradox isn't whether or not it's physically possible to have an infinite hotel, or for it to be completely filled. The paradox is about what happens if you could. This is a purely mathematical paradox because it's just a way to visualize how you can fit more numbers into an infinite set without changing its size. Also, this is the kind of paradox that is only called a paradox because it's unintuitive, not because there's actually a logical contradiction - because there's no contradiction here.
Also where did they even afford that many rooms 😭
@@benjaminhill6171why dont the new guy just go to the last unoccupied n+1 room instead of makimg everyone move
@@josephoyek6574 Great question! All the rooms are already full at the start, so there actually isn't an empty room at any "n+1" place. That's why it's unintuitive: there's no room at the start, but just reordering the people opens up a room. It's an analogy to show that if you add things to an infinite set, that set doesn't get any bigger.
If we take the lamp on/off one literally while considering the constant speed of light, then having the light half on would be a measurable result if the light is turning on and off at as the rate at which it is alternating between on and off is at the speed of light, at which point, it can no longer be turned on or off any faster. So we'd see a dimmed light, as though it were half on. Phenomena we can observe and measure at least seems to have some finite limitations like the plank length, and the speed of light. Gravity was a constant according to Newton's observations, until we could model it as a curvature in space and time caused by mass thanks to relativity. We don't even have a fundamental explanation as to "why" mass curves space and time. Gravity still isn't fully understood.
Numbers, and models and math are all abstract. The reality may be indefinite; a bunch of larger or smaller infinities, and so I'm inclined to think that math and physics are always going to be shy of total consistency.
I think your explanation of Cantor's diagonal argument doesn't hold up. Why can we map the reals to the naturals in the first place (what if we run out of them)? Also how did you map them? And why shouldn't we be able to do the Hilbert's hotel thing and move all the reals one space down to make room for more?
These things can be explained/worked around, but you left out the parts of the proof that do that.
The parts you did keep don't really follow a logical flow either. Why are we even constructing a new number? You explain it, but only after making it...
I'm not sure which bit of it you're questioning, but the point is that you can't map the naturals to the reals, no matter what way you try to link the numbers together.
The reason for not explaining the way they are mapped is because it has to be arbitrary, or the proof has to work no matter what way they're mapped. if you said "well 1 will align with 0.0000001, 2 with 0.0000002" and so on, obviously that won't work for listing all real numbers, but you've only proved it for that specific arrangement, not any arbitrary arrangement.
The point of the diagonal is that you *always* can add one more real number, even an infinitely long one to an infinitely long list. if your list is infinitely long, you've listed all the natural numbers already, and theoretically that should also mean you've listed all the reals. but if you can add another, or even infinitely more, that means you don't have all of them, so the mapping doesn't work.
In regards to the hilbert hotel thing, yes, you can keep adding numbers to the beginning of the list!
But, since you're doing the exact same checking and number creation every time you add a number, you're not actually going to make any progress in listing all of the real numbers, and you're not going to hit a point where you have all of them and the diagonal strategy doesn't work, even after an infinite number of reals added.
Hopefully that makes a little more sense, I feel like I said number and infinity a few too many times :P
The whole point of the proof by contradiction is that it started with the assumption that we could map the reals to the naturals, and then proved that assumption wrong, because we created a real number that had no mapping. If it were possible to do the map in any way, we could not create that number. The mapping method doesn't matter because the process of generating the new real number works for any mapping method.
There are always more numbers in general even between 0 and 1 then there are integers. Integers don't use decimal points
@@elderfrost9892 I think you're just mapping the numbers wrong. If you match the naturals to reals like:
1 = 0.1
2 = 0.2
3 = 0.3
...
9 = 0.9
10 = 0.01
11 = 0.11
12 = 0.21
...
99 = 0.99
100 = 0.001
That'll work just fine.
If you want to pick a random infinite string, you must do it for both sides:
...36492749 = 0.887592738...
...47494693 = 0.474926957...
...39572074 = 0.164959378...
Naturals (like 3) can be thought of as having an infinite series of leading zeros (...000003). This is exactly how we think about reals, just reversed. Reals (like 0.5) have an infinite series of tailing zeros (0.500000).
If you think about it like this you'll realize that the size (cardinality) of the set of naturals and the set of reals are exactly the same.
There are absolutely not more reals between 0 and 1 than natural numbers. Both are uncountable.
