I think the analogy to quantum mechanics is totally reasonable, and i'm a physicist. there was another example of this in the history of physics iirc. The wave model of light was attacked because the wave model of light gave a prediction which was considered "obviously absurd": If light bahaves as a wave, then at the center of the shadow of a circluar obstruction should contain a bright spot if the shadow is sufficient distance from the obsruction. But it was considered obviously absurd that there should be a bright spot in the center of the shadow! But here's the thing, no one had actually done the experiment! And when you DO the experiment, you see a bright spot right in the center of the shadow! I took several images in my lab last week where I did exactly that.
The transition at 26:15 was so smooth I thought some Hilbert's Hotel placed an ad on your video XD you should find a hotel named like that and charge them for marketing services
Hilbert's Hotel begins with a violation of the social contract (or at least social custom) by forcing people out of an occupied hotel room to accommodate a new guest. I challenge Craig to choose something that doesn't manipulate the psychology of people.
The Trick is not just a violation of a social contract/custom, but illegal in most places. Otherwise, you could pull the trick even with a real Hotel. After vacating the last room and placing a guest from another room in there, just start back at room 1. Like with Hilbert's Hotel, you are always vacating a room with an occupant, who then becomes a guest without a room, so the number of guests without a room never becomes 0. All you are doing is temporally storing guests in the hallway. The number of guests in the hallway is X + Z, where X is the number of vacant rooms and Z the number of guest that have arrived after the hotel is Full. (Perhaps -Y if any guests left).
Hi Alex, been a big fan of yours ever since I saw that video where you asked Darth Dawkins Jack's question, triggering the classic "What if a mind is omniscient?" response. Gold. And even though I think I agree with your position more than Craig's, there are some points I want to bring up. 1. Differences in cardinal arithmetic It seems to me that if sums are just unions (which seems right), why aren't differences just associated with set differences? In other words, if A = {3,4,5} and B = {1,2,3} then A-B is just "everything in A that's not in B", yielding: A-B = {1,2}. More importantly, if you have a finite set V, and U is a subset of V then the cardinality of V-U is just the cardinality of V - cardinality of U. Example: V = {1,2,3,4,5}, U = {1,2,3} Then V-U is {4,5}, |V-U|=2, |V| - |U| = 5 - 3 = 2 Similarly then if you look at infinite sets, there's still a notion of "set difference" that everyone seems to agree with. {natural numbers} - {even numbers} is odd numbers, and also {natural numbers} - {all natural numbers greater than 3} is just {1,2,3}. So we can't appeal to set differences being "not well defined" in transfinite sets because we know what they result in. We also can't appeal to the operation not being "well defined" in terms of transfinite sets because we know the cardinalities of the answers for both examples we gave. There's nothing prima facie wrong with talking about |A-B| for transfinite sets. So I don't think you can appeal to that. What's left then is that we have a problem with cardinal numbers as cardinal numbers, not based on the set operations. And that problem is that there is no one-to-one correspondence between sets and cardinalities. All natural numbers -> N0 All even natural numbers -> N0 All natural numbers greater than 3 -> N0 and so on So the problem is that |A-B| could refer to an infinite set - itself which is 0 It could refer to an infinite set - all but 3 of its elements which is 3 etc. This is a "well defined" issue due to the fact that there's no canonical representation of an equivalence class of sets via a cardinal number in the same way that let's say "2(mod)3" is a numerical representation of {...,2, 5, 8, ..., 3k+2, ...} in the integers. And what that tells me is that we need to tighten up the theory a bit. I think that we need to put an algebra on sets robust enough that the cardinalities inherit an algebra robust enough to solve these issues, else we aren't really going to have a handle on what's going on. I think measure theory is some kind of blueprint for this. At this point I just think that thinking of infinite sets in terms of cardinalities is just sloppy. Craig is wrong to say that there's a "formal contradiction" there. The technical thing he CAN say is that is that "cardinal difference as a binary operator on sets is not a well defined function". That's the mathematical expression for his claim, which is right, and a problem, just not the one he thinks.
I think all that Alex meant there is that the operation is not quite the same as the one you'd see in an arithmetic signature. I figure he's aware that there's a subtraction operation you can run on sets. But it's defined differently and can deal with infinities, while the classic "-" can't.
@@mar98co1 right, it's cardinality subtraction, which works fine if you use it on sets, but terribly if you use it on cardinal numbers, which is an issue.
On the Planets analogy: I started counting their orbits a finite time ago, and have observed that Jupiter orbits 2x as often as Saturn, so that when someone tells me "they have been doing that for T amount of time" the correct thing to believe is that Jupiter has completed 2*T more orbits then Saturn. And you can make T is big as you like, and the gap between them is correspondingly bigger and bigger. And yet, if you say "they have always done this", I must somehow think they have completed the exact same number of orbits (instead of Saturn's being infinitely less than Jupiter's). And I must deny my own experience of watching them for as long as I like (potentially forever) and *recording* that one is orbiting twice as often as the other....
Great conversation, Joe and Alex! Unfortunately, I think it’s had an unintended effect on me as I’ve now convinced myself that a full Hilbert hotel, like any full finite hotel, can add a new guest only by actually adding a new room to the hotel. Where am I going wrong with the following? Represent any hotel by a set whose members are rooms with value X if occupied and 0 if vacant. So a full Hilbert hotel is represented by Hf = {X, X, X, X, …}. When a new guest arrives at the full hotel, the manager does “something” (we are told it’s just shifting all the guests) that frees up the first room. After the shift the Hilbert hotel with the vacant first room is represented by Hv = {0, X, X, X, …}. But Hv = {0} ∪ Hf, which looks like the old full Hilbert hotel with a new vacant room added. So should we really believe the story about the shifting guests, or did the manager really have a new room added to accommodate one of the existing guests that got shifted?
Interesting question! I think the first issue is your use of sets. A set, by definition, has no order and contains no duplicates. So {1,3,3,2,3} = {1,2,3}, since the duplicates and order of elements are ignored. Your example sets would thus be {X} and {0, X}, which is not what you meant. I think you mixed this concept with the concept of a regular list (ordered, duplicates allowed, often represented with square brackets or parantheses). Let's stick with the list notion and close to your notation (0 and X might be signs on the door which the guests can flip) and see what we get. Let t=i represent the point item at which the hotel's state is shown. At first, no guests are checked in. t=0 [0,0,0,...] Then, an infinity of guests, numbered 1, 2, 3 etc. arrive and check in simultaneously (we don't need to worry about the time component here). t=0 1 2 3 | | | V V V [0,0,0,...] t=1 [X,X,X,...] Now, the manager does something and we arrive at this: t=? [0,X,X,X,...] So it seems like we did this: [0] + [X, X, X, ...] = [0,X,X,X,...] And we *could* do that to achieve the result. But there is another way, which is the traditionally less intuitive one. Let's look at the process step by step. t=1 [X,X,X,X,...] t=2 1 2 3 4 ^ ^ ^ ^ | | | | [0,0,0,0,...] t=3 ->1 ->2 ->3 ->4 : : : : [0, 0, 0, 0, ...] t=4 1 2 3 4 | | | | V V V V [0, 0, 0, 0, ...] t=5 [0,X,X,X,...] Again, without worrying about the time component, we can see that this process works equally well. The misleading part is, since the Xs all look identical, it only seems as though we have just attached something new "to the left". When we look at what is happening in between the states, it's clear how the shift happens. Hope this helps! :) If you have any more questions, hit me up/reply to this comment ^^
@@MyMusics101 Thank you for the reply and for correcting my sloppy notation and terminology. I think my issue with the HH is really of my own making, in that I think I’m adding an additional condition beyond what is in the typical scenario. If the HH is just a sort of container that can hold an infinite number of generic rooms, then clearly it can accommodate any number of new guests even when full. No problem there. In thinking about whether the strangeness of Hilbert Hotels implies a strangeness for something like a beginningless past I wanted to remove the generic quality of each room to see what happens. I think my example was not successful. Although this also needs improvement, the following scenario might be better: Suppose each room of the hotel has a unique room number (nothing unusual there) and let’s say that when a new guest arrives at the hotel she also registers her room number with the local post office to receive mail. Say there is a particular HH whose room numbers consist of all the natural numbers, 1, 2, 3, …. etc.. and that this hotel is full. If a new guest arrives, the manager can have all the guests move over one room as usual, but if the current guests now insist on taking their room numbers with them to continue receiving mail (because mail forwarding is not an option, for whatever reason we want to make up), the first room will become vacant as usual but it will have no room number, and no unique natural number can be assigned to that room number. The manager could assign the number 0 to that room, and the guest could register room 0 with the post office, but there is a question now of whether this really is the same hotel as before since there was no room 0 in the original hotel. In this sense, the hotel acts more like a finite hotel and perhaps some of the strangeness is gone. Anyway, I’m not sure if this is bringing any clarity about the possibility of an infinite past, but it doesn’t hurt to throw around ideas ;) Thanks again for listening!
@@alexanderdelaney3978 The Trick is in the "something" aka shifting. To understand the trick when applied to a hotel with a finite number of rooms, all you have to do is once you reach the last room you continue with the first room (or any other random room for that matter). By doing so, you can always create an empty room, and it would seem as if you could have an infinite number of guests. Except with a finite hotel you will eventually run out of hallway space, because that is the actual space, you are filling.
15:30 there was a young man from fratton who went to church with his hat on for he always said with my hat on my head i'm damn sure it won't be sat on.
can you say "ALL the rooms are full"? surely it doesn't matter if they are finite or infinite if the rooms are ALL full then you can't move anyone into an "unoccupied" room?
Zero has a lot of the same "problems" that infinity does, but tangentially to what Alex said, our system for arithmetic is constructed to accommodate zero, so it seems more intuitive. If we take the dominos.. It's "unintuitive" that a line of infinite dominos falling won't have a last domino fall. But it seems completely intuitive that a line of zero dominos falling won't have a last domino to fall. We can't count (in the arithmetic sense) _to_ infinity, nor can we count _to_ zero. (Though we could conceivably count _at_ infinity or _at_ zero, depending on how you want to define "count") Also, I'd love to hear the argument for creating a single hotel room ex nihilo being metaphysically possible, but creating an infinite number of hotel rooms being metaphysically impossible.
