Russell's Paradox - a simple explanation of a profound problem
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- Опубліковано 7 вер 2022
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This is a video lecture explaining Russell's Paradox. At the very heart of logic and mathematics, there is a paradox that has yet to be resolved. It was discovered by the mathematician and philosopher, Bertrand Russell, in 1901. In this talk, Professor Jeffrey Kaplan teaches you the basics of set theory (a foundational branch of mathematics dating back to the 1870s) in 20 minutes. Then he explains Russell’s Paradox, which is quite a thrilling thing if you are learning it for the first time. Finally, Kaplan argues that the paradox goes even deeper than Russell himself realized.
Also, I should mention Georg Cantor, Gotlob Frege, Logicism, and Zermelo-Fraenkel set theory in this description for keyword search reasons.
My teacher told me that "all rules have exceptions" and I told her that that meant that there are rules that don't have exceptions. Because if "all rules have exceptions" is a rule then it must have an exception that contradicts it.
That's a good one.
Very True, However; it is a universal Constant that "He who has the Gold, Makes the Rules". Without Exception!!!
needed a minute to figure out how that worked haha
but if a rule has the exception of not having exceptions, it is still a rule with an exception; right?
@@harrymingelickr883 _Except_ there are societies that have rules but don't have gold, _and_ there are societies that have rules but don't _value_ gold. 😉
I started reading Russel’s “the limits of the human mind” and I found out mine lasted one paragraph.
Lmao
The more you know, the more you know what you don't
So true, I am a licensed engineer and have forgotten more about technical subjects than most people know- but education makes one a humble person- just knowing how much knowledge is out there of which 99.9% the average person does NOT know
@@louismartin4446 "of which 99.9% I don't know" you should have said. Why would the ignorance of other people make you humble?
@@louismartin4446how did I know you were an engineer?
Unlike many of your commenters, I don't have anything pithy to say about your presentation. I had never heard of Russell's Paradox or anyone else's Paradox. All I can do is tell you how much I appreciate how you described it. I did have to go back and review a couple of sections near the end, but I got it!
You are passionate about sharing your knowledge with everyone who cares to learn. Even, and perhaps especially, people incarcerated in prisons. You are a gifted teacher, so thank you for sharing your knowledge with ALL of us.
Well said!!!
Boo urns. Have pithy on us!
Weirdly said!.... Now I'd like to learn why @janathonbeton2002 is in prison 🧐
@@squirrelbait2004 I'm guessing it's because @sqrlbain2004 can't spell my name ...lol
@@squirrelbait2004 BC he belongs to that set.
This guy mastered writing on a window to a level i've never seen before
There is a software which change left and right, so he can write normally on the glass like on a school board.
A smart phone does this as well.@@zente16
@@zente16 whats that software
@@EthanWTF Literally ANY video editing software can flip an image...
he was in the Navy, they use the mirror image style to document and track the battle situation, it is not difficult to learn, you just have to practice for a few weeks.
I asked my girlfriend if we could have sets and she told me no because I didn't contain myself.
Lol, I love the humor you two have.
Niiiiiiiice
You're under arrest.
I sat here wondering how to respond to such brilliance for like 5 minutes
- We already have sets at home.
- At home: { }
I really didn’t expect LeBron James to be so crucial to the fundamentals of set theory. What a legend.
4-time NBA champion LeBron James*
@@nim127 Oh yes, thank you. Sorry
How does this effect his legacy?
@@jackthomas3483 It seems like is legacy is a set!
@@jackthomas3483affect. But big props for not misusing "impact" like everyone else.
The effect of LeBron losing in the playoffs was it affected his legacy compared with Michael Jordan.
On the predicate paradox: The main issue you seem to be grappling with on this is functionally comparable to the old, simpler paradox: "This sentence is false." If it's false, it's true; if it's true, it's false. So which could it be?
The most descriptively accurate answer I can think of is that it is neither, because it has no constant referential point upon which to base its definition. What can the sentence even proffer within it as "false"? What truth is it trying to debunk? None, because no such truth was extrapolated. Its only point of reference is itself, but it _ipso facto_ eliminates that point by labeling it false, thus leaving it a useless self-contradictory abstraction, vacuous of point, logic, sense or reason.
And keep in mind that for definitions literary or otherwise, _constant_ referential points are not to be underestimated in their essentiality. Without them, the means to describe them become variable and generalized to the point of uselessness. Consider, for example, the set that contains all sets, [X]. Okay- does that set include itself, [X] + [X+1]? Does it include that set, as well, [X] + [X + 1] + [X+2]? You'd have to keep on reiterating the addition of the set within itself ad infinitum, but doing so leaves you with an infinitely escalating value - and if your set contains an infinite value, can you really say you have a definition for it, considering the whole point of these sets was as a means to define whole numbers and now you have to find a single whole number for a sigma function?
This doesn't mean that math is broken, it only means that generalized categorizations give naive (heh) interpretations of mathematics that don't hold up without much greater scrutiny. If Zeno can be wrong about his ideas on motion being an illusion and Euclid can be wrong on his ideas of geometry, so can some professors be wrong about their ideas on sets. Nobody ever said this math stuff was easy, unless they did, in which case they can file under [set x: x contains all people who are shameless liars.]
just wow.!!!!, you just broke the iteration of this amazing professor I would love to see you do it in a video as good as this one.
it's just a play on words, sort of like values@@hespa8801
Nice description.
Sometimes you can string together words that look cool (this sentence is false) but in reality are just silly words that end up being meaningless or incoherent, logically useless. Words twisted back on themselves.
Maybe sets should be accounted for through the passage of time. It seems like their also may only have meaning for our minds which is subject to time.
I'm learning basic programming (C#) so your comment reminded me of something adjacent. If you define some object such as a string to be some... well... string of letters, that is fine, it can even be a pre-existing string (analogous to sets containing other sets), but you cannot fundamentally define the string object to be itself, as 'itself' doesn't exist yet, it can be defined as null, but cannot be defined as itself or any variation on itself. I find this quite interesting, as this paradox appears to subtly arise even before the introduction of the "this sentence is false" style paradox. It seems that it is ok to say "this sentence is true" because the action of declaring it doesn't invalidate its 'inital state'; the sentence agrees with itself. conversely, the paradox "this sentence is false" invalidates its initial state as it doesn't agree with its own definition. But the problem with both of these sentences is that they are evaluations on a sentence that is still being constructed. It is fine to define a statement that is altered by a different statement, like defining "bool A = true" followed by "A = false" to change its state, but saying that "bool A = A" or "bool A = !A", analogous to saying "this statement is true" and "this statement is false" respectively, is impossible.
Writing this now, I am just realising that you can extrapolate this to compound sentences. "this sentence is a statement, and that statement is false" is allowed (afaik) because the statement has been defined in the former half of the compound sentence, and has been then made false by the latter, which is fundamentally disconnected from the former. You can also say "this sentence is a statement, and it is false", but this opens two possible interpretations. Is the "this sentence" false, or is the statement false? I put this down to the vague nature of the sentence itself, but I'm not sure.
If I had to guess, this intepretation suggests that a set that contains all sets that contain themselves (I'll define as V) must not contain itself, as the set constructor logic for V cannot have V as an input as it hasn't been completely defined according to its conditions. This (I assume) would hold for any set constructor that, upon full compilation, satisfies its own conditions.
I've tried watching this twice now and I realise that I am a member of the set of people who don't care enough about Russell's Paradox to watch to the end.
I speak German and understand the letter Russell wrote to his colleague. the level of confidence he put into his writing that his recipient will just understand him amazes me.
yeah, I can barely read 90% of that handwriting 🤣
Interestingly, depicted are only the first and last page of the letter and the actual paradoxon is not described on these pages (he adds the formulaic representation of "abovementioned contradiction" in the post scriptum, but that's it). Pity, I would've liked to read the original wording and I'm far too lazy to hunt down the letter myself 🤣
@@sourcererseven3858 what did he say in the letter
@@sourcererseven3858 Yeah, because google is such a chore.
But it's cool as he is so respectful and still shoots a hole in the theory with his paradox.
It's important to keep in mind these men have dedicated their lives, decades, studying this. It's a form of language therefore shorthand is precise and expresses those years of knowledge.
It's just one of those random variables that got lucky. I wanted everyone in my class to side with me so we could together over rule the system. But I wasn't so lucky and I spend most days in detention.
As a child I spent weeks writing "S, P, AO, Agent" and whatever else, under words for a language class (this was in a different country so abbreviations may not carry over) - its been 2 decades since, and today is the first time I have seen it used to explain something. It saved me 60, or maybe 90 seconds. Time well spent!
At my age (77), I am not going to wade through 18,643 comments to check if someone else has made the same comment as I am making here! I apologise in advance, however, if that is, in fact, the case.
When I first came across Russell's Paradox, more than 50 years ago, I explained it to myself as follows: if A is a set, then A is not the same thing as {A}, the set containing A. A set, in short, cannot be a member of itself, and the Paradox arises because the erroneous assumption is being made that a set can be a member of itself - your Rule 11.
On the few occasions in the last 50 years when I have thought about this again, I have come to the same conclusion.
I concur with the other comments about the quality of your presentation. Well done!