But the hotel can't fit an uncountable infinite amount of new guests
The problem with the Hilbert's hotel paradox and many other infinity paradoxes is that they violate a fundamental law of numbers. We have been told over and over again that infinity is NOT a number, but rather a concept. But when we say that Hilbert's hotel has an infinite number of rooms, we treating infinity as a number! It only is a paradox because we are violating the rules.
Fuck yeah.
Exactly
Cardinals. It's denoting the size of a set. By your logic you can't say the size of the set of integers is infinite, because infinity isn't a number.
@@CosmicButterfly2 Exactly!
@CosmicButterfly2 no. Most of these paradoxes are nonsense they make and then break their own rules. They're just dumb word games. If a set contains infinite people, you couldn't add infinite people to it because they would already be in it.
Ross-Littlewood paradox is just Grandhi's series 2.0: instead of series alternating between 1 and -1, this one alternates between 10 and -1
in both cases the sum cannot be defined properly, Ross-Littlewood just needs an extra step to see that
Grandhi's series alternates between 0 and 1.
It is said to diverge, but not to infinity.
The numbers in Ross-Littlewood do diverge to infinity.
@@michaelmicek and yet you can group them in a way that shows an unidentified sum, which is why paradox exists
Fun counterexampke to ross little wood, add 10 balls, numbered, remove the smallest number divisible by 3, repeat. After the super task is complete there will be infinite balls, every ball not a multiple of 3
The point of the ross little wood formulation is that you are removing *the entire set of natural numbers* from the jar,, the same set you are adding to it. If you only remove a subset of it then there's no paradox.
@@unadulterated i know, its a cardinality bijection thing. Im just giving an example of a sanity check for the sequence, how this kind of operation is incredibly malleable to leave you with any amount leftover
I feel bad for the janitor in hilbert’s hotel
For the dartboard paradox, with infinitely small points, when split into a quadrants, how does the premise of a dart being able to hit the precise center or even just the precise divide between two quadrants affect it, wouldn’t that mean there’s a non zero chance of it landing outside just one quadrant, meaning it wouldn’t be precisely 1/4, but infinitely close to 1/4?
The probability of the dart hitting the dividing line is 0, so the probability of the dart hitting the interior of the quadrant is the same as the probability of the dart hitting the interior or the border. 1/4+0=1/4
Hilbert's Hotel's Murphy's Law:
Even though they can accomodate an infinite amount of guests they never seem to have enough money to fix the ice machine
3:15 your method to generating a new number will trend to numbers of only 8's and 9's
Yeah, you're supposed to stop the proof as soon as you've got one number not on the list, contradicting the hypothesis.
The proof method works just as well in base 2 as it does in base 10. In fact, I think base 2 makes it easier to see.
Pi is the ultimate paradox
Good video, BUT click bait title!!!
Knowing infinity, it obvious there are infinitely many.
Banach-Tarski paradox, or Pythagoras theorem which is also link with infinity (infinitely small)
Some more infinity paradoxes
Consider "the set of everything." By the diagonal argument (on subsets rather than on digits) one can prove that it doesn't coontain the power set of "the set of everything", hence "the set of everything" doesn't contain everything.
Solution: modern set theory gives axioms that give a restriction to what can be a set, denying "the class of everything" as a set. So it has no power set.
According to (modern) set theory, cardinal numbers and ordinal numbers are both sets. A cardinal number is defined to be an ordinal number alpha that is equal to the lowest ordinal number beta such that alpha can be mapped injectively into beta. Then the smalles transfinite ordinal number often is denoted w (in fact we call it omega.) It is also a cardinal number, and as a cardinal number it will often be denoted Aleph_0. But, w^w is countable while Aleph_0^Aleph_0 is uncountable.
Explanation/solution: when doing ordinal numbers, alpha^beta is an ordered set. But when doing cardinal numbers, alpha^beta is just a set (and we forget about order.) So w^w differs from Aleph_0^Aleph_0.
Banach - Tarski Paradox. It is possible to 'cut' a solid ball into a finite amount of pieces and reassemble those pieces such that you assemble two balls, each being a perfect copy of the original ball.
However, each of those 'pieces' is in fact an infinite scattering of points, rather than a solid piece of the ball.