The Hilbert's hotel is basically just overcomplexly saying that the properties of infinity are weird. Basically that ∞ + 1 = ∞ or ℵ₀ + ℵ₀ = ℵ₀ Which are both a bit unintuitive, yet were both mathematically proven - so there actually exists a specific concrete proof of it being true
I think a better way of thinking of Hilbert's Hotel is to think of a guest being born in the hotel and by that time the hotel builds them another room (recently reading Senlin Ascends so my first thought was the Tower of Babbel). Because if it's simulating the universe, it should be self-sufficient (let's pretend in the thought experiment that the people are like some animals that can give birth without a male counterpart [parthenogenesis]).
Good content as always, Joe. I have a potentially (but not actually ;) ) dumb question, bear in mind I'm no kalam expert. On the discussion around 1:21:00 - to my mind, I can agree that in an endless series of events of counting, the answer to "how many number are there such that they will be counted?" is aleph null. But to me, the symmetry breaker is that the endless series has not yet been crossed, whereas a beginning-less series has been crossed. We will never get to aleph null in the future. But someone proposing an infinite past seems to be saying we have already somehow crossed that gap, which seems impossible. Am I missing something? Maybe this has already been discussed and I'm missing it.
Haven't watched the whole discussion yet, but this doesn't seem to be much of a problem to me. After all, there was never a point in time where we actually crossed anything. If the past is infinite, then it has always been this way and thus, we never were in a "there have been a finite number of moments before this point in time" situation. The two cases are symmetrical in the sense that, for the future, we can say that we *will* forever remain in the "*finitely* many moments counted" state. Whereas in the past case, we can say that we *have* always been in the "*infinitely* many moments counted" state. (Counting moments meaning moments having passed here, obviously, because otherwise we would have to try figuring out what infinite number of moments exactly has passed, to which the answer would (presumably) always be "a countable infinity many".)
@@MyMusics101 So, I re-read the Malpass/Morriston paper last night, and maybe my question has less to do with the paper and more to do with traversing the infinite (which they explicitly don't deal with in the paper). I think I agree with every single sentence in your reply, I'm just not seeing the relevance to my comment. It seems to me that (given an infinite past), the answer to "has the daily negative-integer-counter filled out all the negative integers?" is obviously yes. But how can this be, when it seems that everyone agrees that you can't fill out the natural numbers by progressively counting into the future? aleph null doesn't have an immediate predecessor, which is often the reason given for why I can't reach it in the future. But symmetrically, if it doesn't have an immediate predecessor, then how could we have filled out the negative integers through past counting down?
You are correct, anyone who says we can get to today through an infinite past is talking nonsense. Imagine a iine of dominoes with no beginning or first domino and there is a domino at the end of the table and if it falls it will fall off the table, The only way a domino falls if the preceding domino knocks it down, Now if the chain of dominos extends infinitely backwards with no first domino, will that domino at the end of the table ever fall down and be knocked off the table. The answer is obviously no. Not only will it not fall off, none of the dominoes will fall down, because there is no first domino to knock down any other domino. So the problem is obvious with a beginnless series of events or eternal past, you will never arrive at any point in time.
I can see that the original Craig's video is misleading at least and outright lying at most. Take for instance the guests leaving the hotel. They say: "An infinite number of guests left the hotel, yet there are NO FEWER guests in the hotel." I think that is deliberately said in that manner to cause the highest confusion. As we instinctively expect that when some people left the number of guests must change. If we would word it in this way: "An infinite number of guests left the hotel, yet there are still an INFINITE NUMBER of guests in the hotel." It wouldn't feel as much confusing as more people would instinctively recognize that we are dealing with infinities and then all kind of weird stuff can occur. I don't like this kind of deception. I would say it is outright lie. With infinities we cannot pinpoint some exact number so we cannot say if it is or isn't the same number of guests. For anyone with basic knowledge about sets theory must realize that and given Craig is using it as an argument I believe he has to have decent knowledge about the theory.
2. I have to disagree about Dretzke's argument being a counter to Craig's argument, but I think in order to make this argument I'm going to have to steelman Craig a bit because he argued this is a sloppy manner. When I look at Craig's successive addition argument, the core of the intuition actually goes like this: 1a. To give a historic account of the present, you have to give a historic account of the event preceding the present one. 1b. To give a historic account of the event preceding the present, you have to give a historic account of the event before that. 1c. By induction, to give a historic account of the present, you have to give a historic account of each successive event preceding the present (successive addition) 1d. This is also important, but not explicit in the premises, but you would actually run this argument again for every 2 preceding events in the event that something causal happens at t-2 to affect the present t, but it doesn't manifest in t-1. And then you run it again at 3 prior events, and so on and so forth. Basically the present is the cumulative outcomes of all events which causally effect it at prior points in time. 2. If the past is infinite, then given any past event you count back to, there will be a greater number of prior unaccounted for events, than events you've accounted for. C. If the past is infinite, no historic account of the present is possible. So why, given 1, do we think 2 is true? Simple. A. Let k(t) be a counting function that spits out a natural number k after t time. B. For any k(t), there exists a natural number 10^k(t) > k(t) that you haven't counted to yet. C. As t->00, consider the difference in what you've counted vs what you have left to count: 10^k(t)-k(t), we know that this value approaches 00. D. So paradoxically, as you count back, you're actually going to miss MORE things for each number you count. You actually get further away from having a historical account of the present, rather than closer (as you'd expect by looking to the past). And the "yeah but we'll get there eventually" line isn't going to work. Why? Let's suppose you count to 1, well there are 10 events you could have counted to, and you're missing 9. Yeah yeah OK but we'll count to 10 eventually! Well what happens by the time you get to 10? Well now there's 10^10 things you could have counted to, so you're missing almost 10 billion things. 1->10 10->10^10 10^10->10^(10^10) etc. So the "we'll get there eventually" clause doesn't help if, by the time you do eventually get there, your problems multiply faster than the ground you've gained. And this isn't a trivial matter since we're dealing with past events. What this is basically saying is: By the time we've accounted for 10 events in the past, we now have 10^10-10 events that could have influenced those events, so in appealing to those events to count for the present, all we're going to find out is that there's now billions of events before that, that we haven't a clue about, which means we really have no idea of the causal history of these 10 events which were supposed to influence the causal history of the present. Ah! We'll get to those 10^10 eventually, no problem! OK and then by the time you do there are going to be 10^(10^10)-10^10 events before the 10^10th past event whose causal history determines the 10^10th event, so in appealing to the 10^10th event to explain the 10th event, AGAIN we're actually no closer to explaining the 10th event whose job it was to explain the first event. So we've now counted back 10^(10^10) events and basically have nothing to show for our trouble, except exponentially multiplying problems. Point is, just because you've counted N0 elements, and your initial set has N0 elements, doesn't mean you've exhausted your set. So while I do believe that George counts N0 things, I do NOT believe that George counts every natural number. That's the double edge sword of cardinality. Best way to deal with this argument is causal finitism. Only finitely many causally dependent things in a causal history, an infinite number of causal histories. Done. Then you still have an infinite past without the multiplying complexity of unaccounted for prior events, since each causal history doesn't depend on the previous one, so if you fail to count it, oh well. So long as you can give an account of all the elements in your causal history, you can give an element of the present.
Ok, I'm only 25 minutes in, but Malpass is pushing my buttons again, so I might as well just briefly post a few issues I'm having, but I'll keep watching for sure (maybe they even address some of these as the conversation goes on): 1) Saying “this is just how infinities are” is WAY too quick. Wittgenstein, for example, insisted that there is no such thing as “all the natural numbers”. Such a phrase, he thought, is meaningless. Though we use infinities in the same grammatical forms as finite numbers, we are actually referring to *rules.* So, for example, given how addition by 1 works, I could always tell you the next Natural Number. But it does not follow that it makes any sense at all to speak of “all” the Natural Numbers and call that a set with infinite cardinality. At the very least, we shouldn’t take for granted that that is a coherent thing to say. Neither the experiments nor the mathematics of quantum mechanics require us to deny anything in our basic intuitions. Some ways people _talk_ about QM seem absurd, but I don’t understand why that doesn’t just count against those so-called “interpretations” and in favor of ones that cohere better with everything else we know. John Bell certainly advocated for speaking more sensibly about QM, and he gave examples that could work just fine. 2) “Full” means that every room has a person in it. Shuffling them around just changes *which* person is in which room. It does not create more rooms. This is not just some common sense intuition, it is part of what it *means* for “every” room to have a person in it. Another way to look at it is to say that we are not allowed to add any more rooms to the hotel, and yet, after the shuffling, we will have a room/person combo (ergo, a room) that we did not have before. That’s a contradiction. Why are we not allowed to add any rooms? Because that’s part of the analogy. It’s the whole point. You are stuck with only those same rooms that have been occupied all along, and you shuffle people around and end up with an extra room (or many extra rooms). So you both do and don’t add rooms to your building. 3) “Greater than” and “equal to” at the same time is a straightforward contradiction. “Greater than” is, by definition, an “inequality”. This is no small point. Malpass is right to think that accepting Hilbert’s Hotel means we need to swallow a situation in which it can both be true that the number of guests is greater than it was _and_ that it is equal to what it was, and that is a contradiction; not a mere weirdness. I'll stop for now. Haha. It's just that Malpass has this way of just blithely speaking [what seems to me to be] patent nonsense, and it sets off alarm bells in my head.
1)But if you're gonna be a finitist, then Alex's other objection seems to land, what do you make of the "infinite afterlife"? 2,3) I think the same as the "-" point applies. We're talking about set theory, so we probably shouldn't be thinking about the arithmetical greater than and equal, but rather supersets and bijection. This operation we're doing in the Hilbert hotel is constantly done in set theory, it's well known that sets with infinite cardinality can have the same cardinality of some of their own subsets. And N ⊂ M and |N| = |M| just isn't a contradiction, provided they're infinite sets. Loosely ℵ0+ℵ0= ℵ0, such is the nature of infinite sets. If you don't like that answer that's fine, but keep in mind that to object to transfinite set theory altogether, besides the fact that you're joining a niche minority in mathematics, even those that do take finitist and ultrafinitist views understand there are no contradictions. The problems raised are rather about definitions and counterintuitiveness. Most mathematicians take finitism to be a worthwhile endeavor since its results are important, for example, they tell us a lot more about what we can do computationally. But like I said, almost none takes the position that there's an actual problem with transfinite mathematics.