I think my favourite example of this is "this sentence is a lie". It's the example that helped me to grasp the paradox.
I'm no logician, but I think the answer to that paradox is just to say "nuh uh, that's not really a proposition." So it's better to say that there are two sentences, Sentence A, and Sentence B. Sentence A = "Sentence B is false;" Sentence B = "Sentence A is false." That way, you get around the self-referential problem.
And what if Pinocchio said, "My nose is about to grow."
@@lokidecatPinocchio’s nose only grows when he intentionally tells a lie. So saying something false does not cause his nose to grow. So therefore Pinocchio’s nose will not grow after saying “my nose will now grow”.
You just invoked my memory of how Spock defeated the Normal android in Star Trek TOS.
My favourite is: all generalisations are wrong.
For a 57 year old man who cannot even recite his times tables (my head just doesn't do maths), I'm stunned I actually followed that, I really did!!
That speaks volumes about this guys ability to convey information. I applaud you Sir, especially for the ability to hold my attention for the entire video. I quite enjoyed that!! I've no idea what use it is to me personally, but it was fascinating!
So are you really not good at maths or has it just been explained poorly to you in the past ...
@@GrantDayZA probably a bit of both swaying more toward poor teaching. Ive always been very good at art from being a kid. Don't get me wrong here I'm not saying love me love me I'm thick! I have a BA (Hons) degree in the social sciences, but honestly I've never been able to recite my times tables. If you fire one off I can tell you the answer eg. 7x8 or 9x6 etc. I just can't recite the whole tables they way your taught to more or less sing them if you get my meaning. I got by for a while but when they got to algebra and sticking letters in that was it, I just lost the plot and switched off and had a giggle instead. I enjoyed Pythagoras like working out areas was easy, and the simple letter stuff like 3 × X = 9, but when those equations got a bit serious my head just switched off!! In retrospect, I wish I had a maths head as now theres so much Id like to ask questions on but feel I cannot explain myself because I'll look stupid. I'm absolutely fascinated by Jeremy Strides math on Coral Castle, but I'm lost when he talks about prime sets etc. Anyways, I'm waffling now. But yeah, I just don't have a brain that handles maths, but.... We can't have everything can we!! Gotta work within your limitations so I'll keep on trying lol. Have a good day
What a lovely comment for me to read! You've made my day. Thank you!
I was thinking pretty much the same thing. I've often told people that I am "math stupid", and blunder through anything that involves math. Jeffery's presentation was both captivating and inarguable. At least I think it was - LOL. But you shouldn't take my word for it; I'm math stupid.
BTW - I've never watched any of Jeffery's videos before today. He reminds me of a mix of both Sheldon and Leonard from the Big Bang Theory. "Sheldon and Leonard" is a set. Heh. See? I learned something. :D
@@nichebundles7246 I did took set theory in the university explained by a very good professor and I have to say that the way Jeffrey was able to explain all of this in only half hour, while keeping me focused (maybe helped by the use of LeBron and Garfield) was just flawless!
PS. I was also looking at Sheldon's eyes at some point of the video
Thank you for the brilliantly clear, insightful and extensive exposition of Russell's Paradox! Thank you too for not mentioning the dull, trite and deeply unhelpful 'Barber' analogy along the way either!
I took a set theory class about a year ago, and this was beyond interesting. I was immediately asking about infinite sets the first day. Something seemed wonky. I get it based on real life, set is basically perception and allocations within it, and how things apply to singular vs multiples. Here's a goofy idea, would an empty set be able to equate to potential energy? It's a set with no content, but holds "reservation", it has potential
Perhaps you can help me out since you have studied this more than me.
This doesn't actually seem like a paradox to me, here's why.
This is a set
{ Cat }
This isn't a set
{ Cat
I argue that a set doesn't exist until it is created, because the set doesn't yet exist while you are creating it, it can't contain itself.
It is only after you create a set that it can be added to a set, and once you add the set to the new set it isn't the same set anymore and thus it doesn't contain itself.
Am I missing something here?
Honestly, there's a lot beyond my understanding. So it was weirdly reassuring to hear about the genius guy whose brain just straight-up blue screened because of this paradox.
How could it be reassuring? Because if set theory was “beyond his understanding” then something tells me this dude is not gonna be hospitalized over reading a letter he doesn’t understand.
Fun fact: basic set theory was part of Mathematics education in elemetary school in germany of the 1970s (and not a small one) . My estimate is, this forever reduced Germanys BIP by 2-3%.
Your honest comment is genuinely reassuring, because weirdly you solved the paradox. If there are not things beyond understanding, then the concept of understanding itself becomes nonsensical.
The narrative construct of reality is a instance of mathematical induction, moving from the known to the unknown. Reality is chaos and the unknown, determinism is an emergent property of the process of understanding. So if you take away hope and possibility (which resides in the unknown) you take the life out of reason and the reason out of life.
Fcvk I just blue-screened myself 😂
Your head was designed with paradox-absorbing crumple zones.
@@jacobwiren8142 it’s like the “designed-to-be-dropped” cartilage system never disappears. It merely adapts to be hit you take.
As soon as you got to explaining to the paradox, I knew exactly what the issue was because it's conceptually identical to several other paradoxes I've studied, including the Liar's Paradox and the Grandfather Paradox. I've noticed that this sort of problem tends to arise in almost any kind of abstract, self-referential system, if you dig deep enough.
I solved the paradox to my satisfaction by simply pretending we lived in a quantum universe which has settled down into three dimensions (three mutually perpendicular straight lines whose point of intersection is a single point whose most basic characteristic is extension, which is the lie we define by giving it points connected in a straight line and which is a set filled with all measurable elements or points; it isn't stable and wobbles around a bit, and the "barber" all the while exists in an indeterminate number of point states until it is measured and comes into being when it actuates the 3D point at which the three lines of points become one point from an infinite set of points...the geometric approach...
thank you!
@@choamlockstep I have the same concept, except that I consider datums of existence to be measured in dimensional probabilities, instead of explicitly naming them as quantum. Those two ideas may be the same or not; I haven't thought about it enough. However, I do agree that datums "settle down" into specific dimensions only when required to maintain probabilistic relationships with other datums. Thus, measuring particles within a 4D domain can cause those particles to seemingly pop in and out of existence, when in fact their presence within a particular point in the 3D domain only occurs when absolutely necessary. Perhaps this relates to quantum phenomena because observation seems to be a key factor in "forcing" the probabilistic relationships to adhere to the constraints imposed by observation?
Well done you
yep, it's like an infinite logic loop without a break condition.
The root issue is self-referencing, as noted by Douglas Hofstadter in his famous book "Gödel, Escher, Bach": any language that allows objects to make reference to themselves will contain a form of Russell's Paradox.
The one thing I would add to that, Conrad, is that as I said above, Language is not mathematics, and language can often be used to create a paradox, but, in truth, mathematics is an accounting not a linguistic description. Thus, the paradox is in our description not in fact.
@@jameskelso839 Indeed, in the case of set theory I think the paradox stems from trying to define a mathematical object ("set") out of nowhere by simply putting some words together, instead of having the concept arise bottom-up (which esentially is what the ZF scheme of axioms tries to do). But I also think that the issue is not limited to Mathematics; for instance, in Psychology, I notice a remarkable absence of a rigorous definition for "behavior" in the literature; I suspect that any attempt to define it via some sentence will be met with another instance of Russell's Paradox.
Well, I can't argue with that at all because behavior is an individual trait that no two persons have in common and there are those in my past who have convinced me that they have no idea what good or bad behavior is. I will absolutely say that I have studied mathematics a lot, and Cantors rules a widely used and there is literally no evidence of them ever failing in a calculation. No matter what any philosopher says about Russell the only thing that guy did was send a brilliant mathematician to therapy because he caused him to doubt his life work of furthering Mathematics. I have 4 terms of calculus behind me, and set theory works just the way Cantor set them out.@@conradolacerda
@@jameskelso839 Don't think that's correct. Gödel and Tarski proved Russel's Paradox holds for math and all formal systems. Math can never be both consistent and complete, nor can it prove the truth of itself.
If physics one day becomes complete, each event and each entity in the universe will have a one to one correspondence with a mathematical entity. But the description of the universe is contained within the universe, so it is self-referential, and hence results in the paradox which allows for events we cannot explain.
Whoever talks about Russell’s library catalogue paradox automatically gets my subscription!!
I was sent to Roman Catholic schools. One of the high points was at 'Junior High' school at about age 13 when a teacher (who in retrospect was probably an atheist, although he concealed it well), looked at the class and said: 'God can do everything, right? God is ominpotent.' We had all been programmed that this was the case, so we all said 'Yes, sir.' Then his voice got much quieter and asked us:
'If God can do everything, can he create a rock so heavy that he can't lift it?'
There was silence. Total silence. While the foundations of all the simplistic dogma that we had been taught were blown out of the water by elementary logic that even thirty 13-year olds could grasp.
Thank you, Joe, if you ever see this.
By the way, the question of whether "is not true of itself" is true of itself is equivalent to whether "this statement is false" is true, which is perhaps the most well-known paradox ever.