Achilles Paradox. Consider a race between Achilles (who was able to run very fast) and a turtoise (who was quite slow), with the turtoise starting ahead. Each time Achilles covers the distance between him and the turtoise, the turtoise also moved a little bit. Hence Achilles has to cover another distance to catch up the turtoise. This repeats infinitely many times. "So Achilles can't win the race."
It took quite the time before this paradox was finally solved. First we had to define a notion of "converging limits" before finally reason well why Achilles is able to come ahead of the turtoise. The point where Achilles will pass the turtoise is the limit point of the set of points where either is at any of the mentioned moments. These are the moment the race started, the moment Achilles has reached the starting point of the turtoise, the moment Achilles has reached the point where the turtoise was at the moment Achilles reached the starting point of the turtoise, etc.
Proper video on this topic should be infinitely long. It is impossible to explain every infinity paradoxes in finite amount of time. Or it create new paradox.
I hate it when the word "paradox" is used to describe something that's merely a bit unintuitive to some people.
That's not what "paradox" means.
When was the last time you looked up the word "paradox"?
Paradox does in fact have a definition as being unintuitice
There are several meanings for / types of paradox.
"A paradox is a logically self-contradictory statement *or a statement that runs contrary to one's expectation*."
The word is probably one of the most overused, long with the words "unsolved" (generally, not in maths) and "mystery". Like saying the Bermuda triangle is mysterious. It isn't.
i only could follow the diagonal argument a little
2:50 this one doesn’t make sense to me because even if the new real number is not found anywhere in the current real numbers index there will always be another natural number to reference that index. In this wise I can see it being paradoxical but not in the way the problem is stated. In my interpretation of the paradox it’s more like how can a set that can be infinitely broken down also itself be considered infinite logic would dictate the real numbers must be a larger infinity but because there will always be a natural number index that can increase with every real number in that index both must be the same magnitude, or maybe I’m misunderstanding something.
7:15 I also am having a problem with the gabriel’s horn not because the proof isn’t understandable but because despite pi being represented as a solid number solution it has infinite places after the decimal meaning that itself must represent an infinite volume as it’s the same a saying 3 + 1/10 + 4/100 + 1/1000 etc, with every decimal point being the equivalent of adding a fraction of volume equivalent to number over place if that makes sense, so though it would appear to be finite and is indeed useful in finite calculations it is still infinite. Or once again I may be misunderstanding something, I’m really not any sort of mathematician.
8:28 this one doesn’t seem like a paradox to me since every iteration is the equivalent of saying “balls in vase=balls in vase+9” because every iteration, even if you pull out the index value ball, you are still leaving 9 balls in its place with every index. So the real problem would look like “balls in vase=balls in vase + 10 - 1” which is just “balls in vase=balls in vase + 9” because no matter what index we are at we are still adding 10 and removing 1 so it will never and could never empty completely.
For pi, yes you are misunderstanding decimals.
No matter how many numbers come after the decimal point in pi, it will NEVER be greater than 3. This is obvious when you consider that 1/3 is 0.333 repeating infinity, but we know for a fact that 3/3 is 1. So no matter how many 3’s there are after the decimal point, they will not make 1/3 infinitely large.
As for the jar, you’re misunderstanding infinity. Infinity is not a number. You cannot add 9 to infinity and make it bigger, it doesn’t work like that. The reason it’s a paradox is because if you remove the ball labeled 1 the first time you add 10 balls, and remove the 10 the 10th time, infinite times, then you have removed every single number. It doesn’t matter that 10 times infinity is added, because 10 times infinity is still infinity. You can’t do algebra with infinity because it’s not a number.
Though paradoxically, if when you add the 10 balls you remove a ball that is even, like 2, then once you’ve removed the infinite amount of balls, there’s still the infinite amount of numbers in the jar that are odd.
@@UnluckyLilly still having trouble understanding. If Pi will never be larger than 3 (which it is but I understand your point) then the volume as well could never be greater than x amount meaning as pi can’t represent infinite volume in this circumstance then the volume itself can’t be infinite.
The jar explanation still doesn’t make sense. Infinity isn’t a number but a representation of uncountable numbers, possibly unending. So yes adding numbers to the concept of infinity doesn’t change the concept, just as subtracting numbers from it wouldn’t change it, but the uncountable number it’s added to would change by 1. That’s not entirely the argument I want to make anyways but it seems to me like a more grey argument than anything else. I think that it goes along with the larger and lesser infinite concepts you are subtracting a lesser infinity from a larger infinity which would still be infinity. But I’m not all caught up on infinity theory so again I could be mistaken.