@@mar98co1 1) If I believe in an "infinite afterlife", all I'm saying is I will never die. My age will always be finite. 2) I get how it works in Set Theory, and I'm not saying (necessarily... most days of the week) that there's something wrong with Set Theory internally. But there can be contradictions when applying it to real situations. To say "the number of people now both is and isn't greater than what it was" is a contradiction. I find it interesting that you say "it's often just about counterintuitiveness or definitions". Counterintuitiveness I don't really care about, but definitions... they are the reason why a married bachelor is a contradiction. It's no good to say "the problem only arises when we take the definitions of the terms into account"!
@@Mentat1231 But you will live forever, and I'd ask you, how many moments will you live in total? i.e. what cardinality does the set of moments you lived have. But if you are skeptical of our friend ℵ0, which would seem the standard reply, then the question won't make much sense in the first place I suppose. "most days of the week" cracked me up :D Right, just because we have a mathematical model, doesn't mean we can bring it to metaphysics (or maybe it does). Yes, phrased like that, it's a contradiction. But depending how we interpret that to mean, it might not be: -the number of people there were before is n (∈ ℕ), now it's m, and n
@1:14:50 Why do theists need an actually infinite future? I've never conceived of an afterlife in that way, but as an unending, but always finite, existence into the future.
We are talking about an endless future which, at each point in time, is finite, but is nevertheless endless. By an 'infinite future', we mean an endless future; and our arguments apply to an endless future. What you described is what we are talking about :)
An hour in, and they get to one of my main objections: If we say you can't get to infinity by starting now and counting forever, then why isn't it also true that you can't get to now by having counted forever? After all, the present stands relative to the past series in the exact same situation as the future perfect stands in relation to everything before it.... Malpass basically puts the burden on the objector to show that the reason why you can't do the one is reversible so that you can't do the other. So, what is that reason? Malpass says the reason the Counter can't start now and eventually "have finished" is because there is no immediate predecessor to aleph-null. So, let's see if the objector can turn that around: When we say "aleph-null has no immediate predecessor", we are talking as though there could be an end to an unending count, but that we can't actually get there because of this disjoint. Of course, there is no such end (that's what "unending" means); but if Malpass can phrase it that way, then so can the objector. So, imagine the guy who's counting down "all the negative numbers" and finishes today. What about the beginning of his beginningless count? is it not also disjointed from aleph-null by not having an immediate successor? If it is allowable to say that the *reason why* the Counter cannot start now and "have finished" later, is *because of the gap* behind aleph-null, then what forbids saying that the reason the Counter cannot have always been counting and yet finish now is because of the gap in front of aleph-null? Think about it: If the problem is a gap between the "all of them" number and any other number, then that gap can get pushed back as far as you like (even infinitely), but it's still there! What argument can you possibly give to show that it some how *goes away* or ceases to be a gap? To put it another way, if you say the Counter cannot _fill out_ the set of all Natural Numbers because there is a gap before the number that means "all of them", then who cares which order I fill them in?? if I'm counting backward and just finish today (-2, -1, 0, done!), then I have filled out the Natural Numbers finally today without worrying about any gap. It is inconsistent to say both "you cannot fill it out because of the gap, and you can fill it out despite the gap". Malpass would be the one with the burden of proof to show why one is forbidden and the other isn't. The way out, of course, is to say the gap problem is NOT the reason why the Counter can't get to infinity. He can't get there because there is no such thing. There is no highest natural number, nor are there actually infinitely many natural numbers, there is just a rule for getting the next higher one. But, that would entail Finitism.....
The universe did not have a beginning nor does the universe have a past. Time is an illusion. What does full mean in this context anyway... One person per room? Or an infinite amount of persons in each room?
@@jackx341 The big bang is only evidence for an explosion. It is as far as our measurements currently take us but it does not say what happened prior to the big bang. Thus, it is not the beginning of the universe but it is called the beginning of our universe because of our experimental limitations. Causality allows us to call a prior cause the past but if causality is cyclical then this ordering is arbitrary.
"Time is an illusion." You have to be literally brainwashed to hold such a radical position! Time is one of the most obvious truths we as human beings, which we experience on a daily basis, know about reality. The fact that most scientists hold time to be an illusion is actually very powerful evidence that scientists can be wrong rather than evidence that time does not exist.
I never understood how Craig doesn't get the counting all the naturals thing. Like, consider how mathematical induction works. You prove a theorem for your starting point 0, then prove that it holds for n+1, and that's it, you're done. You've proved it holds in all the naturals. Seems we can apply the same idea: John will count 0, and John will count n+1. It just seems trivial from this that John will count an amount that is the same cardinality as the naturals, just like for induction, if the predicate holds for 0 and it holds for n+1, then it just holds for all the naturals. Craig would have to say there is some n such that John will not count it, ever (rather than equivocating with, there is some n that John HASN'T/will not HAVE counted).
What Craig says is that the ability to count *any* number does not entail the ability to count *every* number. He's right about that. It is an illicit move to go from "any" to "all".
@@Mentat1231 But I don't see how that's relevant to the question as to whether John will count every number. Here we're clearly not saying that John will count any number, we're saying he'll count every number. From: there does not exist an n such that John will not count n, it just follows via equivalence that John will count every n. That's another way to think about that which just seems straightforward.
@@Dan_1348 Whatever point you pick in time, he will always be counting a finite number at that time, and it will always be easy to see what the next one will be that he has so far "missed out".
@@mar98co1 I could give intuition pumps (like the Tristram Shandy story from Russell), but I think it's pretty straightforward: The move from "there is no number that he will not eventually get to" to "he will eventually finish counting them all" is an illegitimate move. As Craig said in his discussion with Malpass, it is akin to a Composition fallacy. From the fact that there are no members you will not eventually get to, it does not logically follow that you will eventually finish them all.
@@Mentat1231 I don't think it's a move to "he will eventually finish counting them all" though because that would be equivalent to Craig's shift to "will have counted". Alex's point is, as I understand it, that if Fred keeps counting forever, then if we pick any number n, Fred will count n, so in that sense Fred will count every natural number. As to "there is no point at which Fred will have counted all natural numbers", I think Alex agrees with that. EDIT: Alex describes it at 52:40.
It's from his Theological-Political Treatise. For a reference, you can do command F and search 'what altar of' in the following: returntocinder.com/source/SPIN.TPT
I'm only ten minutes in, but it's looking like you are granting straight away that having Cantor's property directly implies that new empty rooms can be generated. I don't have a good understanding of that and would like to see that fleshed out a little more, I don't quite see it yet. It seems to me that you remove someone from a room in the process of preparing to move them to a different room. Surely this is what actually generates an apparently empty room. Then you go around shuffling other people from full room to full room, never finding another empty room to put that extra person into... even if you do this shuffling procedure infinitely, you always have one empty room and one person who has been displaced from their room. That's the state after every shuffling step, the same as after the first step. Keep doubling the room number, and you'll never find an empty room up there to put the displaced person into. So, the inference from Cantor's property to new rooms seems to me to involve some sort of intuition that if you keep looking down an infinite line of full rooms, you'll get to an empty room. If you have an infinite set of X's, there must be a non-X in the set somewhere. That you'll run out of finite numbers if you keep iterating a growing maths function forever.
Luckily you aren't leaving it there, I see you go on to say that fullness is maintained, but you can still add people. You don't need to find an empty room for that. I'll keep watching :)
I'm still not clear on it. Without understanding the issue, my intuition tells me that if these procedures with infinite sets of numbers generates new numbers, then the same procedures with an infinite set of full rooms should generate new full rooms - not empty ones you can put people in.
@@HeyHeyHarmonicaLuke there's no need to shuffle people around for infinity. Every guest in the hotel just moves to the next room simultaneously, so that room 2 - infinity are filled, and room 1 is left empty.
@@HeyHeyHarmonicaLuke it's not the case that the hotel is generating a new room whenever you move the guests around, it is just opening an already existing room up for another guest. Since for every room in the hotel will always have another room next to it (R+1 applies to every room), there will never be a point where a guest is unable to pack up and move to the next room. This means you can move guests over as many times as you'd like to open up any number of rooms.
@@HeyHeyHarmonicaLuke The key here is to have a starting point. If the hotel rooms went from -infinity to +infinity, and then you told all the guests to move down one room, then no rooms will have been opened up, because the guests are not moving away from any starting point. If instead you said: every guest in a room below R1 move down a room (R-1) and every guest in R1 and above move up a room (R+1), well then you have successfully freed up 2 rooms, R-1 & R1.
Consider Hilbert Jr's hotel. It has more than 5 rooms and more than 5 guests. So it's full I guess (that seems to be the logic in Hilberts hotell)? Now another guest wants a room, simple enough. Every current guests just moves up one room and they all fit. And now to the crazy part, it's still more than five guests in the hotel. So "more than 5" + 1 is still "more than 5". Insane, obviously "more than five" isn't a real concept. But wait there is more, now "more than 5" guests come to the hotel. No problem, just move all the guests to the room double it's size. And it is still "more than 5" guests in the hotel. Bonkers
Also, assume every guest in an even numbered room leaves. There is still "more than five guests" left. Now everyone but 3 leaves, and only 3 remain. So "more than 5" - "more than 5" is both "more than 5" and 3. A contradiction
@@Oskar1000 More than five doesn't specify a number uniquely . a is more than 5 and b is more than five does not imply a=b. That doesn't mean "more than five" isn't a "real concept". It's just that the property doesn't uniquely determine a natural number. Nor does it say that if you have two sets with the property of having more than five members then you can find a bijection between the sets. So your example is not analogous to saying two sets has the property of being countably infinite. In that case you could find a bijection between the sets. And you have the counter intuitive properties (depending on your intuitions) that an extra perpended member of a countably infinite set is still a countably infinite set.
@@HyperFocusMarshmallow Infinity doesn't specify a number either. Mathematicians warn against using it that way. I don't think showing a bijection is enough to prove that one set isn't larger than another. (Even if it does prove they have the same cardinality). Imagine this scenario: 1) You put down one black ball numbered 0 into an infinitely big vase. 2) Then for each natural number you put down a white ball (with that number) in the vase. 3) You remove all the white balls in the vase. On my hypothesis there is one ball left, the black ball. But we can show a bijection between the white balls and "The white balls plus the black ball". Let W be the set of white balls. Let B be the set of white balls plus the black ball. For all w in W we can find a corresponding b in B by the relation w-1. For all b in B we can find a corresponding w in W by the relation x+1. So we shown both that one set has 1 more member and that they have the same cardinality. My point is that in both the "more than 5" and the infinite we lose some information that makes it impossible to do the maths precisely. If we add that information back both Hilbert's and Hilbert Jr's hotel seem quite intuitive.