Not in this case. Let us call the original statement S1. So "this statement is false" = "S1 is false" = statement S2. S1 and S2 are not the same statements (e.g., S1 is the statement, "apples are never red", which is obviously not the statement S2: "the statement "apples are never red" is false"). So if S2 is true, it does not mean at all that S1 is true (they not being the same statements) and in fact bolsters S1's falsehood, not imply that S1 is true. So S2 is NOT self-contradictory and there is no paradox here. You made the mistake of equating S1 and S2 and thus the truth of S2 implying S1 is also true (which it does not as explained above), thus leading falsely to a paradox (i.e., S2 is self contradictory), which is not there.
@@shan79a I think you misunderstand what I am talking about. It is not the term "Is false", which I can apply to any statement S1 to get statement S2= "S1 is false" . It is the statement S3 = "This statement is false", where "this statement" refers to S3, so contextually S3 = "S3 is false".
"This statement is false".
If the statement is false, then it is actually proven to be true. If it is true, then it no longer satisfies the condition of being false, which would mean that it is actually false. But if it's false, then that means it's true. And on and on
4:53 yeah. this is nonsense. in the moment in which I refuse the set-theory in and itself as a highly hypothetical, theoretical construct that it is, I don't have a paradox. It is that simple. the redundancy is self-evident from the start. you cannot simply walk past, that something unimaginable IS something - and then be astonished, when you run into a paradox. What a stupid theorem. utterly useless.
@@alexeytsybyshev9459 The statement "this statement is false* is a vacuous statement as it is not saying that anything particular is false, but an empty/null concept/entity is false, i.e., a nothingness is false. Such extreme boundary-condition logic (referring to a nothingness as opposed to something concrete) can never occur in any type of logical statement in any field, and thus is meaningless and worthless to consider as leading to a self-referential paradox.
Never thought I could have such an enjoyable time watching a 30 min video on advanced mathematical theory. I chuckled and even laughed multiple times. Well done sir
same here
I as well and I'm math phobic.
Wow, it was that long? I didn't even notice.
Advanced?
I couldn't help but laughing every time he said "LeBron James, 4 time..."
I also just want to appreciate, for a moment, the skill and practice involved in you writing that stuff backwards.
Bonus points for doing so left-handed.
I think he writes normally and then flips the video
@@bobia8885 Awwww! Don't ruin the illusion for me! =)
Thank you for this. What got me here is my quest to understand Robert S. Hartman's formal axiology. Glad I found your channel.
As someone who has always sucked at math, I'm actually shocked that I pretty much understood everything you said.
Me too, because he uses images like a basketballblayer and cats.
So you do not suck in math ... you suck in logic !
Then you are set.
@coldshot1723, you did not suck at math. It was your teachers that sucked at teaching you. In high school, Albert Einstein had the same problem as you did, but fortunately for the World, he realised that it was his teacher that sucked at teaching, and not he at learning
@@newmankidman5763 Absolutely, I fell into the same dilemma, having a shitty High school match teacher
I have no interest in math and somehow i just watched this whole 28 minute video on something ill never use. Youre a great content creator, bravo.
I’m pretty sure this entire video only exists because someone bet you that you couldn’t say “LeBron James, four time NBA champion” a hundred times in a math video.
Dear Jeffrey kaplan I have just found out ur channel and I must say that I really enjoyed ur very this video. Thanks for the wisdom that you have shared here.
Much love and respect.❤️
This problem of self reference, infinite recursion, strange loops, or whatever one chooses to call it comes up again and again. Gödel’s incompleteness theorem is essentially another form of it, Hofstadter has made a career writing about it, and classical philosophers knew all about it and expressed it in many ways that we might boil down to the Liar’s Paradox or the most efficient form, “this statement is false”. They’re all logically-topologically equivalent. Good presentation for lay people, I like your channel and have subscribed. Going to check out your other videos. Cheers.
Yes, but no, but yes, but no, but yes, but no adinfinitum.
It strikes me as analogous to dividing by zero.
@@BoxdHound It has a certain undefined quality about it.
You can also create a problem of undecidability. You can have statement "this statement is true". It can be proved to be true and it can be proved to be false. If you put the statement into a set with some true statements and then you say "all the statements in the set are true" then you have another undecidable statement.
My answer to the learner asking these questions would be to go dig a hole. Then another hole. And then another. And keep doing that until he figures out it is stupid to dig holes just to dig holes, and stops.
I love the bridge between the linguistics and mathematics. I too believe that math is a branch of logic and that there are many parallels between language and math. Great Video!
I would say there is very little in aim and intention. The correspondence is obvious, but a mathematical language sounds like a mathematical romance, dull and ugly. Stay in your own field, and do well by your own harvest and herd.
Well you kind of need language to explain math to someone. That includes computers.
9:59 10:00 10:00 10:01 true
language and mathematics are symbols. symbols represent things or ideas. we use symbols to communicate ideas to each other. so they are both imperfect symbols. we’re not perfect beings, having not discovered the perfect language with which to communicate. i believe bertrand found a glitch in our matrix. but: it’s ok. the matrix still works …
When I was about 7 years old. I came across a calculus text book and opened it. Of course I did not understand it, but after looking through it, I said to my dad- dad that looks like someone thought up their own math and wrote it down. I'm 46 so I noticed this argument 39 years ago. Imaginary numbers. It's all you need to know to realize math starts as an idea before it becomes math (differential equations is another example). The only math is 2+2 =4. All other "math" was an idea before it was math. They can certainly be mutually exclusive- math and language- but they are also undoubtedly connected. Astronomy is where we see the biggest "man-made math" in my opinion. Certain terms, the best example being parsecs, is a made up term using made up measurements. However, lastly, all math at one point was "invented" one day in the past. One day in the past someone said a circle is 360 degrees and we all said -okey doke.
To anybody who's a bit frustrated with the lack of resolution(though I mean it is a paradox regardless), I wish Jeffery explained it in this video but see why not. The logician and language philosopher Wittgenstein had a solution inherent to his logic that possibly(read this multiple times because I am not saying he definitely did) nullified this, and the paradox of predication, given that some other axioms of his logic are true. The argument is both atomist and constructivist but would never allow for this specific paradox to happen.
It occurs leading up to 3.333 in the Tractatus Logico-Philosophicus, I will put the relevant sections below for clarity:
"3.322
Our use of the same sign to signify two different objects can never indicate a
common characteristic of the two, if we use it with two different modes of
signification. For the sign, of course, is arbitrary. So we could choose two
different signs instead, and then what would be left in common on the
signifying side?
3.323
In everyday language it very frequently happens that the same word has
different modes of signification-and so belongs to different symbols-or
that two words that have different modes of signification are employed in
propositions in what is superficially the same way.
Thus the word ‘is’ figures as the copula, as a sign for identity, and as an
expression for existence; ‘exist’ figures as an intransitive verb like ‘go’, and
‘identical’ as an adjective; we speak of something, but also of something'
happening.
(In the proposition, ‘Green is green’-where the first word is the proper
name of a person and the last an adjective-these words do not merely have
different meanings: they are different symbols.)
3.33
In logical syntax the meaning of a sign should never play a rôle. It must be
possible to establish logical syntax without mentioning the meaning of a sign:
only the description of expressions may be presupposed.
edit: removed 3.33 for clarity
3.333
The reason why a function cannot be its own argument is that the sign for a
function already contains the prototype of its argument, and it cannot contain
itself. For let us suppose that the function F(fx) could be its own argument: in that
case there would be a proposition ‘F(F(fx))’, in which the outer function F
and the inner function F must have different meanings, since the inner one
has the form ϕ(fx) and the outer one has the form Ψ(ϕ(fx)). Only the letter ‘F’
is common to the two functions, but the letter by itself signifies nothing.
This immediately becomes clear if instead of ‘F(Fu)’ we write
‘(∃ϕ):F(ϕu).ϕu = Fu’.
That disposes of Russell's paradox."
So basically 3.322 says that just because we use the same sign to refer to 2 different objects does not mean that those objects are the same thing. Pretty straight forward, like how pronouns are the same word being used to refer to entirely different people, signs can refer to multiple things without having any relation to each other.
3.323 is saying both that words as symbols for meaning are arbitrary(not tied to the meaning) and that the rules of the proposition can not only change a symbols meaning but we can have the same word act as a different symbol, which means that the sentence ‘Green is green’
3.332 Is when he makes the jump that signs in a proposition CAN NOT contain themselves because in being in a different role in the proposition, the set proposition(or predicate, or original set) would be out of scope and would become a different symbol all together. A proposition can never, in his logic, cross into being contained with itself because once it does it becomes a fundamentally different proposition.
3.333 This is the biggest move to solving the paradox at hand, this allows for a sign and a symbol to be recursive while the fundamental meaning to be illusive and logically impossible to be contained within itself. In his example the notation is freaky so lets just say that in the proposition ‘F(F(fx))' the outer F and the inner F might look similar but are completely different concepts that just so happen to be using the same sign and symbol, similar to how in set theory {1} is not equal to 1 because it's identity is in being a set in which 1 is contained.
So under Wittgensteinian logic, Russell's paradox - like everything else to Wittgenstein - is nothing more than the misinterpretation of the meaning of signs and symbols as being the same just because those symbols and signs are the same, and by all means this would explain how a lot of common errors are made, that they are conflations of symbols. "Let R be the set of all sets that are not members of themselves" becomes completely illogical because it is impossible for that set to contain itself in the first place.
this must be the most liked answer here :)
I can attest that the set of people watching this video is spread out across time, because I am watching this video from the future.