The bit he jumped over in the Cantor's theorem part of his video is that the "set" of natural numbers, which is infinite, is a function. The set of real numbers is a function too. Functions are how we get or return those numbers out of infinity. The natural numbers look like f(n) = n, so that every number is in a room and if you want to get that number you go to that room; f(1) returns 1, f(2340) = 2340.
You can perform operations on real numbers function (uncountable set) that produces a number not contained in its set -- that's the diagonal bit -- but you can't perform those operations on countable sets like the set of natural numbers. Again, its important to stress that it's not the number that is limited it is the function or, if you rather, the set. Yes, you can make a larger natural number, by adding or multiplying, but you can do that to your real numbers as well so those parts of these sets are "one-to-one." The difference is that in the function of the real numbers you can derive a new number and thus that set is, cardinally, larger than a set which is limited to the elements it already contains. You could swap any amount of digits by any amount of numbers in the natural number function but you'll only return another number already contained in the set--the function can't produce a new number because of the function/set.
Intuitively, it makes sense that a two dimensional infinity is smaller than a three dimensional one, and that the three dimensional one is infinitely larger. Despite an unlimited amount of numbers available to us in two dimensions, we can't map any numbers to all the ones in the third dimension that aren't already mapped to the other two dimensions.
@@anhi399 that actually does make alot more sense than just saying one infinity is larger than another. I can understand how one function can’t get a return from another set, that makes complete sense. Thank you for explaining it!
It is very well known in maths that there are far more numbers than integers, because there are many more ways of being specific over whatever range you care to name
Another paradox: Why do I feel infinitely feel more smart when I'm actually infinitely more dumb lol
Those who are truly smart understand there is always something more to learn. Those who arent smart will think theyve got an understanding of everything already.
Hilbert’s Hotel paradox - The concept calls for an unreal premise to begin with with an incorrect set up. An Infinity Hotel can never be fully occupied so the question is invalid.
google ‘thought experiment’ brother. also google ‘paradox’ lol.
Hilbert’s hotel “paradox”: infinite rooms are… infinite! And the number of guests they can accommodate is… infinite! Mind blown 🤯
Math is just counting an infinite amount of zeros.
All negative and positive numbers cancel each other out. Zero flips the number line from negative to positive (-0+). Infinity flips it from positive to negative (+0-). Which is how we can fit infinity into a finite space.
Hilbert slayed with that hat
0:50 TF? If there are infinite rooms in the hotel, it can never be full
yes it can, if all the rooms are occupied
@@MikeyMobes but then, noone else can join because it‘s full
@@magnuslunzer2335the countable rooms are full that's the closest to it if not an infinite room can't be full if its full then it can't accommodate any new person
@@yungdkay1008 But it‘s a countable infinite amount because it‘s N_0. All numbers from N_0 never fit in N_0 is technically what this „paradox“ is saying.
fr
if all da rooms occupied but da rooms are infinite then theres no state where its full
This video is look more entertaining then your other videos. I liked the sarcasm in this video and also the animation you put in which make video easy to understand. Kudos to you 👏
Can't you disprove Cantor's diagonal argument by putting natural numbers on the left side of the table with an infinite number of zeros in front of them and then do the same thing proving their are more natural numbers than natural numbers?
Adding zeroes in front of a natural number doesn't turn it into a different number lol, it's still the same number.
@@unadulterated no i mean starting with a list like
0001 - 1
0002 - 2
0003 - 3
0004 - 4
And doing the same thing along a diagonal from the top right, generating the number 1112, and like said in the video, add it it to the list, then repeat to infinitely generate another unpaired number?
The reason Cantor's arguement can't be applied to the natural numbers is due to how the natural numbers are defined.
Natural numbers are inherently finite. There are an infinite number of them of course, but any specific natural number has a finite amount of digits (excluding the trailing 0s because left side trailing 0s don't actually change anything).
Attempting to do the diagonalization method on the full in-order list you mentioned (assuming it is infinitely long) will result in an 2 followed by an infinite string of 1s (i.e. ...111111111112, where the 1s go on forever).
This number is NOT a natural number. By definition, it cannot be, as it does not have a finite number of non-left trailing 0s, but instead an infinite amount of 1s.