@@Oskar1000 A set can't have 1 more member and have the same cardinality. Cardinality is defined as a bijection, and there ain't no bijection if the two sets have different amounts of members, by definition. I think you're mixing up arithmetical and set-theoretical operations. When you add one member to an infinite set, the set's cardinality remains the same. That means the same as saying that the amount of things in the set(amount of members) remains the same. If you find that ridiculous that's another matter, but mathematicians that reject transfinite mathematics are quite the minority, you just need to use the right tools.
Have you finished reading "the kalam cosmological argument, philsophical evidence for the finitude of the past" (Paul Copan) It's probably the best compilation of the philosophical literature on the argument.
@@matthieulavagna Why would you take philosophical arguments as "evidence" in light of our best scientific knowledge saying otherwise? Would you take Zeno's paradox to say that I can't walk across to the other corner of the room?
@@dr.shousa because philosophy is more reliable then science. Science changes overtime, philosophy does not because logic is infallible. Once something is demonstrated, it's demonstrated forever. I don't see why you're talking about Zeno's paradox. It's completely disanalogous to the kalam, since the intervals are potentially infinite and unequal whereas the events in the kalam are actually infinite and equal.
malpass just says absurd stuff, actual infinites are not counterintuitive, they are impossible and illogical. its just a way of getting away from the truth, to just call something counter intuitive which you could say to anything if you want.
Hi, I'm obviously not an atheist but I am a mathematician. WLC does not understand Cantor and infinities. Completed infinities are not only possible but absolutely necessary. Without them we cannot have basic maths. The whole thing collapses into absurdity. If anyone wants to see how let me know. But please, please start looking into why Cantor said what he said and how he proved that actualized infinities MUST exist mathematically.
I want to see how, please! I find it very difficult to believe actual infinity is a coherent concept. It's always seemed obvious to me, even as a child the absurdity of actual infinity. For example, I had this moment, when I thought to myself the idea of God recalling his entire past one moment at a time and how logically impossible that seems. I thought to myself a form of: "If God lived an infinite amount of moments in the past, how could he get to today?" It made no sense to me then and as an adult, I now accept WLC's conception of God's being timeless without creation and in time since the creation.
@@danglingondivineladders3994 That's an interesting idea and I have thought about it myself but I'm pretty sure it'd still be classified as potentially infinite. I can give an example of a ruler that represents the number line. We can continuously divide the ruler up between 0 and 1 meters into smaller sub-units but we will never get to the point where the ruler is divided up into an actually infinite number of parts. A finite object should not be classified as having an infinite number of parts but rather as an undifferentiated whole and only after it is cut up can we consider its parts as distinct entities. Otherwise, we don't have an actual infinity but rather a finite-sized object that could be cut up into a potentially infinite number of parts!
@@michaelsayad5085 well I would say that Cantor's diagonal argument is the basis of completed infinities and he uses infintesimal numbers to establish his point. it really is about the amount of numbers in between 0 and 1.
@@michaelsayad5085 those are two different things. WLC misrepresents completed infinities in order to say the past must be finite. there are many reasons to think the past is finite without asking people to trust their unverified intuitions instead of a 100% proven theorem. He is provably wrong and if we accept that completed infinities are impossible then we cannot have a number line, geometry or any sort of coherent system. just use any of the other reasons to believe in a finite past as I do.
The part around 35 min where Joe argues that even though we’re removing the same quantity from the same quantity and yielding different results, because we’re using different operations it’s not a problem, seems confused. For example, if we have 10 balls numbered 1-10, and remove all the odd numbered balls, we are left with 5 balls. If we instead remove all the balls numbered greater than 5, we’re also left with 5 balls. We removed the same quantity from the same quantity, but using different operations, and ended up the same quantity. So why should we expect to end up with differing quantities when we use transfinite numbers? It seems to me that the only answer is “well it’s because they’re transfinite numbers, you have to check your intuitions at the door.” And that’s the problem.
Thanks for the comment my dude! Here’s a clarification of my argument. There doesn’t seem to be anything problematic about removing an identical quantity from an identical quantity and yielding, as a result, different quantities because, in some cases [namely, those cases involving denumerable collections], different *ways* you remove elements of a set can differentially determine the cardinality of the resultant set. And, importantly, there is absolutely nothing inexplicable or mysterious about this - there’s no puzzlement, no mystery, and consequently - for me, at least - no intuitive absurdity. When I reflect on the cases, they seem perfectly (indeed obviously) benign rather than absurd, since I have a perfectly good explanatory story to tell concerning how different results are yielded depending on the way in which elements are removed. And so my point here is really meant to help people see why I don’t find it at all problematic or unintuitive, in those cases involving denumerable collections, that one can remove subsets of identical cardinality from the original subset and yield resultant subsets of different cardinalities. No one is saying that you have to check your intuitions at the door; instead, I’m explaining why I don’t find it unintuitive in the slightest. (Indeed, I find it intuitively obviously benign and intuitively obviously non-absurd.) I don’t claim everyone must share my intuition here. But I can at least try to explain to people what considerations undergird my intuitive judgments. And one such consideration is that it’s perfectly explicable, with no mystery at all, why we can yield sets with different quantities depending upon which precise denumerable sets we remove from an original denumerable set.
@@MajestyofReason first of all, thanks for the reply. I really wasn’t expecting that. I watch a lot of your content and respect your work, so this is meaningful to me. I would like to get your thoughts on what I said about finite sets though, my example of a set of 10 things. The reason I find the existence of actual infinities so intuitively absurd is because of how different infinite quantities act when we consider them in the real world. If the hotel has 10 people and odd numbered rooms are vacated, then 5 remain. Same when the rooms greater than 5 are vacated. This holds for any finite number of guests. But suddenly when we perform both these operations on an infinite number of guests, we get different answers. The reason this violates my intuition is because the people are the same, the rooms are the same, the mathematical operations are the same, etc., but because the number of guests we start with are infinite, the results are different. It just doesn’t seem to me that such a thing could plausibly occur in the real world. To be clear, you don’t think that there is a contradiction in guests leaving the hotel as Craig’s video says, right? I think there is, and if I’m right, doesn’t that mean it’s logically impossible for any guests to leave once an infinite number of guests occupy rooms? And wouldn’t this be true even if the hotel manager put the infinite number of guests in only the odd numbered rooms and left the even numbered rooms vacant, for example?
@@mjdillaha I think what Joe means is that for him the people are not the same, the rooms are the not the same and the mathematical operations are not the same either. Which is perhaps why Joe doesn't share your intuitions on the Hilbert Hotel paradox. ie: Joe has different intuitions when it comes to how finite subtraction works compared to infinite sets. Joe doesn't think there's a contradiction because the mathematical operations are not the same: different subsets are being removed, which is why we see different results. So the differing results are actually to be expected.
@@roger5442 If this is correct, then I don’t see why we can’t invent other systems of mathematics that describe absurdities like this and hand-wave away contradictions by saying we can tell a story about it. For example, I could say that if we have 10 people, and all the even numbered people leave, then we have 5 left. But if the first 5 leave, then we have 10 people left. When you point out that there’s a contradiction, I respond “well they’re different people, they left in different ways. This is fine given the axioms that this system is based on.” I don’t think I’m speaking out if turn by saying you’ll definitely take issue with that explanation. These axioms don’t describe the real world at all, and worse, they contradict the math that describes the real world. That is why I think we’re warranted in saying that actual infinities cannot exist.
@@mjdillaha I don't think Joe is necessarily saying he's correct and you're incorrect. He's saying that your appeals to your own intuitions aren't persuasive to him since Joe has different intuitions that tell him there's nothing mysterious or weird about the results. With your example above the oddity there is that you've said you've removed people 6-10 but then say people 6-10 are still there. ie: you are removing and not removing the same subset of people 6-10. ie: You start with a set of 10 people. You remove subset 6-10, and say your set still contains subset people 6-10. Certainly I would agree that appears odd. However this oddity doesn't appear to occur in the Hilbert's Hotel. When you remove subset people 6 and above then you are only left with subset people 1-5. The result is expected. And when you remove subset people odd numbers (1, 3, 5 etc) then you are still left with the subset people even numbers (2, 4, 6 etc). Again expected result. The results are different because we are not removing the same quality of subsets. One subset you create contains people 6 and above vs the other case the subset contains only people odd numbered. I think this is the misleading part of the Hilbert's Hotel argument - it wants us to think the subsets are of the same quality - but they are not. Besides - I often think Hilbert's Hotel is flawed in the sense it's talking about concrete physical objects like people that can moved around, where as no-one is saying the past events are physical objects that occupy space all at the same moment in time that can moved around.
Sound quality is terrible along with his accent. Jerky choppy speech. So having trouble hearing/ understanding Malpass. He did nothing to clarify the absurdity of infinite past events. I didn' t need it. Infinity has no concrete manifestation which requires limitation.
I think the analogy to quantum mechanics is totally reasonable, and i'm a physicist. there was another example of this in the history of physics iirc. The wave model of light was attacked because the wave model of light gave a prediction which was considered "obviously absurd": If light bahaves as a wave, then at the center of the shadow of a circluar obstruction should contain a bright spot if the shadow is sufficient distance from the obsruction. But it was considered obviously absurd that there should be a bright spot in the center of the shadow! But here's the thing, no one had actually done the experiment! And when you DO the experiment, you see a bright spot right in the center of the shadow! I took several images in my lab last week where I did exactly that.
I am such a Malpass addict!
Hey, Joe! Thanks for the discussion and for having Alex on again. Here you go, have a comment for the algorithm. Thanks for what you do!
The transition at 26:15 was so smooth I thought some Hilbert's Hotel placed an ad on your video XD you should find a hotel named like that and charge them for marketing services
Hilbert's Hotel begins with a violation of the social contract (or at least social custom) by forcing people out of an occupied hotel room to accommodate a new guest. I challenge Craig to choose something that doesn't manipulate the psychology of people.
Stand back guys, we got a 450 IQ libertarian here.