This reminds me of the "failure paradox" as well.
In a nutshell; if one sets out with the goal to fail, then they can only succeed. Because if they fail then they succeeded at failing which invalidates the failure, but if they fail at failing then they succeeded at failing which is still a success.
But is the goal to fail at anything? or one thing
@@vanceoz4080the goal is just to fail - I don’t think the paradox works when you assign it to something
❤
Not true. For example if I want to fail on a math test I can just not answer any of the questions and with certainty I will fail the math test.
@@linsqopiring6816 it doesn’t work when you apply it to another action.
As a mathematically challenged person, this video made my head hurt and I had to rewatch several parts, several times. But, I understood much, perhaps even most, of what you said, and must agree with the other comments before mine, that you are a gifted teacher. Thank you for sharing this.
My brain ONLY works mathematically. So, of course, I have trouble relating to most of the world. So, in that sense, you are lucky to be able to relate to so many people.
The fact you want to learn even though you're not mathematically minded is awesome in of itself
@@Pepespizzeria1 Anyone can learn from a great teacher. Even a genius can fail with a poor teacher. Compared to other nation I think that the USA ranks 47th below the country with the best educational system. Finland has been rated number 1 for many years. USA spends $800 billion a year on defense to have the best military on the planet. I guess the military is what America values the most. It's NOT you, it's currently the teachers.
@@Pepespizzeria1: Good attitude. I bet that it's not Jonathan Bakalarz's fault that he feels that way.
Well this never bothered me (even though it should), mostly because these types of paradoxes exist in all sorts of places. For example, the simplest one I can think of is the statement “I am lying.” If you’re lying while saying that, then you’re telling the truth. If you’re telling the truth while saying that, then you’re lying. Anything that is both self-referential and a negation can create a paradox/contradiction.
The paradox is lovely, as it proves that mathematics is - as Kant said - a result of human thought. Just add the rule: a set which doesn't contain itself, cannot contain itself. Rule 12. Done. It derives from basic logic, as your video explains. There's no paradox, just one rule missing.
Great video. I think it perfectly illustrates the fundamental flaws inherent to viewing language as a logical system. Even Wittgenstein, once considered the greatest champion of linguistic logic, decided later in his life to abandon that path. Language is not and never will be logical, because the purpose of language is not description but communication. All language is at its core more concerned with forming connections and being useful than with being accurate to reality
I was sifting through the comments because he never restated is original claim, “math is a human construct”?
You put words to my thoughts. Math is objective, and any apparent paradox speaks to the limitations of our tools, namely language and human thought.
I must disagree. The purpose of language is not communication. The purpose of language is deception. There is no reason for language to be as complex as it is if it is designed to communicate ideas. It only needs the complexity we see if its goal is to deceive.
@@rainey82 NO! Math is not objective! Math is completely subject to human minds. If human minds (or some other complex mind) do not exist then math does not exist. Being dependent on a complex mind by definition makes it subjective. You clearly do not understand the definitions of objective and subjective.
@@acvarthered Fair enough. Math is a language. Our understanding of the universe, described through math, can be complete or incomplete. The principles of the objective universe are objective.
@@acvarthered So did you just communicate or deceive?
I was a math major a million years ago. I wish you were one of my math profs! You are a great teacher!
Wow it took 2 genius mathematicians to figure out "I'm lying" is a paradox 😂
I'm watching this in the middle of the night from India it's winter here and it's soo cold here and watching this video and being amazed by this
stfu who cares
It's the same as the liar paradox, "this sentence is false". Whenever you allow self-reference in a logical system (where true/false are the expected outcomes), you enable the paradox. "Sentences can refer to themselves", "Sets can contain themselves", "Predicates can refer to themselves" - - these are all equivalent, and all problematic in a T/F logical system. The solution, as Russell and others proposed, is to not allow self-reference (or self-containment), which makes sense because self-reference creates endless loops for the T/F evaluation, as you aptly demonstrated. The solution is to show how self-reference, though feasible semantically, isn't logically valid in subject/object relationships - to get into that is beyond the scope of this comment. Another solution is to allow self-reference but to specifically handle endless loops as "undefined", i.e. have 3 possible outcomes: T, F, Undefined. Great video but I disagree if you have hit on anything new using Predicates.
He wasn't claiming he hit on something new by showing predication is structurally equivalent to set theory. He was claiming that predication is how we naturally think. And that the paradox arises from the way we naturally use predication in language (and maybe in thought), and can't be solved way by saying "self-reference isn't allowed". Because in language, self-reference is allowed. And the rules of language are observations, not rules we can change.
@@thenonsequitur Yes, Jeffrey's proposal is that the problem is real because it's noticeable in real language - the use of predicates, which is a new twist on the problem. However the problem with predicates is no more real than the problem with "this sentence is false", which has been around for a long time, and which he didn't bring up in the video, oddly... The video is initially about logic. When you bring up logical problems in language, you need to address the relationship between logic and language... what's valid/acceptable in one system is not necessarily so for the other. "This sentence is false" is a misconstruction (in logic) or a syntactic curiosity (in linguistics).
@@brynbstn yeah this video, while interesting, felt anti climactic for me. He seemed to leave out some important bits to make it as mind blowing as it seems to be for him
@darren collings ... and like a paradox, you can tie up more than one boat
Here's an example to clarify that the relationship between language and logic is not fully correspondent: "Four plus nine is one". This a completely valid statement, syntically. Is it correct? "No" most would say because 4+9=13. However it depends on the mathematical domain you are using. You assumed the standard number line. What if we were talking about the numbers on a clock? The point here is that, the logical domain behind a statement should not be assumed. It has to be defined. The syntax tells you nothing of the logical system under question. A valid syntax does not necessitate logical validity.
This man legit put his own death year in the quote what an enigmatic legend I dig this guy
off the grave yeah
I have long known that *I shall die on 21 April 2052* , aged 89; I am so sure, in fact, that it's been up on a poster (containing my favourite quote) I created and stuck in my office 26 years ago. My hope is that if nothing funny happens on that day, some gentle soul -- knowing of the prophecy -- will be kind enough to do me in. There are, after all, many ways to generate a self-fulfilling prophecy.
@@olafshomkirtimukh9935 I will hopefully live tomorrow starting from the day I post this comment. There you go.
@@olafshomkirtimukh9935 Olaf, if this is your true desire, I may be able to help
@@olafshomkirtimukh9935 I don't know if it is true or not, but I have heard it said many times that, before we come to this Earth to live the experience (under the veil), that we have pre arranged the mission, that supposedly we have entered into a contract to fulfill an experience, if this is the case if this is true, was your mission to come here to this world, to wish that your prediction comes true? if so then if your prophecy comes true was that a lesson that you had to experience in this life time to help you gain some form of knowledge for your TRUE self, because if you are trying to will your self into a state of absolute focus, what happens if you pass away a day earlier, or a day later, does this mean that if you don't make your prophesized date that you have not reached a pass in this life, or is it just an experiment where by near enough is good enough, because if I wish you well in your prophecy or desire, I only wish that you can achieve anything that you put your mind too, but not wanting you to die, hopefully you have other reasons for being here and that your prediction date is just a passing interest but not the main focus, because for what it's worth the meaning of life is subjective, to the individual but I truly do hope your lifes experience for your TRUE self is achived before your date of passing over to the bigger classroom, I truly don't wish for you to fail, but I am hoping that your wishes or desires are achieved, either way, the way humanity is travelling along these days towards another world war, presumably to reduce the population, ("Alice in chains" had an album called Jar of flies,) this was based off a famous scientific experiment, and I can't help but think that represents humanity)
a planitary extinction level event may just arrive before hand, to reset the earth like the pins at a bowling alley for the next set of contenders, if this "set of humanity" doesn't make the grade, we may all end up giving our seats away to more deserving passangers.
In my mind, 4 is a magnitude, a frequency if you take it further, a level, a phase, action potentiated. I’m also high
Your, sir, are the best best teacher i have ever seen in my life.
You have a gift. I actually understood this whole thing. I wish I'd had you teaching all of my philosophy and math classes in college.
I wish he would endeavor to explain why Zenos Paradox isnt really a paradox, because it surely is not. That would make an interesting video
If you understood it, you would realize that you can't really understand it ;)
Btw, I like the way you explained it very much. Very clear arguments. I wish all my fundmaental and logic math classes were given this clearly.
I haven't read much philosophy and even that was a long time ago, but I remember Plato going on about the skill philosophers need to practice - abstraction, maybe? Well, he didn't use that word, it was something about learning to see the idea behind the things. The point is that it is a skill to be practiced. So maybe professors don't always explain the best way possible, but you gained the skill anyway - which is useful, since it is rare that math comes the most consumable way possible. My logic prof was also quite dry btw. Well, I rambled, sry.