This applies even if you are smarter about this and randomize the order of natural numbers. You will have an infinite string of random digits, which is also not a natural number.
No
The total volume increases infinitely in the cone, its just increasing by an unimaginably low amount at smaller and smaller decimal places so that it essentially is pi, but it never stops increasing, just as each new digit of pi is an increase from the previous. I think this should be considered infinite, just because we humans dont view the number as large doesnt mean its finite, because it will never stop increasing.
Pi has an infinite number of digits when you write it down but that doesn't mean it's "increasing" lol, it has a definite, unchanging value, the fact that you cannot write every digit down is due to it not being rational but that's a notation problem and not a magical property of pi.
6:39 smoke alarm battery ..
Cantor’s diagonal argument just cooked my brain bruh it did not make sense at all
Just means using only real integers is far more limiting than using all the decimals in between (in essence)
I know the lamp paradox as the runners paradox, a runner runs half the distance of the distance left within 1 minute, so he runs for an infinite time.
Zeno’s paradox
Exactly this. The answer is that you consider all the moments up to the two minute mark, but you don't define what happens at the two minute mark. It's like saying "Walk one mile north Monday, three miles West on Tuesday, eight miles north on Wednesday, where are you on Thursday?"
I doubt this is actually as similar to Zeno's Paradox as you might think. I disagree that the moment of Achilles finally arriving is not defined, because that paradox is only unintuitive, and there's no direct contradiction there. He definitely reaches the destination. But in the example of the lamp, there *is* a direct contradiction in trying to find the solution. There certainly was a last step - we know, because after two minutes it's definitely over - but if we assume the lamp is on or off then our assumption can be disproven by the Archimedean Principle, that there's always another counting number. The lamp paradox goes one step farther than Zeno's Paradox.
@@benjaminhill6171 There is no last step of the lamp, because there's always a step afterwards, right? The paradox is "let's do an infinite number of things in a finite length of time." That's the paradox.
It's no more paradoxical than saying "1+1-1+1-1....." and then asking what the answer is.
It's *similar* to Zeno's paradox in that in both cases, the problem tells you to look closer and closer to the end of the process without looking at the end of the process. In the case of Zeno's, we say "use the calculus of limits and *define* the answer to be the limit of the progression." But the light bulb has no limit because it doesn't converge to an answer.
Yes, after two minutes, it's definitely over. *But* the problem doesn't specify what happens after you've done the infinite number of changes. It simply says "you're getting closer and closer to being done, working more and more, now what happens when you *are* done?" I could give exactly the same "paradox" and add "and at two minutes, turn it off." And then there wouldn't be a paradox, nor would it contradict the process leading up to it. There is no "last step" because you do an infinite number of steps.
@@benjaminhill6171 There is no last step of the lamp, because there's always a step afterwards, right? The paradox is "let's do an infinite number of things in a finite length of time." That's the paradox.
It's no more paradoxical than saying "1+1-1+1-1....." and then asking what the answer is.
It's *similar* to Zeno's paradox in that in both cases, the problem tells you to look closer and closer to the end of the process without looking at the end of the process. In the case of Zeno's, we say "use the calculus of limits and *define* the answer to be the limit of the progression." But the light bulb has no limit because it doesn't converge to an answer.
Yes, after two minutes, it's definitely over. *But* the problem doesn't specify what happens after you've done the infinite number of changes. It simply says "you're getting closer and closer to being done, working more and more, now what happens when you *are* done?" I could give exactly the same "paradox" and add "and at two minutes, turn it off." And then there wouldn't be a paradox, nor would it contradict the process leading up to it. There is no "last step" because you do an infinite number of steps.
I don't play darts often. One time showed up at cousins house and they were playing darts. Take a turn and first shot hit a bullseye. Shocked as I don't recall if I had ever done that before. Threw a second dart and it stuck into the back of the first dart. It was an "I am in the matrix" moment. My cousins were in disbelief.
The real paradox is all of these brilliant people wasting their time on irrational thoughts.
Some of this stuff comes up in science and often you don’t know what will until you actually do some math.
Quantum Mechanics has completely nullified everything represented in the video. There are not infinite divisions, infinite steps or infinite amounts of time. Those are Classical descriptions, which don't exist in the real world.
I feel like hilbert's paradox works because we say that it works. How can a countable set of things be infinite? If you go backwards and empty the hotel using n-1 do we eventually get to 1 person occuping the 1 room in the infinite hotel making infinity =1?