Ya found me. The smartest guy in the backwoods baptist youth group somehow always spots me
The Trick is not just a violation of a social contract/custom, but illegal in most places. Otherwise, you could pull the trick even with a real Hotel. After vacating the last room and placing a guest from another room in there, just start back at room 1. Like with Hilbert's Hotel, you are always vacating a room with an occupant, who then becomes a guest without a room, so the number of guests without a room never becomes 0. All you are doing is temporally storing guests in the hallway. The number of guests in the hallway is X + Z, where X is the number of vacant rooms and Z the number of guest that have arrived after the hotel is Full. (Perhaps -Y if any guests left).
@@andrewtate5252 I'm pretty sure they were joking
Hi Alex, been a big fan of yours ever since I saw that video where you asked Darth Dawkins Jack's question, triggering the classic "What if a mind is omniscient?" response. Gold. And even though I think I agree with your position more than Craig's, there are some points I want to bring up.
1. Differences in cardinal arithmetic
It seems to me that if sums are just unions (which seems right), why aren't differences just associated with set differences? In other words, if A = {3,4,5} and B = {1,2,3} then A-B is just "everything in A that's not in B", yielding: A-B = {1,2}.
More importantly, if you have a finite set V, and U is a subset of V then the cardinality of V-U is just the cardinality of V - cardinality of U.
Example: V = {1,2,3,4,5}, U = {1,2,3}
Then V-U is {4,5}, |V-U|=2, |V| - |U| = 5 - 3 = 2
Similarly then if you look at infinite sets, there's still a notion of "set difference" that everyone seems to agree with. {natural numbers} - {even numbers} is odd numbers, and also {natural numbers} - {all natural numbers greater than 3} is just {1,2,3}. So we can't appeal to set differences being "not well defined" in transfinite sets because we know what they result in.
We also can't appeal to the operation not being "well defined" in terms of transfinite sets because we know the cardinalities of the answers for both examples we gave. There's nothing prima facie wrong with talking about |A-B| for transfinite sets. So I don't think you can appeal to that.
What's left then is that we have a problem with cardinal numbers as cardinal numbers, not based on the set operations. And that problem is that there is no one-to-one correspondence between sets and cardinalities.
All natural numbers -> N0
All even natural numbers -> N0
All natural numbers greater than 3 -> N0
and so on
So the problem is that |A-B| could refer to an infinite set - itself which is 0
It could refer to an infinite set - all but 3 of its elements which is 3
etc.
This is a "well defined" issue due to the fact that there's no canonical representation of an equivalence class of sets via a cardinal number in the same way that let's say "2(mod)3" is a numerical representation of {...,2, 5, 8, ..., 3k+2, ...} in the integers. And what that tells me is that we need to tighten up the theory a bit.
I think that we need to put an algebra on sets robust enough that the cardinalities inherit an algebra robust enough to solve these issues, else we aren't really going to have a handle on what's going on. I think measure theory is some kind of blueprint for this. At this point I just think that thinking of infinite sets in terms of cardinalities is just sloppy.
Craig is wrong to say that there's a "formal contradiction" there. The technical thing he CAN say is that is that "cardinal difference as a binary operator on sets is not a well defined function". That's the mathematical expression for his claim, which is right, and a problem, just not the one he thinks.
I think all that Alex meant there is that the operation is not quite the same as the one you'd see in an arithmetic signature. I figure he's aware that there's a subtraction operation you can run on sets. But it's defined differently and can deal with infinities, while the classic "-" can't.
@@mar98co1 right, it's cardinality subtraction, which works fine if you use it on sets, but terribly if you use it on cardinal numbers, which is an issue.
@@logos8312
1+∞ = ∞
1+∞-∞= ∞-∞
1=0
checkmate :D
LMAOOO I had no idea Alex engaged with Darth Dawkins. I wonder what that apologist is up to nowadays x3
@@gabbiewolf1121 ua-cam.com/video/mxdeIaXuPQ4/v-deo.htmlfeature=shared
Cheers!
On the Planets analogy: I started counting their orbits a finite time ago, and have observed that Jupiter orbits 2x as often as Saturn, so that when someone tells me "they have been doing that for T amount of time" the correct thing to believe is that Jupiter has completed 2*T more orbits then Saturn. And you can make T is big as you like, and the gap between them is correspondingly bigger and bigger. And yet, if you say "they have always done this", I must somehow think they have completed the exact same number of orbits (instead of Saturn's being infinitely less than Jupiter's). And I must deny my own experience of watching them for as long as I like (potentially forever) and *recording* that one is orbiting twice as often as the other....
Great conversation, Joe and Alex! Unfortunately, I think it’s had an unintended effect on me as I’ve now convinced myself that a full Hilbert hotel, like any full finite hotel, can add a new guest only by actually adding a new room to the hotel. Where am I going wrong with the following?
Represent any hotel by a set whose members are rooms with value X if occupied and 0 if vacant. So a full Hilbert hotel is represented by Hf = {X, X, X, X, …}. When a new guest arrives at the full hotel, the manager does “something” (we are told it’s just shifting all the guests) that frees up the first room. After the shift the Hilbert hotel with the vacant first room is represented by Hv = {0, X, X, X, …}. But Hv = {0} ∪ Hf, which looks like the old full Hilbert hotel with a new vacant room added. So should we really believe the story about the shifting guests, or did the manager really have a new room added to accommodate one of the existing guests that got shifted?
Interesting question!
I think the first issue is your use of sets. A set, by definition, has no order and contains no duplicates. So {1,3,3,2,3} = {1,2,3}, since the duplicates and order of elements are ignored. Your example sets would thus be {X} and {0, X}, which is not what you meant. I think you mixed this concept with the concept of a regular list (ordered, duplicates allowed, often represented with square brackets or parantheses). Let's stick with the list notion and close to your notation (0 and X might be signs on the door which the guests can flip) and see what we get.
Let t=i represent the point item at which the hotel's state is shown.
At first, no guests are checked in.
t=0
[0,0,0,...]
Then, an infinity of guests, numbered 1, 2, 3 etc. arrive and check in simultaneously (we don't need to worry about the time component here).
t=0
1 2 3
| | |
V V V
[0,0,0,...]
t=1
[X,X,X,...]
Now, the manager does something and we arrive at this:
t=?
[0,X,X,X,...]
So it seems like we did this:
[0] + [X, X, X, ...]
= [0,X,X,X,...]
And we *could* do that to achieve the result. But there is another way, which is the traditionally less intuitive one. Let's look at the process step by step.
t=1
[X,X,X,X,...]
t=2
1 2 3 4
^ ^ ^ ^
| | | |
[0,0,0,0,...]
t=3
->1 ->2 ->3 ->4
: : : :
[0, 0, 0, 0, ...]
t=4
1 2 3 4
| | | |
V V V V
[0, 0, 0, 0, ...]
t=5
[0,X,X,X,...]
Again, without worrying about the time component, we can see that this process works equally well. The misleading part is, since the Xs all look identical, it only seems as though we have just attached something new "to the left". When we look at what is happening in between the states, it's clear how the shift happens.
Hope this helps! :) If you have any more questions, hit me up/reply to this comment ^^
@@MyMusics101 Thank you for the reply and for correcting my sloppy notation and terminology. I think my issue with the HH is really of my own making, in that I think I’m adding an additional condition beyond what is in the typical scenario. If the HH is just a sort of container that can hold an infinite number of generic rooms, then clearly it can accommodate any number of new guests even when full. No problem there. In thinking about whether the strangeness of Hilbert Hotels implies a strangeness for something like a beginningless past I wanted to remove the generic quality of each room to see what happens. I think my example was not successful. Although this also needs improvement, the following scenario might be better:
Suppose each room of the hotel has a unique room number (nothing unusual there) and let’s say that when a new guest arrives at the hotel she also registers her room number with the local post office to receive mail. Say there is a particular HH whose room numbers consist of all the natural numbers, 1, 2, 3, …. etc.. and that this hotel is full. If a new guest arrives, the manager can have all the guests move over one room as usual, but if the current guests now insist on taking their room numbers with them to continue receiving mail (because mail forwarding is not an option, for whatever reason we want to make up), the first room will become vacant as usual but it will have no room number, and no unique natural number can be assigned to that room number. The manager could assign the number 0 to that room, and the guest could register room 0 with the post office, but there is a question now of whether this really is the same hotel as before since there was no room 0 in the original hotel. In this sense, the hotel acts more like a finite hotel and perhaps some of the strangeness is gone.
Anyway, I’m not sure if this is bringing any clarity about the possibility of an infinite past, but it doesn’t hurt to throw around ideas ;) Thanks again for listening!
@@alexanderdelaney3978 The Trick is in the "something" aka shifting. To understand the trick when applied to a hotel with a finite number of rooms, all you have to do is once you reach the last room you continue with the first room (or any other random room for that matter). By doing so, you can always create an empty room, and it would seem as if you could have an infinite number of guests. Except with a finite hotel you will eventually run out of hallway space, because that is the actual space, you are filling.
One might say that set theory can be very Cantor-intuitive
My only problem with hilberts hotel is a gravitational one
15:30
there was a young man from fratton
who went to church with his hat on
for he always said
with my hat on my head
i'm damn sure it won't be sat on.
can you say "ALL the rooms are full"? surely it doesn't matter if they are finite or infinite if the rooms are ALL full then you can't move anyone into an "unoccupied" room?
Zero has a lot of the same "problems" that infinity does, but tangentially to what Alex said, our system for arithmetic is constructed to accommodate zero, so it seems more intuitive.
If we take the dominos.. It's "unintuitive" that a line of infinite dominos falling won't have a last domino fall. But it seems completely intuitive that a line of zero dominos falling won't have a last domino to fall. We can't count (in the arithmetic sense) _to_ infinity, nor can we count _to_ zero. (Though we could conceivably count _at_ infinity or _at_ zero, depending on how you want to define "count")
Also, I'd love to hear the argument for creating a single hotel room ex nihilo being metaphysically possible, but creating an infinite number of hotel rooms being metaphysically impossible.
The Hilbert's hotel is basically just overcomplexly saying that the properties of infinity are weird. Basically that
∞ + 1 = ∞
or
ℵ₀ + ℵ₀ = ℵ₀
Which are both a bit unintuitive, yet were both mathematically proven - so there actually exists a specific concrete proof of it being true
Happy to see another discussion between you two!
Is hilbert's hotel a noun or verb? It's not finite, so is it active concept of counting rather than a count or countable?
I think a better way of thinking of Hilbert's Hotel is to think of a guest being born in the hotel and by that time the hotel builds them another room (recently reading Senlin Ascends so my first thought was the Tower of Babbel). Because if it's simulating the universe, it should be self-sufficient (let's pretend in the thought experiment that the people are like some animals that can give birth without a male counterpart [parthenogenesis]).