I think in order for your words to have value they have to mean something, have some sort of understandible interpretation or definition. So following that idea to say something "is warm is warm" is the same thing as saying "is warm" and if it isnt than we dont know what it is it means nothing, its useless. CS Lewis wrote in his book "the problem of pain" about the idea of what he calls if i remember right a "absolute contradiction" essentially the idea is that something with absolute power couldn't do, and not do something at the same time in the same way, its impossible. the reason why isn't because he lacked power (he has absolute power) but is because we can say jibbersh and think things that cant reflect a possible world even with absolute power. Words and thoughts usually have a imaginary representation sometimes they dont even have that.
Thank you for the fantastic video i appreciate it!
Also as sort of rebuttable to the example that the phrase "typically comes at the end of a sentence" "typically comes at the end of a sentence" you actually aren't referring to the predicate with another predicate because they aren't the same thing, they are the same words used two different ways. The subject is the first phrase, the words used to describe a facture of the phrase happened to be the words chosen for the video. He could've said "the phrase "typically comes at the end of a sentence" is most often at the end of sentences" it's a nature of "A" that "B" is usually true. Is what is actually going on not "it's a nature of "A" that "A" that "A" is usually "A". Just because a descriptor happens to also be the name of the tool doesn't make it the same things. It's the same description for two different things, not a predicate for predicate, at Least in the way I understand the word. If it was as he said I don't think we'd be able to make sense of the two sentences together
If it were up to me, I would make a distinction between actual sets and hypothetical sets. While for mathematical reasons, you may want to have an empty set... that can't actually physically exist and is therefore a hypothetical set. So I would propose a different or modified set of rules for actual sets v.s. hypothetical sets. You can say "1.) A set can contain anything." But is that true of actual sets? No. Because actual sets can't "contain" something that doesn't exist and therefore can't be "contained" by it. So "Things I can't imagine" falls into that category. It can't ever be an actual set. But maybe for theoretical reasons, it would still be helpful to explore hypothetical sets. So we would have a a different set of rules that accepts the fact that paradoxes can occur. Why does it matter if its only hypothetical? In that event, the paradox is only hypothetical too.
What’s really surprising is how perfectly he’s able to write backwards
what if he writes normally and the video is mirrored?
@@renge598Yeah the first few minutes of the first video of his you see are fun.
"How does he do that? Ohhh, mirroring."
Look at his shirt and the way it is buttoned. Two sides overlap opposite to what you normally see. That is a good solid clue that the video was reversed.
He doesn't. In the video his watch looks like it's on his right-hand, it looks like he's writing left-handed, his shirt lapel is reversed, his jacket buttons are reversed. Which means...the video is flipped horizontally before posting.
@@renge598 thats fucking genius
Your passion for teaching is exemplary. If half the teachers around the world had this type of energy and devotion, students would stay riveted. This is what we need. Also, if I name my cat "Is a cat," Then "Is a cat" is a cat!
thats a set of sets
..well.. the really fun part it isent at all a paradox...
its a problem all/most programmers has solved multiple times...
the destinction between data-(sets) and meta-data-(sets)...
its where Le Browns confusion comes from...
ex
{x: x data sets} X can be say shoe sizes, and the data for X is 30, 31, 32 etc...
the metadata X is a description for the data X nothing more.. hence no paradox...
say we make a dataset of cat colours, the colours are gray, white ...
the meta data is cat colours, and the data is gray, white etc
the meta data is obviously never a part of any set,
but the set can point to itself and it creates no paradox
Cat colours: gray, white, {Cat Colours} aka as self reference wich is used in objective programming
we can also use it to create collections of sets thats contains references to themself or other sets that contains referenses to it (aka Cycles in programming) without creating any paradoxes...
ex
Variable:INT A, B, C
Dataset:OBJECT-ARRAY AA:OBJECT, BB:INT, CC:FLOAT
Data A:1, B:2, C:3
Data AA:AA, BB, CC ( This dataset contains a cycle 'AA' )
Data BB:A, B, C
Data CC: 1.0, 1.5, 2.3
No, because in that case "is a cat" is not a predicate, it's a noun.
You're brilliant! The cat named "is a cat" is a member of the set of all cats. Therefore, "is a cat" is an element contained within the set {x : x is a cat} ...Kudos, @copasetic87
we actually need all the teachers to be like that. And they should be. And many many many actually are already. lmao "is a cat" is not a cat!
Argentinian writer Jorge Luis Borges generated a similar paradox in a piece of paper, simply writing in one side: "what is written in the other side is true" then in the other side wrote: " what is written in the other side is false".
Voliá! here we have a similar paradox
As a backend kotlin/java dev, apparently I knew a lot about sets and predicates without knowing I knew a lot about sets and predicates. Interesting video!
I mean we deal with them all the time in the forms of arrays and whatnot right?
😅
I spent most of my 20s arguing about the saying “there’s an exception to every rule”. If there’s an exception to every rule, then the rule that there’s an exception to every rule can’t be true because there would be an exception to the rule that’s there’s an exception to every rule, meaning not all rules have exceptions, meaning there isn’t an exception to every rule. That’s about as far as my logical abilities can carry me down a path similar to this
I just posted a comment mentioning that very rule.
You failed to notice the companion rule which exists solely because of this rule.
This rule must not have exceptions.
A rule without exception there is no circumstance in which it needs or could ever have an exception ergo it is the exception to the rule that all rules have an exception ‘including this one’.
Also the exception to rule eleven is that sets can contain themselves except where they reference self non containment.
The truth is the set neither contains itself nor doesn’t contain itself, the opposite is true of the set of sets that contain themselves because that one both does and does not contain itself.
It’s really confusing because it either doesn’t contain itself making it a set that doesn’t contain itself or it does contain itself making it a set that does contain itself both are true but never at the same time it’s in a superposition of two states both are equally true until you pick one and then only that one is true.
The exception to rule eleven follows the opposite concept antiposition ( or whatever you want to call it) where an object cannot be inside or outside and should you try to choose one you must always be wrong.
Do to its state of antiposition the waveform (schroedinger’s cat reference) cannot collapse so it cannot be resolved to inclusion or exclusion.
And yes writing this is giving me a headache I hate conceptual paradoxes.
There are conditions that affect every rule. Perhaps that partially addressed your paradox?
I am sorry i havent found you sooner. Excellent presentation. As a grad student id ponder the property of being Richardian in Kurt Godels incompleteness theorem as I was driving and almost have a collision on the road. The Russell paradox also crept into the metaphysical boundaries of the classical world meeting the quantum world - the so called "classical - quantum barrier", quantum entanglement and superposition of states both satisfying classical rules of energy and momentum conservation. E. Spence Browns essay "Laws of Form" stimulated the necessary condition to understand set theory antimomies, and i wondered then if we could think in 5 dimensions rather than a simpler SPACE-TIME of an observable 4-D universe, would the Russell paradox and Godell's incompleteness theorem would just melt away. I dont yet know the answer but set theory is a road not taken by me yet. Thanks for he tutoring. B B.- Calif.
The predicate paradox is resolved because of "gerunds". When you contain in quotes "is not true of itself", it is no longer a predicate, but becomes the object and a noun (gerund). I think the same is true of a set and set theory, because when you change the set it becomes a new set and an equivocation of the set itself.
A set doesn't exist until it is created so a set can't contain itself without creating a new set.
No paradox.
@@CivReborn Yeah, I think that's what I said. Which is why P11 was invalid and needed to be removed.
Being told I don't have to remember certain things is surprisingly comforting. What a wonderful video. I truly enjoyed your presentation of Russell’s paradox..
Wait, am I supposed to remember that?
You took 30 min to explain something that my prof couldn’t explain in 2 hrs, lifesaver 🙏
Literally be explained in 5 mins its not that mind blowing this bored me hard
@@toddallan7086 okay todd allan
This despite the fact he actually over-explains at almost every point along the way.
No, he didn't.
@@tognah6918 lmao
I honestly thought I would be out of my depth here. I am no maths genius, I hardly know my times tables. But I understood this!! Wow
Thank you, I understood all except for your example of Le brone, I had no idea what you were talking about.
As someone who is much more linguistic in my thinking than mathematical, this was a great explanation.
Was it though?
I agree! We can easily (or almost) understand the paradox! Great video!
While the style is good... his information is wrong.
@@JamesJNothingIsTooSensitive how so
@@ame_lia To quote my OG comment I made when I first saw this video:
Your very first premise is wrong. Even if math itself is a product of the human imagination, that doesn't make mathematical truths subjective.
The *_units used_* are subjective, but the truths themselves are objective, as shown by the ability to come to those truths no matter what type of mathematical system you choose to use.
The *_system_* is subjective, but the truths discovered by that system are still *_objective_* no different than measuring distance. You can shoose to measure in inches, or centimeters... or even use cubits or any other mesaurement you choose, but the distance remains an objective distance. Only the representation of that distance is subjective.
Since your very first premise is demonstrably incorrect, I'm going to assume the rest of your arguement is as well, although I will still watch the rest of the video to be certain. As such my comment may get edited as I see more of this video.
Edit: Holy shit, 7 minutes and 21 seconds in and this is downright *_riddled_* with erroneous claims and misinformation about mathematics and sets. This is... fucking hell this is bad.