Yeah, no shit. Infinity only exists because we say it does. None of this is actual math, it's just a thought experiment
Well that wouldn't work because removing 1 person at each step will leave out infinitely many still in their rooms. Unless we do that infinitely many times but I don't think that makes sense.
I like that the number of ways of arranging zero things is one, not zero.
@@PhilipHaseldine
To see why this is true, we need to make an observation: Rearranging a collection of objects is to give me them a new order. Thus, we can associate a tuple to each rearrangement.
Example: Consider the set S = {a,b,c} The arrangements of S would be represented by 3-tuples
Here are all of them:
(a,b,c) (a,c,b)
(b,c,a) (b,a,c)
(c,a,b) (c,b,a)
This definition can work for any set of n elements.
Now, let's apply the definition to the empty set {}. The arrangements of 0 elements correspond to the 0-tuples. There is only 1 of them which is the empty tuple () . It is empty so it's a valid arrangement of {}
Thus, the amount of rearrangements of 0 things is 1.
You know what? We can actually empty the hotel, but not in the way you suggest.
1st, in order for us to do infinitely many steps, we will use the same trick as in the Ross-Littlewood paradox (the vase paradox) . We do step 1, 1 minute before midnight, step 2, 30 seconds before midnight, step 3, 15 seconds before midnight and so on, each time leaving out half the remaining time. This way we can do infinitely many steps in 1 minute.
If in each step, 1 person leaves and then everyone shifts 1 room back, then the hotel will never be empty even after doing infinitely many steps. But if they don't do the shift then *each room* will get emptied eventually at some point, and thus in th end, the hotel is empty.
*Thinking about the process proposed*
I have just realized! In these two scenarios, we have asked in each step the same person to leave, so they should lead to the same result, that everyone will leave. I think I have just created a new paradox.
*Thinking again*
I think I was just wrongly concluding that in the scenarion where everyone shifts, that the hotel won't be emptied because in each step every room stays occupied due to the shifts and thus it seemed to me that they will stay like that after the infinitely many steps. I was wrongly assuming that lim (n→∞) f(n) = f(∞). f(n) prsents the state of a room after n steps.
In the end, the answer to your question is: Yes, we can empty the hotel, but no, this won't lead us to the scenario where 1 person stays in room 1, we transition directly from infinitely many occupied room to 0 occupied rooms.
We WILL be using this for powerscaling 🗣🗣🗣🗣🔥🔥🔥🔥🔥
With Hilbert's Hotel, the hotel can't be fully occupied if there's a room for the last person to move over to. It's not a paradox, it's just a lie.
there is no last person :/
It is. I don't think you've understood it...c.heck out Wikipedia
@@familjenlover6827there could be
The lamp will be off, because your dad yelled at you.
The first one just seems dumb.
"If I have a magic hotel that has infinite rooms, I can have infinite guests."
Wow. What a genius :/
The guy literally proved the infinite number of guests can be greater than the infinite number of rooms lol
@@interloper9589 He didn't prove it though.
This is one of the most famous thought experiments in maths. I guess you haven't heard of it before....
@@GodwynDi Well yeah he didn't. Cantor did.
@@PhilipHaseldine I have actually. I've argued with professors about it. I disagree with the phrasing of the premise behind the experiment. And I have rarely had professors agree. This one, and the balls into the bin are the only ones I take issue with.
9:29 it is 1/∞²
That’s what I immediately thought as well.
5:14 He's the only sigma here
I did not expect to hear that
The interesting thing about paradoxes is, they all seem to contain infinity. Because infinity is incomprehensible. It's sucha thing that it could not physically exist, but the concept can. It's freaky.
Hilbert’s Hotel is the easiest way to show normal people that mathematicians are not as smart as they want to come across as.
Seriously. How is the hotel full if you have infinite rooms 🤨
@@Y_tho yeah, also they try to say that “infinity” is synonymous with “all” and “never ending”, yet this dumb thought experiment (that’s all it is) requires you to treat “infinity” as a finite number that you can place into an equation. And for any geeks who are sure I’m being obtuse, I’ve got a cute hotel room where each of the four corners of the room are smaller than 90 degrees.
I used to have a paradox but then I realized I only needed one.
xD
i usually hate math videos but these are always great before bed
@0:35
If all the rooms up to infinity are “filled”, how can N be moved to N+1…?