First line of the Craig video asks if the universe had a beginning or is eternal into the past. False dichotomy - which category does God fall into?
Good content as always, Joe. I have a potentially (but not actually ;) ) dumb question, bear in mind I'm no kalam expert.
On the discussion around 1:21:00 - to my mind, I can agree that in an endless series of events of counting, the answer to "how many number are there such that they will be counted?" is aleph null. But to me, the symmetry breaker is that the endless series has not yet been crossed, whereas a beginning-less series has been crossed. We will never get to aleph null in the future. But someone proposing an infinite past seems to be saying we have already somehow crossed that gap, which seems impossible. Am I missing something? Maybe this has already been discussed and I'm missing it.
Haven't watched the whole discussion yet, but this doesn't seem to be much of a problem to me. After all, there was never a point in time where we actually crossed anything. If the past is infinite, then it has always been this way and thus, we never were in a "there have been a finite number of moments before this point in time" situation. The two cases are symmetrical in the sense that, for the future, we can say that we *will* forever remain in the "*finitely* many moments counted" state. Whereas in the past case, we can say that we *have* always been in the "*infinitely* many moments counted" state.
(Counting moments meaning moments having passed here, obviously, because otherwise we would have to try figuring out what infinite number of moments exactly has passed, to which the answer would (presumably) always be "a countable infinity many".)
@@MyMusics101 So, I re-read the Malpass/Morriston paper last night, and maybe my question has less to do with the paper and more to do with traversing the infinite (which they explicitly don't deal with in the paper). I think I agree with every single sentence in your reply, I'm just not seeing the relevance to my comment. It seems to me that (given an infinite past), the answer to "has the daily negative-integer-counter filled out all the negative integers?" is obviously yes. But how can this be, when it seems that everyone agrees that you can't fill out the natural numbers by progressively counting into the future? aleph null doesn't have an immediate predecessor, which is often the reason given for why I can't reach it in the future. But symmetrically, if it doesn't have an immediate predecessor, then how could we have filled out the negative integers through past counting down?
You are correct, anyone who says we can get to today through an infinite past is talking nonsense. Imagine a iine of dominoes with no beginning or first domino and there is a domino at the end of the table and if it falls it will fall off the table, The only way a domino falls if the preceding domino knocks it down, Now if the chain of dominos extends infinitely backwards with no first domino, will that domino at the end of the table ever fall down and be knocked off the table. The answer is obviously no. Not only will it not fall off, none of the dominoes will fall down, because there is no first domino to knock down any other domino. So the problem is obvious with a beginnless series of events or eternal past, you will never arrive at any point in time.
I can see that the original Craig's video is misleading at least and outright lying at most.
Take for instance the guests leaving the hotel. They say:
"An infinite number of guests left the hotel, yet there are NO FEWER guests in the hotel."
I think that is deliberately said in that manner to cause the highest confusion. As we instinctively expect that when some people left the number of guests must change. If we would word it in this way:
"An infinite number of guests left the hotel, yet there are still an INFINITE NUMBER of guests in the hotel."
It wouldn't feel as much confusing as more people would instinctively recognize that we are dealing with infinities and then all kind of weird stuff can occur. I don't like this kind of deception. I would say it is outright lie. With infinities we cannot pinpoint some exact number so we cannot say if it is or isn't the same number of guests. For anyone with basic knowledge about sets theory must realize that and given Craig is using it as an argument I believe he has to have decent knowledge about the theory.
"You're the caretaker. You've always been the caretaker".
2. I have to disagree about Dretzke's argument being a counter to Craig's argument, but I think in order to make this argument I'm going to have to steelman Craig a bit because he argued this is a sloppy manner.
When I look at Craig's successive addition argument, the core of the intuition actually goes like this:
1a. To give a historic account of the present, you have to give a historic account of the event preceding the present one.
1b. To give a historic account of the event preceding the present, you have to give a historic account of the event before that.
1c. By induction, to give a historic account of the present, you have to give a historic account of each successive event preceding the present (successive addition)
1d. This is also important, but not explicit in the premises, but you would actually run this argument again for every 2 preceding events in the event that something causal happens at t-2 to affect the present t, but it doesn't manifest in t-1. And then you run it again at 3 prior events, and so on and so forth. Basically the present is the cumulative outcomes of all events which causally effect it at prior points in time.
2. If the past is infinite, then given any past event you count back to, there will be a greater number of prior unaccounted for events, than events you've accounted for.
C. If the past is infinite, no historic account of the present is possible.
So why, given 1, do we think 2 is true? Simple.
A. Let k(t) be a counting function that spits out a natural number k after t time.
B. For any k(t), there exists a natural number 10^k(t) > k(t) that you haven't counted to yet.
C. As t->00, consider the difference in what you've counted vs what you have left to count: 10^k(t)-k(t), we know that this value approaches 00.
D. So paradoxically, as you count back, you're actually going to miss MORE things for each number you count. You actually get further away from having a historical account of the present, rather than closer (as you'd expect by looking to the past).
And the "yeah but we'll get there eventually" line isn't going to work. Why?
Let's suppose you count to 1, well there are 10 events you could have counted to, and you're missing 9. Yeah yeah OK but we'll count to 10 eventually! Well what happens by the time you get to 10? Well now there's 10^10 things you could have counted to, so you're missing almost 10 billion things.
1->10
10->10^10
10^10->10^(10^10)
etc.
So the "we'll get there eventually" clause doesn't help if, by the time you do eventually get there, your problems multiply faster than the ground you've gained. And this isn't a trivial matter since we're dealing with past events. What this is basically saying is:
By the time we've accounted for 10 events in the past, we now have 10^10-10 events that could have influenced those events, so in appealing to those events to count for the present, all we're going to find out is that there's now billions of events before that, that we haven't a clue about, which means we really have no idea of the causal history of these 10 events which were supposed to influence the causal history of the present.
Ah! We'll get to those 10^10 eventually, no problem! OK and then by the time you do there are going to be 10^(10^10)-10^10 events before the 10^10th past event whose causal history determines the 10^10th event, so in appealing to the 10^10th event to explain the 10th event, AGAIN we're actually no closer to explaining the 10th event whose job it was to explain the first event. So we've now counted back 10^(10^10) events and basically have nothing to show for our trouble, except exponentially multiplying problems.
Point is, just because you've counted N0 elements, and your initial set has N0 elements, doesn't mean you've exhausted your set. So while I do believe that George counts N0 things, I do NOT believe that George counts every natural number. That's the double edge sword of cardinality.
Best way to deal with this argument is causal finitism. Only finitely many causally dependent things in a causal history, an infinite number of causal histories. Done. Then you still have an infinite past without the multiplying complexity of unaccounted for prior events, since each causal history doesn't depend on the previous one, so if you fail to count it, oh well. So long as you can give an account of all the elements in your causal history, you can give an element of the present.
Ok, I'm only 25 minutes in, but Malpass is pushing my buttons again, so I might as well just briefly post a few issues I'm having, but I'll keep watching for sure (maybe they even address some of these as the conversation goes on):
1) Saying “this is just how infinities are” is WAY too quick. Wittgenstein, for example, insisted that there is no such thing as “all the natural numbers”. Such a phrase, he thought, is meaningless. Though we use infinities in the same grammatical forms as finite numbers, we are actually referring to *rules.* So, for example, given how addition by 1 works, I could always tell you the next Natural Number. But it does not follow that it makes any sense at all to speak of “all” the Natural Numbers and call that a set with infinite cardinality. At the very least, we shouldn’t take for granted that that is a coherent thing to say.
Neither the experiments nor the mathematics of quantum mechanics require us to deny anything in our basic intuitions. Some ways people _talk_ about QM seem absurd, but I don’t understand why that doesn’t just count against those so-called “interpretations” and in favor of ones that cohere better with everything else we know. John Bell certainly advocated for speaking more sensibly about QM, and he gave examples that could work just fine.
2) “Full” means that every room has a person in it. Shuffling them around just changes *which* person is in which room. It does not create more rooms. This is not just some common sense intuition, it is part of what it *means* for “every” room to have a person in it. Another way to look at it is to say that we are not allowed to add any more rooms to the hotel, and yet, after the shuffling, we will have a room/person combo (ergo, a room) that we did not have before. That’s a contradiction. Why are we not allowed to add any rooms? Because that’s part of the analogy. It’s the whole point. You are stuck with only those same rooms that have been occupied all along, and you shuffle people around and end up with an extra room (or many extra rooms). So you both do and don’t add rooms to your building.
3) “Greater than” and “equal to” at the same time is a straightforward contradiction. “Greater than” is, by definition, an “inequality”. This is no small point. Malpass is right to think that accepting Hilbert’s Hotel means we need to swallow a situation in which it can both be true that the number of guests is greater than it was _and_ that it is equal to what it was, and that is a contradiction; not a mere weirdness.
I'll stop for now. Haha. It's just that Malpass has this way of just blithely speaking [what seems to me to be] patent nonsense, and it sets off alarm bells in my head.
"You can't subtract in transfinite arithmetic" and yet people can leave Hilbert's Hotel.
1)But if you're gonna be a finitist, then Alex's other objection seems to land, what do you make of the "infinite afterlife"?
2,3) I think the same as the "-" point applies. We're talking about set theory, so we probably shouldn't be thinking about the arithmetical greater than and equal, but rather supersets and bijection. This operation we're doing in the Hilbert hotel is constantly done in set theory, it's well known that sets with infinite cardinality can have the same cardinality of some of their own subsets. And N ⊂ M and |N| = |M| just isn't a contradiction, provided they're infinite sets. Loosely ℵ0+ℵ0= ℵ0, such is the nature of infinite sets. If you don't like that answer that's fine, but keep in mind that to object to transfinite set theory altogether, besides the fact that you're joining a niche minority in mathematics, even those that do take finitist and ultrafinitist views understand there are no contradictions. The problems raised are rather about definitions and counterintuitiveness. Most mathematicians take finitism to be a worthwhile endeavor since its results are important, for example, they tell us a lot more about what we can do computationally. But like I said, almost none takes the position that there's an actual problem with transfinite mathematics.
@@mar98co1
1) If I believe in an "infinite afterlife", all I'm saying is I will never die. My age will always be finite.