The problem with the bit about predicates is that, when they're being used as the subject of a sentence predicates cease to be predicates. A predicate is a predicate because it's performing the function of a predicate in a sentence. And it is _only_ a predicate _because_ it is performing the function of a predicate in a sentence. At all other times that string of characters is just a string of characters. You can put quotes around the string of characters and use it as the subject of a sentence, but that makes it a different string of characters and the subject of a sentence, it does _not_ restore the quality of 'being a predicate' to the string of characters. This means that the example predicate 'is a predicate' which was used in the video is not, in fact, true of itself in the example sentence given because in the sentence " 'is a predicate' is a predicate." is not actually a true statement. 'is a predicate' is not, in that context, a predicate.
The real irony is that the point which was being made is still valid, Kaplan just used the wrong example. "This sentence is a string of characters." In this example the predicate 'is a string of characters' is true of itself and thus predicates are, as the video was trying to demonstrate, capable of being true of themselves. There were even a few other examples given later in the video that are properly true of themselves, one of which was quite similar to the one I came up with. And yet the guy chose to focus on a predicate that was demonstrably not true of itself to demonstrate that predicates can be true of themselves.
Honestly though, that seems about right for the guy who made a whole video on a topic that's about as useful to talk about as why mathematicians unilaterally decided that any number to the zeroth power is one and any number to the first power is itself. Neither of these things make any logical sense when you break down the math behind them. And yet they are still defined as true because otherwise a great many very important mathematical operations would fall apart.
I like the point you made. I would like to add that rule 11 only said that prediciates CAN be true of themselves. This is that same logic that should be applied to his set theory, that sets CAN contain themselves, but they dont have to.
I wonder, though, why not make a rule that says sets of sets of sets that do not contain themselves cannot be made. (What is that? A set twice removed?) That is the stated exception to rule number 1.
"are still defined as true because otherwise a great many very important mathematical operations would fall apart." Perhaps it is that fact that lends voracity to those statements, i.e. raising a number to the power of zero and one are defined in the way that they are, and why they have remained as such. If you can demonstrate new math where those are not true statements, and that new math exceeds the usefulness of all the math built around those simple facts as they're stated today... then more power to you.
@@thom1218 The problem with the power of zero and the power of one is that by the rules of exponents those aren't even valid operations to perform. Perhaps there's some logical language in which you can define exponents such that raising a number to the power of zero equals one and raising a number to the power of one equals the number you started with, but by the commonly available definitions of exponents, multiplying a number by itself the number of times indicated by the exponent, there is no actual mathematical procedure to perform for the powers of zero and one. You can't multiply a number by itself zero times, nor does multiplying a number by itself once make sense when multiplying it by itself is raising it to the power of two, not one.
I'm not saying I have some new math where it makes sense. I'm saying these represent similar situations where what makes sense doesn't line up with the way things has to work for the systems of logic and math to operate correctly.
Precisely what I thought: a predicate can't be a predicate if it doesn't have a subject. So unless you make up a rule that whatever string that was a predicate in another sentence, keeps the value of being a predicate in a new one, then the logic would cease there.
@@jver1384 His problem was that he got too self-referential
Your explanations are so helpful and easy to understand that even a non native speaker like myself can follow.
I think maybe the way to escape this paradox is not to eliminate some rule, but to accept that sometimes things can be both true and false at the same time 🤔
If only my teachers were like this. "You don't need to remember this" was my internal mantra throughout school, and my grades reflected that. Now, if my teachers said that all the time, I would have way higher grades.
Awe yuh, the ol' -- "its my teachers fault that my grades were so terrible!"
You will find at least one of these under every maths related vid on utube
@@whatshumor7639 So how, (even if he could have done that), would that have made it his fault that you refused to learn anything. It is not your teachers job to force you to learn things. It is you and your parents job.
Schools are not supposed to babysit you.
I went to school and aced every maths related and physics class I took from the first grade on. No one, not parents or teachers forced me to do that
@@whatshumor7639 Oh, I get it, it was a joke! Good job!!!
You’re a brilliant teacher; it’s not an easy feat to deliver such an entertaining intro to set theory in several minutes! Looking forward to checking out your other vids. Thanks!
Our shanti Mantra
पूर्णमिदम् पूर्णमद:.....
This is explained in only four lines.
Google it.
How is he good? He is literally terrible in every way. He has no clue what he’s talking about and litterally talks way too much. He talks for the sake of talking in this video.
The ‘I AM that I AM’ statement is awesome in this sense and the dialog with Christ and Pilat later on will be significant, “when men speak Truth they are speaking of me” and “I hate witness to the Truth”.
If I recall correctly, Borges loved this paradox but he talked about a a catalog that contains every catalog, it's in "Babel's library" (sorry for my English translation).
Your deadpan delivery of hilarious lines makes my brain happy
really? thats funny, because it has the exact opposite effect on me... but anyways heres some shiny keys 🔑🗝 look how they shine 😲
That's enjoying the humor while being too engaged to laugh. All those happy little endorphins just pile up on one another.
@@Sergiodeus172May I see the shiny keys again. They make me happy.
Being told I don't have to remember certain things is surprisingly comforting
It’s been a while since a 30min video was too short. This I hope you’ll agree is true of itself.
Fantastic explanation
This world needs teacher like u, ur millions of clones r needed but all the clones should hv the same teaching depth like u.
🙏
Many moons ago, I was a Physics major. My school offered several electives, one of which was Number Theory. I sat through Day 1 but decided to transfer to a different elective. At the time, I just wanted to learn how to build quality things. Now, as an oldster I can better appreciate what Professor Kaplan teaches. Thank you!
I hated Number Theory because it didn't apply to physics - or anything really. Now, Number Theory is essential to our encrypted communications!
I've never laughed so hard learning about math and/or logic, and I'm a university professor, so I've learned a lot of both.
What have you learned ?
May the love and the peace of Jesus be with us.
@@nickelchlorine2753 Neither. Professors teach, don't learn.
May the love and the peace of Jesus be with us.
@@nickelchlorine2753 When did I asked you a question ⁉️
May the love and the peace of Jesus be with us.
@@jwu1950 right there. But may the love of the lord and light be with us.
@@thej3799 Peace. Don't forget peace.
May the love and the peace of Jesus be with us.
This is an excellent explication of Russell's paradox and, despite the idiocy of most of the commentators here, an important idea in mathematics and philosophy. Thanks for posting.
Maybe I didn't fully unterstand the point, but I'm not sure why people get confused with the effects of diverging/fluctuating logic without a resting state or whatever you want to call it.
If a wheel turns clockwise, the top will go right, the right will go down and so on until you have a full revolution, starting from the beginning. A steam engine will move the piston to the right when it's on the left and to the left when it's on the right. Almost all moving things are designed with this principle in mind since it makes them predictable.
Or in set theory, I assume it should also be possible to create a simple flip flop by having a set of all things that are not contained in that set. {x: x is not in the set} (Also, does this set contain more negative than positive integers?)
Alternatively, you could have a diverged set which growths with each call with: {1, x: x are all integers that are less or equal to the largest number in the set + 1}
And hence we have arrived at regular programming.
You’re a phenomenal instructor/teacher! I took basic college algebra just because it was a requirement for my Bachelors… I have always hated math. The way you explained this was EXCELLENT and I totally understood it.
Thank you for a perfectly clear explanation. I am in my 50's now and value it so much when I can find a concise and intelligent explanation for something I should have learned a long, long time ago but just didn’t. It's like scratching a 30 year itch.
Subscribed. Appreciated.
However, this video's explanation was *Not* concise, and it's *Not* something you should've learned a long, long time ago. :)
Agreed on the quality of the presentation. Though I've studied this problem, Kaplan's presentation brought some aspects into focus. But the real questions raised have not been discussed yet, such as the relationship between set theory and ordinary language, in general and on the point of this paradox. Kaplan says "just saying things about things." But that's exactly what happened in the video itself, it was all "saying things about things." So what is the difference? Needs to be a sequel, I guess there is.
Ok this video is imperfect many places, and my comments were imperfect too, so I just leave this here, not in my Notes, because it is not worth it, it is manily wrong and I just leave it here like that idk
I have an important question, can all sets be depicted on an infinite 2d plane, as lines encircling objects? That would make it easy to comprehend and analyze.
4:50 "Sets can even include objects that cannot be imagined"
That is a very bold statement, since imagination is the Superhuman limit.
5:40 Set Builder {x: x is a cat} = A set of all xs such that x is a cat.
7:30 Rules of Set Theory.
1.Unrestricted Composition - If you can imagine a set of something, such a set exists.
My objection to this rule, is that there should be stated the restriction that an object can only be Inside or Outside a set. In other words, a set must be disjoint from everything not included in it.
(From this follows that a set cannot contain itself:
Let us imagine a set containing LeBron and itself. When we go one depth level into it, we have LeBron inside of the set, and outside of the set. The same element, both in and out.
(This eliminates all self-containing sets except the set that contains ONLY itself. Is that set problematic? Idk.)
OH WAIT GODDAMN, IMAGINE THIS {{LeBron}, LeBron} this set seems legit, did my argument just fall? (Along with the 2D-plane-representation assumption?))
2.Axiom of Extensionality - If two sets contain the exact same elements, they are the same set. The order of the elements does not matter, duplicates do not matter.
19:40 Well, if we implement the restrictions to rule 1 in the first place, then dropping rule 11(Sets can contain themselves) does not drop rule 1.