That is only possible if you switch to different, a larger infinite-hotel……therefore it’s Not the same hotel…so it is just word-trickery…
I don't understand why you say that the person in room N can't move to room N+1.
@@michaelmicek
Because by definition, all the rooms are occupied ( or listed)….. that is what the first type of infinity means….ie to be “counted or listed”
If it’s all listed, then how can you have a unlisted room…?
In order to move to N+1, it’s switching to a different hotel , ie a different set..
@dumblr
In order for N to move to N+1, you must have a empty room (or another decimal place; ie 2.11. Vs 2.111)
In that case, it’s just word play. (Room)-occupied doesn’t mean it’s “All” occupied, it just means “only the listed rooms” are occupied!
If I say there are no numbers between 2.1 and 2.2…
You say: 2.12 exists (the extra room).
I say: 2.12 was never listed. If every decimal point was not listed, then the hotel was never “occupied” to start with…
@@jimliu2560 The guests all move simultaneously. They each leave their room at the same time, and enter the next room, which is empty by then.
In the thought experiment, it doesn't even have to be simultaneous per se.
The guest in room N just knocks on the door of room N+1 and tells the occupant that everyone is being moved to the next door down.
Or to the room with twice the current number, in the case where the bus with an infinite number of guests arrives.
The point is that that's the nature of infinity.
Unlike a finite hotel, where moving everyone down one requires a space at the end, the infinite hotel doesn't have an end, so there's no problem.
I really don't understand the criteria mathematicians use to call something a paradox, like the first one just sounds like a logical error, how can an infinite something be full? This is not a paradox is a logical contradiction, the same thing with other two "paradoxes" that can be basically reduced to you can't complete something doing only half the work left. It just sounds that they made up a formula and because it doesn't fit something, instead of the formula being wrong or something, no it's a paradox
Some people need to realize that paradoxes aren't necessarily formulated to 'prove' anything, but rather to show a gap in our understanding of the subject on hand.
For instance, people like to clown on Zeno's paradox for being 'stupid' since Achilles obviously would outrun the tortoise. However, with what was known to the Greeks at the time, one could reasonably land at the conclusion that Achilles would never outrun the tortoise. The paradox shows us that there was a gap in our knowledge of infinity.
Paradox only exist in language, never in reality. Languages are only medical to the extent they are descriptive of reality, so things that cannot occur in reality but are still phrased grammatically correctly can be a paradox. Also statements that only reference themselves, like "this sentence is a lie". No external validity makes any internal logic meaningless.
@@vincentb5431 dude i think the greeks understood Zeno's paradox just fine lol, it's not like a bunch of ancient greek imbeciles stumbled upon it by accident and went "well no point in trying to catch up with a tortoise ever again"
@@unadulterated Yeah, the greek paradox is every bit as valid as any of these.
Wouldn't the concept of a supertask itself be a paradox since it's a task repeated an infinite number of times in a finite amount of time?
The problem with infinity is: It does not exist. At least not that we know of. And paradoxes involving things that don't exist do not exist themselves. You can make up all kind of crazy sh*t but in the end all you did is making things up.
@dumblr We invented math to describe our reality. And it turns out that you can calculate something new to make a prediction, and then when you test the prediction in real life, the results actually match the theoretical prediction. So you can predict reality using only math, but that doesn't mean that everything you can calculate actually matches reality. It's like language. I can use language to describe reality, and I can make logical predictions with it (if Peter is the only child of Mary and George, and Peter has no children, I predict that Mary and George do not have grandchildren), but I can also use it to make up a fairy tale that has nothing to do with reality.
Indeed. It seems to me that applied mathematics is always in the business of mapping things from reality and remapping things, extracting properties and methods that were not yet there in the open from the outset. "Infinity" does simply not exist, whereas actual members of mapping do exist, i.e., all members of groups that are not infinite. There is no practical use in mapping "infinity", unless it is meant to indicate a direction. It will never be reached, because it does not exist, and vice versa. It is useless to try to play with it in analogous ways as we play with non-infinite mapping scenarios. Encountering paradoxes is therefore to be expected.
The mere fact that twice infinity equals once infinity already says it all. I'm convinced that it can support some approaches in mathematics (mainly infinitesimal algebra), but not as a philosophical topic.