2) I get how it works in Set Theory, and I'm not saying (necessarily... most days of the week) that there's something wrong with Set Theory internally. But there can be contradictions when applying it to real situations. To say "the number of people now both is and isn't greater than what it was" is a contradiction.
I find it interesting that you say "it's often just about counterintuitiveness or definitions". Counterintuitiveness I don't really care about, but definitions... they are the reason why a married bachelor is a contradiction. It's no good to say "the problem only arises when we take the definitions of the terms into account"!
Yet one more based thing Wittgenstein says. More I hear of the guy, the more I like him.
@@Mentat1231 But you will live forever, and I'd ask you, how many moments will you live in total? i.e. what cardinality does the set of moments you lived have. But if you are skeptical of our friend ℵ0, which would seem the standard reply, then the question won't make much sense in the first place I suppose.
"most days of the week" cracked me up :D
Right, just because we have a mathematical model, doesn't mean we can bring it to metaphysics (or maybe it does).
Yes, phrased like that, it's a contradiction. But depending how we interpret that to mean, it might not be:
-the number of people there were before is n (∈ ℕ), now it's m, and n
@1:14:50 Why do theists need an actually infinite future? I've never conceived of an afterlife in that way, but as an unending, but always finite, existence into the future.
We are talking about an endless future which, at each point in time, is finite, but is nevertheless endless. By an 'infinite future', we mean an endless future; and our arguments apply to an endless future. What you described is what we are talking about :)
@@MajestyofReason thx... obviously way out of my depth here lol.
@@AlexADalton No worries! I hope you are nevertheless served by the convo! Much love
You attract very strange people in the comment section. :) Anyway I'm a Malpass addict, too.
We're all strange in our own ways :)
Much love
@@MajestyofReason I thought I was strange, but now it's hard to compete.
An hour in, and they get to one of my main objections: If we say you can't get to infinity by starting now and counting forever, then why isn't it also true that you can't get to now by having counted forever? After all, the present stands relative to the past series in the exact same situation as the future perfect stands in relation to everything before it.... Malpass basically puts the burden on the objector to show that the reason why you can't do the one is reversible so that you can't do the other. So, what is that reason? Malpass says the reason the Counter can't start now and eventually "have finished" is because there is no immediate predecessor to aleph-null. So, let's see if the objector can turn that around:
When we say "aleph-null has no immediate predecessor", we are talking as though there could be an end to an unending count, but that we can't actually get there because of this disjoint. Of course, there is no such end (that's what "unending" means); but if Malpass can phrase it that way, then so can the objector. So, imagine the guy who's counting down "all the negative numbers" and finishes today. What about the beginning of his beginningless count? is it not also disjointed from aleph-null by not having an immediate successor? If it is allowable to say that the *reason why* the Counter cannot start now and "have finished" later, is *because of the gap* behind aleph-null, then what forbids saying that the reason the Counter cannot have always been counting and yet finish now is because of the gap in front of aleph-null?
Think about it: If the problem is a gap between the "all of them" number and any other number, then that gap can get pushed back as far as you like (even infinitely), but it's still there! What argument can you possibly give to show that it some how *goes away* or ceases to be a gap?
To put it another way, if you say the Counter cannot _fill out_ the set of all Natural Numbers because there is a gap before the number that means "all of them", then who cares which order I fill them in?? if I'm counting backward and just finish today (-2, -1, 0, done!), then I have filled out the Natural Numbers finally today without worrying about any gap. It is inconsistent to say both "you cannot fill it out because of the gap, and you can fill it out despite the gap". Malpass would be the one with the burden of proof to show why one is forbidden and the other isn't.
The way out, of course, is to say the gap problem is NOT the reason why the Counter can't get to infinity. He can't get there because there is no such thing. There is no highest natural number, nor are there actually infinitely many natural numbers, there is just a rule for getting the next higher one. But, that would entail Finitism.....
~Someone~ should setup a discussion between Malpass and ElephantPhilosophy
*hinthintwinkwinknudgenudgetugtug*
The universe did not have a beginning nor does the universe have a past. Time is an illusion. What does full mean in this context anyway... One person per room? Or an infinite amount of persons in each room?
What about the big bang?
@@jackx341 The big bang is only evidence for an explosion. It is as far as our measurements currently take us but it does not say what happened prior to the big bang. Thus, it is not the beginning of the universe but it is called the beginning of our universe because of our experimental limitations. Causality allows us to call a prior cause the past but if causality is cyclical then this ordering is arbitrary.
"Time is an illusion."
You have to be literally brainwashed to hold such a radical position! Time is one of the most obvious truths we as human beings, which we experience on a daily basis, know about reality. The fact that most scientists hold time to be an illusion is actually very powerful evidence that scientists can be wrong rather than evidence that time does not exist.
Joe, what's your tweeter account name?
I never understood how Craig doesn't get the counting all the naturals thing. Like, consider how mathematical induction works. You prove a theorem for your starting point 0, then prove that it holds for n+1, and that's it, you're done. You've proved it holds in all the naturals. Seems we can apply the same idea: John will count 0, and John will count n+1. It just seems trivial from this that John will count an amount that is the same cardinality as the naturals, just like for induction, if the predicate holds for 0 and it holds for n+1, then it just holds for all the naturals. Craig would have to say there is some n such that John will not count it, ever (rather than equivocating with, there is some n that John HASN'T/will not HAVE counted).
What Craig says is that the ability to count *any* number does not entail the ability to count *every* number. He's right about that. It is an illicit move to go from "any" to "all".
@@Mentat1231 But I don't see how that's relevant to the question as to whether John will count every number. Here we're clearly not saying that John will count any number, we're saying he'll count every number. From: there does not exist an n such that John will not count n, it just follows via equivalence that John will count every n. That's another way to think about that which just seems straightforward.
@@Dan_1348
Whatever point you pick in time, he will always be counting a finite number at that time, and it will always be easy to see what the next one will be that he has so far "missed out".
@@mar98co1
I could give intuition pumps (like the Tristram Shandy story from Russell), but I think it's pretty straightforward: The move from "there is no number that he will not eventually get to" to "he will eventually finish counting them all" is an illegitimate move. As Craig said in his discussion with Malpass, it is akin to a Composition fallacy. From the fact that there are no members you will not eventually get to, it does not logically follow that you will eventually finish them all.
@@Mentat1231 I don't think it's a move to "he will eventually finish counting them all" though because that would be equivalent to Craig's shift to "will have counted".
Alex's point is, as I understand it, that if Fred keeps counting forever, then if we pick any number n, Fred will count n, so in that sense Fred will count every natural number.
As to "there is no point at which Fred will have counted all natural numbers", I think Alex agrees with that.
EDIT: Alex describes it at 52:40.
Where did you get your Spinoza quote from? Cannot find it anywhere.
It's from his Theological-Political Treatise. For a reference, you can do command F and search 'what altar of' in the following: returntocinder.com/source/SPIN.TPT
@@MajestyofReason, thanks, brother in arms. That quote is so good I needed it.
I'm only ten minutes in, but it's looking like you are granting straight away that having Cantor's property directly implies that new empty rooms can be generated. I don't have a good understanding of that and would like to see that fleshed out a little more, I don't quite see it yet.
It seems to me that you remove someone from a room in the process of preparing to move them to a different room. Surely this is what actually generates an apparently empty room.
Then you go around shuffling other people from full room to full room, never finding another empty room to put that extra person into... even if you do this shuffling procedure infinitely, you always have one empty room and one person who has been displaced from their room. That's the state after every shuffling step, the same as after the first step. Keep doubling the room number, and you'll never find an empty room up there to put the displaced person into.
So, the inference from Cantor's property to new rooms seems to me to involve some sort of intuition that if you keep looking down an infinite line of full rooms, you'll get to an empty room. If you have an infinite set of X's, there must be a non-X in the set somewhere. That you'll run out of finite numbers if you keep iterating a growing maths function forever.
Luckily you aren't leaving it there, I see you go on to say that fullness is maintained, but you can still add people. You don't need to find an empty room for that. I'll keep watching :)
I'm still not clear on it. Without understanding the issue, my intuition tells me that if these procedures with infinite sets of numbers generates new numbers, then the same procedures with an infinite set of full rooms should generate new full rooms - not empty ones you can put people in.
@@HeyHeyHarmonicaLuke there's no need to shuffle people around for infinity. Every guest in the hotel just moves to the next room simultaneously, so that room 2 - infinity are filled, and room 1 is left empty.
@@HeyHeyHarmonicaLuke it's not the case that the hotel is generating a new room whenever you move the guests around, it is just opening an already existing room up for another guest.
Since for every room in the hotel will always have another room next to it (R+1 applies to every room), there will never be a point where a guest is unable to pack up and move to the next room. This means you can move guests over as many times as you'd like to open up any number of rooms.
@@HeyHeyHarmonicaLuke The key here is to have a starting point. If the hotel rooms went from -infinity to +infinity, and then you told all the guests to move down one room, then no rooms will have been opened up, because the guests are not moving away from any starting point. If instead you said: every guest in a room below R1 move down a room (R-1) and every guest in R1 and above move up a room (R+1), well then you have successfully freed up 2 rooms, R-1 & R1.
Here we go!
EDIT: That accent sounds South African to my South African ears.
Icelandic perhaps?
Consider Hilbert Jr's hotel. It has more than 5 rooms and more than 5 guests. So it's full I guess (that seems to be the logic in Hilberts hotell)?
Now another guest wants a room, simple enough. Every current guests just moves up one room and they all fit.
And now to the crazy part, it's still more than five guests in the hotel.
So "more than 5" + 1 is still "more than 5".
Insane, obviously "more than five" isn't a real concept.
But wait there is more, now "more than 5" guests come to the hotel. No problem, just move all the guests to the room double it's size.
And it is still "more than 5" guests in the hotel.
Bonkers
Also, assume every guest in an even numbered room leaves. There is still "more than five guests" left.
Now everyone but 3 leaves, and only 3 remain.
So "more than 5" - "more than 5" is both "more than 5" and 3.
A contradiction
@@Oskar1000 More than five doesn't specify a number uniquely . a is more than 5 and b is more than five does not imply a=b. That doesn't mean "more than five" isn't a "real concept". It's just that the property doesn't uniquely determine a natural number. Nor does it say that if you have two sets with the property of having more than five members then you can find a bijection between the sets. So your example is not analogous to saying two sets has the property of being countably infinite. In that case you could find a bijection between the sets. And you have the counter intuitive properties (depending on your intuitions) that an extra perpended member of a countably infinite set is still a countably infinite set.