20:00 Sir Kaplan, do not tell me that the original rules of Set Theory apply IRL. IRL we never have a set containing itself, that is a Strange Loop!
At the end he just said the Grelling-Nelson Paradox.
Great video. I’ll attempt a rebuttal.
I contend that Russel and others eliminating rule 11 (that sets can contain themselves) was not an arbitrary decision for the sake of making things work. There is something inherently wrong with a set containing itself. A set, as rule 2 stated, is defined by what is in it. A set containing itself therefore creates an inherently circular definition. What is in A? Well A has, say, 1, 2, and A. But what is in A?? Well, it’s 1, 2, and A. But that A… what is it??? Well, it’s 1, 2, and A….
Consider that you can never take a set that does not already contain itself and put it in itself. Any attempt to do this simply creates a new set that contains the old set but does not contain itself.
Now you might think, ahh, but when I abandoned set theory and used predication instead, I showed that we can have predicates of themselves and it works just fine!
Actually I don’t think it does. And in fact, even just making predicates about predicates does not work as smoothly as you lay out. To take it further, the objective, non-made up rules of predication are in fact stricter than those used for set theory. Set theory was a journey into abstraction that tried to relax the rules that govern “real life”, not make them tighter.
For example, “is a cat” sounds funny. That may be, but “is a cat” here is different from the predicate “is a cat” that you showed before. Here, what is meant are the sounds made by speaking these three English words out loud. That is quite different than “is a cat” the predicate. “Is a cat” the predicate does not sound funny. It does not make any sound at all. It can be thought of as a function that takes in a single object and returns a Boolean value indicating whether that object is in fact a cat. It is distinct from any representation or expression of itself.
Now, for the meat of the matter. “‘Is a predicate’ is a predicate.” This sentence works because we all know what you mean, but it actually is not a true example of a predicate predicating itself. What you really mean is that the four words, “is a predicate”, when placed at the end of an English sentence with some subject beforehand, serve as the predicate of that sentence. But then this is clearly not an example of a predicate predicating itself. In fact, when you fine tune it enough, it is unclear what it means for a predicate to be the subject of another predicate. If it is a subject, then it is not a predicate.
This is why programming languages that allow you to pass functions as arguments to other functions do not actually pass functions as arguments. What is passed is some representation of the function that serves the purpose you’re trying to accomplish. For example, in PHP, there is a function “is_callable”, which returns whether the argument can be called as a function. But to get a return value of true, you do not pass “a function”, whatever that means. You pass a string containing the name of a function.
If we’re being literal, a function is a mapping from domain onto image. Or we can think of it as a behavior that can be applied to an object. But any attempt to make the function an object in itself, and thus apply it to other functions (or to itself), requires concretizing it in some way, such as talking about the result you get when it is applied to certain objects or talking about what purposes it is used for. But none of these representations are “it” truly. It is then impossible to pass a function, in its true and purest form, as a parameter to another function, and so it is also impossible to truly predicate another predicate. You are always predicating some converted form of said predicate and not the predicate itself.
And so, even the rule that sets can contain other sets does not hold when dealing with predication. Predicates cannot predicate other predicates. Once again, set theory allowed for a relaxation of the rules of predication, not a tightening of them.
So once again, we are saved from your version of Russel’s paradox.
"This sentence is false." I love the Russell paradox. I love all paradoxes. They don't break anything except our ideas of reality. They hint at new truths out there just beyond the veil of our understanding. It is exciting.
I wonder if there may be an allowance in logic for something like a superposition, where a proposition can be an inherent contradiction, true and false at the same time.
I also like me some negative self-references (they are positive and negative about themselves at the same time). There have been experiments with fuzzy locigs, but there is no consensus. I think that paradoxes are still regularily treated with stating some prohibiting rules (do not self-refer, at least not negatively, though you can) in most fields of life.
Ad "new truths": you can infer nothing/anything from a paradox (ex impossibile quodlibet), so one opportunity is show the regular paradoxes of communication (e.g. it can be bad to be good) to win new opportunities of thinking. I think that is not only refreshing, but a good loosening-up-exercise of thinking.
Are these new truths anything to do with the new structures physicists are finding beyond our reality, according to Donald Hoffman?
For real, that's all this is: the old "this statement is false" paradox. Scientists are (and really all of academia is) too "smart" for anyone's good imo.
This video is more proof of God's existence e for me
The correct answer is " no, it is". Or "yes, it isn't". ... I can't recall which
@ Jordan Munroe: I came to the same conclusion myself. It felt like a lifting of a corner of the mathematical logic rug and momentarily glimpsing all the quantum weirdness scuttling for cover underneath.
I feel compelled to quote Vroomfondel from the Amalgamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons, “That’s right, we demand rigidly defined areas of doubt and uncertainty!” (Douglas Adams, Hitchikers Guide)
PS: I love how Jeffery explained the derivation of axioms from rule one. I think Bill who commented above has a point about revealing linguistic limitations. At best, the parallels between "true of" predicates and "contained" objects seem to be only a useful metaphor... instead, perhaps paradox actually has a definable mathematical reality.
Nicely done; this video reminds me of one of my favorite books, Gödel, Escher, Bach: an Eternal Golden Braid (1979, Douglas Hofstadter). A set of things that are allowed to define themselves will always be incomplete or in other words if a class of something is allowed to define itself, then an instance of "something" can always be constructed that leads to a contradiction. I think Russell's Paradox, Gödel's Incompleteness Theorem, and the Halting Problem are all just different instantiations of this same underlying problem.
I frikken love that book.
I have that book, loved its metaphors with achilles and the turtle, but even with such informative and illustrious metaphors to help me understand the subject... man... it is way beyond me. I could try to to understand it but I fear I would suffer the same fate as when Gotleb Frege when he got Russell's paradox in the mail.
Yeah - this is just another version of 'All Cretans are liars', so it's very old.
@@ghwrudi Also reminds me of sitting in class listening to some rule and then shouting out the paradox to get a laugh. This paradox is everywhere whenever you start curving arrows inwards
yes
About 3:33
I know that this isn't too important to the video but I wanted to mention it.
The empty set axiom isn't needed since you could just get that an empty set exists from axiom 8.
The foundation axiom doesn't really prevent much to be honest, it's the selection axiom being restricted to only allowing subsets of existing sets that solves the russell paradox. In fact, from what I heard at least, there's a version of set theory where foundation is actually not true and it's perfectly fine.
I did have some "Set Theory" in upper-division Math classes. But I'm learning a lot here!
As someone who never liked mathematics and didn't even try to understand it, i watched this video with so much focus that now this is the main problem in my brain, which i never thought of before
i know how you feel.
Agreed, tho' in my case I do like math, use math in my profession(s), still I found this video thought provoking. One can read/hear something complex or profound for the Nth time, and there are always unrealized nuances to consider.
YES!!!
Everything is nothing and nothing is everything
How did I suddenly start listening to a lecture? You are a GREAT teacher.
?
@@toby7582 I certainly wasn't planning on hearing a math lecture when I clicked on this, but he is such a good teacher that I stayed.
@@private464 it was pretty good.
I just don't like how in the beginning he's trying to sound like Vsauce.
That gets old pretty fast for me.
But still interesting and impressive backwards writing on that glass or however he's doing it.
@@toby7582 Agreed. It wasn't perfect. Some parts too slow and the end was too fast. I just find it remarkable (literally) that he got me to listen to a whole lecture that I wasn't planning on or even interested in!
@@private464 you weren't interested?
Why click on the video then?
28 minutes of defining sets defining the rules how they work and how there's a paradox. Results I have no idea what talking about and what it means and doesn't matter if it means anything. That's part of the reason I had trouble with more advanced math.
" it depends on what the meaning of is is"
I listened to this to really trying to learn something I did not.
What I think creates the paradox is truth is an unconditional factor in all these arguments. The sets represent conditional factors that can clearly be articulated or symbolized. However, the truth of the argument cannot be symbolized because it is unconditional. In every predicate there is this subluminal assertion of existence often associated with the word "is". But the word "is" is also a condition. Existence and non existence are conditions. Dead & alive are conditions. (i.e. absence or presence of elements within a set). A set of all sets that contain {3}. Set of all sets that do not contain {3}. A set of all sets that simultaneously do and do not contain {3} - (i.e. a conditional contradiction that is true: see Quantum Mechanics).
What Russel has in fact stumbled upon & proved by virtue of the contradiction is that truth is not a state of being (existence or non existence). A cat can simultaneously be alive & be dead. A true contradiction that I believe was proven in the subatomic world. Things being in 2 places at the same time and all the other weird quantum happenings. Hermaphrodites occur in nature. A genetic expression of Russel's paradox?
Hence, what we are referring to as "truth" is unconditional. This is the only way out of the paradox and off the merry go round.
If I have an empty set of conditions, then where is the truth? The set is empty. Therefore, the truth is not a condition. Zero is a "REAL" number the last time I checked.
Perhaps a new set rule is in order to address all these real contradictions. Perhaps sets can contradict themselves-or something along that line of thinking. Why should something be declared incorrect because it creates a paradox of conditions that argue against itself when there is clear evidence such things do in fact happen? Has anyone really proven such results can't be true?