It does though. Conceptually at least. How long is a coastline?
There is an issue. You say infinity doesn’t exist, but how long would it take until my rubix cube eats my grandma and burps louder than any volcanos that ever existed?
@@skellious You are touching on fractals, not infinity as such. Fractals depict issues of scale and repeatability. That is not the same as infinity being something real.
Looking at the comments I'm chuckling at the number of people simply dismissing Hilbert's Hotel out of hand when it's literally one of the most famous thought experiments in maths, like one day a mathematician will wake up and go: "Oh yeah! How silly of us!!" and it will be be all over the news that it's dead easy to solve (Just showing they really don't understand it all).
These are not paradox, this is spitballing out of sheer boredom. This is basically turning math into a fiction.
Infinity has weird properties. Its pretty well known in any math circle.
If you hate this, wait until you learn about philosophy.
@@piercexlr878 You mean circle jerks because that's how these "paradox" are termed
@@skellious Philosophy is WAY more easier than maths. All you need to understand is - "lead by example" or "Practice what you preach", and watch as 90% of "philosophers" jump out the window head first.
@@SuperSky9 No. I mean if you plan to work with infinity. You must accept certain properties that are simply unintuitve and entirely different than our typical number system
What I learn from this is that there's a lotta thought snags you get yourself into when you base your line of logic on totally hypothetical equations.
The Hilbert Hotel is so trash in my opinion. It basically says "If you carefully build 2 infinite sets of numbers so that they have exactly the same size then one can be bigger than the other one.". Sounds like complete nonsense.
What?
It doesn't say that at all. It just says that infinity is infinite. As in you can just keep going endlessly because infinity+1 = infinity
Not ".. then one can be bigger". However, you can construct two infinite sets of equal cardinality such that one is a proper subset of the other. That is what the Hilbert hotel construction does.
@@landsgevaer Ok that explanation does make sense, thanks :)
Literally the opposite of what it's trying to show, which might suggest that it doesn't do it's job very well! If you think of the initial full hotel as containing all positive integers, and then having all negative integers turn up and the hotel being able to make space for them, we see that the set of positive integers and the set of all integers are actually the same size
for the dartboard: if we go to point like targets and pint like darts, the dart tip has no volume so it does not exist, hence it cannot hit the board at that state. if we assume any actual size, the probability is not zero.
none of these are paradoxes 😭😭
The hotel one actually is, it meets the definition
Cool. Were you expecting 1000 likes or sum? Cause who cares what you have to say lmao
@@Glept5542 why did you feel the need to be rude?
@@zackbuildit88 that’s not how UA-cam works. I didn’t mention your username therefore I wasn’t speaking to you…jeez
@@Glept5542 I know you weren't, but I can care about people other than myself. Why did you feel the need to be rude to the original commenter?
my brain hurts
A Glimpse Into Every Infinity Paradox
Is more of an accurate title. How can you call this an explanation?
WOW 😮.. actually, you are right.
The dart one is a prime example of "snarky maths nerds hiding obvious mcguffins in the verbage used to describe the 'problem' and working against finding a realistic solution"
Instead of stating the obvious "hur dur its amazing" trash 'it could land infinitely anywhere on the board (completely ignoring the grain structure of the cork the board is made of) you could easily state it in the much more realistic, not masturbating over your ti-82, way of 'the dart tip diameter is 3mm diameter. What is the probability it will land within the 3mm area you believe it will.'
My purposefully introducing infinity the answer trends towards infinity, therefore it being locked to an infiniti is not that special. It's like saying I have an infinite amount of eggs how many eggs do I have.... It would be amazing if I had two eggs, it would not be amazing if I had an infinite amount of eggs. Other than actually possessing an infinite amount of eggs....
The lamp problem is a unit 1 calc problem. You can't take the limit of an oscillating function. Same reason why lim x->0 of sin(1/x) is undefined.
My favourite 'paradox' is the continuum hypothesis which is likely the simplest to express unaswerable question in ZFC. Though explaining exactly how it works and why it is the case is a lot of fairly heafty maths
Not unanswerable but rather, independent.
The thing about it is, there exist systems for which it is true and there exist systems for which it is false. Both options are acceptable and neither leads to contradictions.
The issue with the dartboard paradox is the fact that the dart will have an area in which the head takes up. Therefore the probability is related to the size of the dartboard in relation to the size of the dart.