@@HyperFocusMarshmallow Infinity doesn't specify a number either. Mathematicians warn against using it that way.
I don't think showing a bijection is enough to prove that one set isn't larger than another. (Even if it does prove they have the same cardinality).
Imagine this scenario:
1) You put down one black ball numbered 0 into an infinitely big vase.
2) Then for each natural number you put down a white ball (with that number) in the vase.
3) You remove all the white balls in the vase.
On my hypothesis there is one ball left, the black ball.
But we can show a bijection between the white balls and "The white balls plus the black ball".
Let W be the set of white balls.
Let B be the set of white balls plus the black ball.
For all w in W we can find a corresponding b in B by the relation w-1.
For all b in B we can find a corresponding w in W by the relation x+1.
So we shown both that one set has 1 more member and that they have the same cardinality.
My point is that in both the "more than 5" and the infinite we lose some information that makes it impossible to do the maths precisely. If we add that information back both Hilbert's and Hilbert Jr's hotel seem quite intuitive.
@@Oskar1000 I think I can agree to that 😊 Thought-experiments can often highlight that we’re thinking about some concept too imprecisely. Cheers!
@@Oskar1000 A set can't have 1 more member and have the same cardinality. Cardinality is defined as a bijection, and there ain't no bijection if the two sets have different amounts of members, by definition. I think you're mixing up arithmetical and set-theoretical operations. When you add one member to an infinite set, the set's cardinality remains the same. That means the same as saying that the amount of things in the set(amount of members) remains the same.
If you find that ridiculous that's another matter, but mathematicians that reject transfinite mathematics are quite the minority, you just need to use the right tools.
First!
Have you finished reading "the kalam cosmological argument, philsophical evidence for the finitude of the past" (Paul Copan)
It's probably the best compilation of the philosophical literature on the argument.
It’s a book right? It’s not on PDF?
@@JoshuaMSOG7 yup it's a book😉
@@matthieulavagna Why would you take philosophical arguments as "evidence" in light of our best scientific knowledge saying otherwise? Would you take Zeno's paradox to say that I can't walk across to the other corner of the room?
@@dr.shousa because philosophy is more reliable then science. Science changes overtime, philosophy does not because logic is infallible. Once something is demonstrated, it's demonstrated forever.
I don't see why you're talking about Zeno's paradox. It's completely disanalogous to the kalam, since the intervals are potentially infinite and unequal whereas the events in the kalam are actually infinite and equal.
@@dr.shousa Yeah, philosophy is infallible, get outta here with your scientism
malpass just says absurd stuff, actual infinites are not counterintuitive, they are impossible and illogical. its just a way of getting away from the truth, to just call something counter intuitive which you could say to anything if you want.
Hi, I'm obviously not an atheist but I am a mathematician. WLC does not understand Cantor and infinities. Completed infinities are not only possible but absolutely necessary. Without them we cannot have basic maths. The whole thing collapses into absurdity. If anyone wants to see how let me know. But please, please start looking into why Cantor said what he said and how he proved that actualized infinities MUST exist mathematically.
I want to see how, please! I find it very difficult to believe actual infinity is a coherent concept. It's always seemed obvious to me, even as a child the absurdity of actual infinity. For example, I had this moment, when I thought to myself the idea of God recalling his entire past one moment at a time and how logically impossible that seems. I thought to myself a form of: "If God lived an infinite amount of moments in the past, how could he get to today?" It made no sense to me then and as an adult, I now accept WLC's conception of God's being timeless without creation and in time since the creation.
@@michaelsayad5085 how many real numbers are there inbetween 0 and 1? there you go.
@@danglingondivineladders3994 That's an interesting idea and I have thought about it myself but I'm pretty sure it'd still be classified as potentially infinite. I can give an example of a ruler that represents the number line. We can continuously divide the ruler up between 0 and 1 meters into smaller sub-units but we will never get to the point where the ruler is divided up into an actually infinite number of parts. A finite object should not be classified as having an infinite number of parts but rather as an undifferentiated whole and only after it is cut up can we consider its parts as distinct entities. Otherwise, we don't have an actual infinity but rather a finite-sized object that could be cut up into a potentially infinite number of parts!
@@michaelsayad5085 well I would say that Cantor's diagonal argument is the basis of completed infinities and he uses infintesimal numbers to establish his point. it really is about the amount of numbers in between 0 and 1.
@@michaelsayad5085 those are two different things. WLC misrepresents completed infinities in order to say the past must be finite. there are many reasons to think the past is finite without asking people to trust their unverified intuitions instead of a 100% proven theorem. He is provably wrong and if we accept that completed infinities are impossible then we cannot have a number line, geometry or any sort of coherent system. just use any of the other reasons to believe in a finite past as I do.
The part around 35 min where Joe argues that even though we’re removing the same quantity from the same quantity and yielding different results, because we’re using different operations it’s not a problem, seems confused.
For example, if we have 10 balls numbered 1-10, and remove all the odd numbered balls, we are left with 5 balls. If we instead remove all the balls numbered greater than 5, we’re also left with 5 balls. We removed the same quantity from the same quantity, but using different operations, and ended up the same quantity. So why should we expect to end up with differing quantities when we use transfinite numbers? It seems to me that the only answer is “well it’s because they’re transfinite numbers, you have to check your intuitions at the door.” And that’s the problem.
Thanks for the comment my dude!
Here’s a clarification of my argument. There doesn’t seem to be anything problematic about removing an identical quantity from an identical quantity and yielding, as a result, different quantities because, in some cases [namely, those cases involving denumerable collections], different *ways* you remove elements of a set can differentially determine the cardinality of the resultant set. And, importantly, there is absolutely nothing inexplicable or mysterious about this - there’s no puzzlement, no mystery, and consequently - for me, at least - no intuitive absurdity. When I reflect on the cases, they seem perfectly (indeed obviously) benign rather than absurd, since I have a perfectly good explanatory story to tell concerning how different results are yielded depending on the way in which elements are removed. And so my point here is really meant to help people see why I don’t find it at all problematic or unintuitive, in those cases involving denumerable collections, that one can remove subsets of identical cardinality from the original subset and yield resultant subsets of different cardinalities. No one is saying that you have to check your intuitions at the door; instead, I’m explaining why I don’t find it unintuitive in the slightest. (Indeed, I find it intuitively obviously benign and intuitively obviously non-absurd.) I don’t claim everyone must share my intuition here. But I can at least try to explain to people what considerations undergird my intuitive judgments. And one such consideration is that it’s perfectly explicable, with no mystery at all, why we can yield sets with different quantities depending upon which precise denumerable sets we remove from an original denumerable set.
@@MajestyofReason first of all, thanks for the reply. I really wasn’t expecting that. I watch a lot of your content and respect your work, so this is meaningful to me.
I would like to get your thoughts on what I said about finite sets though, my example of a set of 10 things. The reason I find the existence of actual infinities so intuitively absurd is because of how different infinite quantities act when we consider them in the real world. If the hotel has 10 people and odd numbered rooms are vacated, then 5 remain. Same when the rooms greater than 5 are vacated. This holds for any finite number of guests. But suddenly when we perform both these operations on an infinite number of guests, we get different answers. The reason this violates my intuition is because the people are the same, the rooms are the same, the mathematical operations are the same, etc., but because the number of guests we start with are infinite, the results are different. It just doesn’t seem to me that such a thing could plausibly occur in the real world.
To be clear, you don’t think that there is a contradiction in guests leaving the hotel as Craig’s video says, right? I think there is, and if I’m right, doesn’t that mean it’s logically impossible for any guests to leave once an infinite number of guests occupy rooms? And wouldn’t this be true even if the hotel manager put the infinite number of guests in only the odd numbered rooms and left the even numbered rooms vacant, for example?
@@mjdillaha I think what Joe means is that for him the people are not the same, the rooms are the not the same and the mathematical operations are not the same either. Which is perhaps why Joe doesn't share your intuitions on the Hilbert Hotel paradox. ie: Joe has different intuitions when it comes to how finite subtraction works compared to infinite sets.
Joe doesn't think there's a contradiction because the mathematical operations are not the same: different subsets are being removed, which is why we see different results. So the differing results are actually to be expected.
@@roger5442 If this is correct, then I don’t see why we can’t invent other systems of mathematics that describe absurdities like this and hand-wave away contradictions by saying we can tell a story about it.
For example, I could say that if we have 10 people, and all the even numbered people leave, then we have 5 left. But if the first 5 leave, then we have 10 people left. When you point out that there’s a contradiction, I respond “well they’re different people, they left in different ways. This is fine given the axioms that this system is based on.”
I don’t think I’m speaking out if turn by saying you’ll definitely take issue with that explanation.
These axioms don’t describe the real world at all, and worse, they contradict the math that describes the real world. That is why I think we’re warranted in saying that actual infinities cannot exist.
@@mjdillaha I don't think Joe is necessarily saying he's correct and you're incorrect. He's saying that your appeals to your own intuitions aren't persuasive to him since Joe has different intuitions that tell him there's nothing mysterious or weird about the results.
With your example above the oddity there is that you've said you've removed people 6-10 but then say people 6-10 are still there. ie: you are removing and not removing the same subset of people 6-10. ie: You start with a set of 10 people. You remove subset 6-10, and say your set still contains subset people 6-10. Certainly I would agree that appears odd.
However this oddity doesn't appear to occur in the Hilbert's Hotel. When you remove subset people 6 and above then you are only left with subset people 1-5. The result is expected.
And when you remove subset people odd numbers (1, 3, 5 etc) then you are still left with the subset people even numbers (2, 4, 6 etc). Again expected result.
The results are different because we are not removing the same quality of subsets.
One subset you create contains people 6 and above vs the other case the subset contains only people odd numbered. I think this is the misleading part of the Hilbert's Hotel argument - it wants us to think the subsets are of the same quality - but they are not.
Besides - I often think Hilbert's Hotel is flawed in the sense it's talking about concrete physical objects like people that can moved around, where as no-one is saying the past events are physical objects that occupy space all at the same moment in time that can moved around.
Sound quality is terrible along with his accent. Jerky choppy speech. So having trouble hearing/ understanding Malpass. He did nothing to clarify the absurdity of infinite past events. I didn' t need it. Infinity has no concrete manifestation which requires limitation.
Why does the past need to be made out of an infinite number of events? Could it not be one infinitely long event?