The fact that he is so involved and tells you the story as if it is conspiracy tale is just amazing
Yes truly amassing!
Dude you need to chill 😂
items amassed = set
{x: is a youtube comment, x: is grammatically correct, x: does not contain malapropisms, x: is read by anyone}
@@scambammer6102 The set of all things that are amassed ?
To those who would like to learn about the context of how that theory came to be along with Godel's incompleteness theorem and the birth of set theory in a rather fun and pedagogic way i would recommend the comicbook "logicomix" that centers around the life , with some apocryphal event, of B. Russell in search of mathematical truth .
apart from that Great video as always from you Jeffrey
I am wondering how many times have you said the word Set vs Sets ! I love the analogy made with the predicate! Great video, I've laughed my head off.. Cheers
Professor Kaplan: A beautifully-presented explanation of the paradox, and that it is not easily resolved. But have you anything further to say about it? Do you have a follow-up? It is fun to consider the nature of paradox, but what next?
One clear result is that there are statements that are demonstrably true, there are statements that are demonstrably false, and then there are other statements. Paradoxical statements are interesting because they are demonstrably neither true nor false. Of course there are other statements without clear truth-status. Once you are using statements whose truthfulness is unclear, you do not have a good rule to define a set. I assume there are even statements for which truthfulness is not decidable. There are certainly statements for which the truthfulness may be decidable but it may be very hard (e.g. Fermat's last theorem, which was unsolved for many years, or some other theorem that is currently unsolved, and we then don't know whether it ever will be or not.) To try to build sets on such rules, one would get potential sets, but one would not find them useful. This does not need to bring us to despair.
I can't help thinking that on some perverse level Russel was pleased with himself that his ideas had the power to literally blow someones mind.
or it was all pointless rhetoric.
@@honeycat535 Sounds like you just prefer to not think about it.
@@somedudeok1451 Sounds like youre relieved someone replied.
@@honeycat535 ??? I replied to you...
@@honeycat535 Aw gee, my little snickerdoodle, you're too precious for words, darling dumpling. They make Midol for what ails you, bless your itty bitty little old heart, pumpykins. Koosh koosh!
If you liked this discussion, I recommend checking out Gregory Bateson's essay "A Theory of Play and Fantasy" from his Steps to an Ecology of Mind (1972). He shows that the paradox runs deeper than even predicate-based syntax/human language. In the process, Bateson shows that Russell violates his own rule (a set may not be a member of itself) by even positing his rule. And here's the kicker: Bateson argues that without such violations/paradoxes, communication as we know it (beyond rigid mood-signaling), would not be possible (including that peculiar game we call logic). This includes non-human behavior such as mammalian play, threats, and other metacommunicative interactions from which the metalinguistic rules and thus spoken language evolved. Bateson ends his essay with imagining what the world would be like without such paradoxes: "Life would then be an endless interchange of stylized messages, a game with rigid rules, unrelieved by change or humor."
God Said, Ha!
wow, I gave it a read and it was 100% worth it, thank you so much for your comment:)
After reading Bastedon's essay, I now understand why Frege had a meltdown after reading Russell's letter lol. Basically our whole world and everything we do falls apart.
But I think most people actually already know this and don't realize it. The answer to "why do I do what I do?" Is very often, "I don't know. I just do it."
I also recommend Gary Gygax.
So I assume chance being at times random has something to do with this. The complexity of the system is such that we cannot fathom the pattern of randomness. Good enough for me to feel like I am alive and not a self-deterministic machine.
Well that seems that problem is not really with sets but with generating functions. Probably somebody had thought about that (and maybe even debunked), but I’ll still describe my idea.
You can build sets as collections of elements, like, we state that for everything that is exists a singleton, and we can add those singletons to create more sets. Some addition processes are infinite but you can’t do anything with that. That’s all sets that we can ever get.
However, if we want to actually use sets, we need to generate them in another way, like using their properties. What I am trying to state is when you are trying to define set using a generating function you deal with sets that can be created as infinite unitings of singleton sets. Some generating functions just don’t work and there is no set corresponding to it.
That actually raises the question, what does exist? What is the cardinality of set of all sets that contain themselves and nothing else? 0?1? Infinity? That’s undefined? What’s the set of sets that are uncountable but lesser than continuum?
Probably you could argue that generating function is just iterating over all things including sets and fit everything checking if that fits, so it is still union of singletons. Well, I guess it’s a good point, however I feel problematic to just iterate over literally everything as literally everything is greater than itself: set of all sets contains all its subsets and thus violates Cantor’s theorem. In other way, after you iterated over any set including set of all sets I take all subsets of the sets you have passed and find one you missed.
That might kill the initial idea of all sets existing before generating. Or maybe set of all sets is a bad generating function - Wikipedia says that in most axiomatic languages it either doesn’t exist or is not a set.
Predicates are more complicated, but i guess they could be considered just a kind of generating functions. So predicate “is cat” corresponds to generating function “set of all cats”, and “is false” to “set of all generating functions, corresponding to sets, not containing themselves”
Two things, #1. Definitions are important. Sets "are a [description] to describe [sets of objects]". A (as the subject) then, by definition are a (description) of [something else]. Therefore, there's always an abstraction layer between the two. You can take one or more sets and use them as the [something else] in the creation of a new set to describe the [something else]. It's important to note, the description (abstraction layer) is infinitely removed from the stuff being described [something else]. Something cannot be something else. In as such, I can discuss a [description] of a [set of descriptions], but here the [set of descriptions] becomes the predicate, whike the [description] is the subject. Which leads us to #2 ["Is a cat" is a predicate.] Not really. ["Is a cat"] is the subject, it's not in itself a predicate. It's a noun phrase. It's an example of a predicate, but an example of a description of a predicate is itself, not a predicate.
I have heard of this Set Paradox before but you did a great job of explaining it and the ramifications. Now you got me wanting to review my Chomsky and perhaps taking a deeper dive into counterfactuals which will further muddy the waters for me.
The reason you can't solve the paradox is that paradoxes aren't real things in the real world, they're the literal manifestation of cognitive dissonance. They're purely cognitive artefacts arising from either naïve construction or naïve analysis.
In general, paradoxes arise from either including something irrelevant or excluding something critical, either in construction or analysis. The Rich Guest paradox includes an irrelevant rich guest, opening what's fundamentally a closed circle. Xeno's paradox arises because it excludes time (in the form of speed; it appears to include time throughout via 'when', but it only treats the spatial distance and doesn't include the notion of distance over time, i.e., speed). The Twins Paradox arises via the exclusion of a third reference frame, which is what's necessary to bring the twins back together.
This particular paradox comes from the exclusion of rigour, in that the natural language construction contains ambiguous or undefined terms.
There are no real paradoxes, they're a sign that something has gone seriously wrong with our thinking.
@@tonymurphy2624 So, what is the problem in the predicate case? Where did it go wrong?
@@KaiHenningsen Would this resolve it?
Axiom 12: A set can simultaneously contain itself and not contain itself. This can be called a higher order set, or an upset.
@@KaiHenningsen I think it goes wrong by describing something as true or false that cannot be true or false without reference to a subject. “Is not true of itself” isn’t true or false because those ideas require a statement to be made and “is true of itself” is not a complete statement
@@brendanh8193 You win!... in a highly improbable upset. 😭
My subscription has been earned. My calculus I professor depicted this paradox without reference to Russell, using the words “autological” and “heterological”. He summed it up by saying that just because the answer to a yes or no question isn’t “yes” we can’t necessarily assume the answer is “no”.
Indeed, the answer is yesn't.
That is a brilliant statement by your professor. It seems to fit with how life actually is. Don't ask me more!
Excellent 👍
I actually had been thinking about stuff like this a lot, especially on Physics and time
But when I saw all my notes and assumptions written down, I just felt afraid of someone declaring me a crazy person
I could sit through a 5 hours math class of this guy, he somehow made a math subject entertaining.
go out in the sun, look at a plant - seriously .
@@RazaXML some people (just like myself) need that 5th grade comprehension to even begin to understand math, so this is actually really valuable for someone like me and others like me.
My math classes were always taught to those who actually understood math, like two people, the ones who didn’t (the rest of the class) were left in the after classes and usually got 1-3/10 grades..
So we always needed to go after class and rewrite the tests, it’s a fucked up system in a lot of Latvian schools, probably a lot more places in the world as well
This isn't maths - where is the practical application? This is a waste of time.
@@jbooks888 math has applications that transcend the merely practical.
It’s a playground of logical thought where black holes are discovered and the contents of atoms and nuclei found.
More than that, math describes and circumscribes the limits of our understanding of what’s *out there*.
I've always been fascinated by the other set in Russell's paradox-the set of all sets that do contain themselves. If you ask which of the two sets that one belongs to, you can show that there's no problem with saying it belongs to either set. There's a certain freedom there that mathematical objects are not supposed to have.
Gödel.
That "this statement is false" is undecidable gets all the attention, but "this sentence is true" is just as undecidable.
that's basically what philosophy offered to all the mathematicians, a back door
thinly veiled mysticism, opium for the few. adults speaking in codes according to complete arbitrariness. you might as well expel hot air the other way round!
LOL. I love your comment. Point well taken. Most of the time Math is about the rules and following them. This may be the only exception.