This discussion is good in summarizing the standard position taught today, but this axiom is really the heart of what makes set theory controversial. The point of set theory is to manipulate completed infinite totalities, collections which are imagined to have been gathered up all together 'once and for all' at the beginning of time, and stripped of the ability to grow further, and now that they've been collected and pinned down, we are now enumerating their properties. This conception is unavoidable in mathematics past the renaissance, and I think it's mostly due to Simon Stevins' completed infinite decimal sequences. Equivalently, this is second order arithmetic, there are certain collections of integers/symbols which we feel are perfectly well enumerated in an infinite time limit, for example, computable reals, all the Gaussian primes, dense collections in function spaces like all continuous functions which take rational values at rational positions, and these countably infinite collections become definite well-defined infinite objects in the limit of infinite computing power and infinite time. These countable structures are what you can (partially) represent on a computer, and they provide all the precise models of any first order axiomatic system. This is extremely important, because it means that this is 'as big as it gets' for purposes of precise (first order) logical reasoning. We can't define mathematical systems of reasoning any larger than what can be enumerated by these countable collections. The proofs by Skolem and Godel clarify how this is so. For Skolem, starting for any model of a given first order language, we choose out of any model we are given a countable collection of witnesses for all the first order statements that are true or false, so a counterexample for every false statement, or an example for any predicate that is satisfied, and now we gather together all these witnesses into a (countable) set. By construction, every element of the original model is indistinguishable in its properties from one of the countably many witnesses we gathered, so all other members of the original set are philosophically extraneous and can be thrown away! Similarly, in Godel's completeness theorem, we create witnesses for every existence proof, one by one, as they get proved, and since the deductions are countable, we end up with a countable model. What doesn't get constructed can be thrown away. This is also the philosophy behind Godel's L, except the deductions go on ordinally long, not just finitely long. The main point of Cohen's work is to explain how to produce sets/real-numbers that are FOR SURE not contained in Skolem reductions, or in Godel style step-by-step productions. that these organized countable collections, because you can create real number digits, which are specifically chosen to AVOID any definite ordered property like being equivalent to some Skolem witness. For example, you could choose the binary digits of a real number completely 'at random', one by one. Such a real number, such a subset of the natural numbers, would still become precisely defined at the infinite time limit of infinitely many coin-tosses, all questions we can ask about this collection should get an answer at some finite time (and does get an answer, this is the fundamental theorem of forcing), but these random collections are fundamentally of a different kind than the ordered collections, so they aren't going to be inside a Skolem reduction. These randomly generated real numbers are defined to avoid all measure zero sets previously existing in a Skolem reduced model, which, incidentally, also avoids all one-point sets, you've created a new real number which isn't defined by any procedure, just by flipping coins. The fact that you can always philosophically do this, you can always adjoin new real numbers, means that the previously existing collection of reals was never philosophically complete. What Cohen actually did wasn't to use randomness, but to define reals which avoid previously definable meager sets, not measure zero sets, so he was imagining playing a game on the digits of the real number, where you specify countably many properties that a definite real number would have in your theory, and Cohen plays to avoid satisfying any of those properties, and because they all define meager sets, he can do it. One equivalent definition of a meager set is that if you are playing a digit game where you choose successive digits of the real number in turn, and player A tries to force the real into the set, and player B tries to force it out, then player B always wins. So Cohen is playing to generate a real number not in a Skolem reduction of Godel's L, so he's guaranteed to win. This is why the concept of powerset is so model dependent, you can introduce as many Cohen reals as you like whenever you like. This isn't true in the same way for countable conceptions of completed infinity. This is why Cohen says the powerset axiom is so powerful. It's not because it's powerful logically, because it's not all that powerful logically, it's because it's philosophically ridiculous to create an axiom which steps you up completed cardinal totalities, when it is so easy to extend even just the real numbers to become larger than the entire universe. The proper view of the real numbers is as a function type, which is never exhausted as a totality by any set. It's like a proper class. It's fundamentally bigger than anything in any set theory, ever. The sets are just various countable structures which are used to create definable collections inside the types, and they can only exhaust orderly methods of producing new objects, they are always surprised by disorderly methods, like Cohen's game, or choosing digits at random.
Countability and Uncountability always confuse me from the beginning of my math journey. In advanced calculus I learned these concepts, and starting to get a little sense of it while learning Lebesgue’s theory. But it just never ends. Whenever we consider operation or collection, the extension to infinite always give those subtlety between countable and uncountable cases. After watching the lecture and reading this comment (which I do not understand at all ), I’m on one hand glad to hear that no one really know the answer, but on the other hand got terrified by how complex this matter is.
@@LillianRyanUhl Happy birthday Richard, 2 months late! God is great, but Richard is trying to be greater! Cohen's book is great. Godel's proof of Incompleteness is wrong, but Gentzen's counterstatement is right! General recursive functions allow 1 = 0. This is the error of Godel's proof. This is no more difficult than the set of natural numbers does not have cardinality a natural number. Maybe we could say the sequence is complete when the greatest is equal to the greater!
5:50 I'm not really a platonist, but I guess they wouldn't be particularly annoyed by this: they'd just say that V is not a set, so "all subsets" means something different for it than for V_alpha (for a particular alpha).
I.e. a set is a class which satisfies a first order logic predicate. Because proper classes usually have higher order predicates, with 2nd order being the coarsest to formalize the incompleteness theorems. Does that make sense?
Powerset, obviously necessary for the Von Neumann hierarchy. 1:30 "What if we remove the Powerset Axiom?" Consider the model Hk = all sets of cardinality < k. Borcherds says that if you take k to be a Regular Cardinal > ω, then this will be a model of ZFC-P. Oh, like Beth5 is Regular! I thought that after taking the union of all Hk's sets we could create a set with cardinality ≥ k, but now I see that when k is Regular, that is impossible! (Irregular Cardinals like Bethω can be created by unionizing Beth0,Beth1,Beth2...) You cannot progress from Beth5 to Beth6 without Powerset! (Though idk how in my example we would even reach Beth5, will we not get stuck at Aleph/Beth0/1?) [Jesus Christ how much I hate the unexplained distinction between Beths and Alephs!] 2:30 ZFC→Con(ZFC-P), the way of showing that Powersets strictly strengthens ZFC that already has been used multiple times in this series. And Borcherds gives no proof yet again. 2:45 "But what is a subset of A?" - "It's a set of some elements of A." "But what is a set?" - And we cannot use Von Neumann this time, because we are yet to cumulate it. "Maybe a subset of A is the set of all elements of A defined by some formula." - It must be a formula in some language then, and a language will have only a countable number of statements, and we want Powerset to be able to give us uncountable subsets. 4:15 ZF bypasses these problems by ignoring them, assuming we know what a subset is. Borcherds says the issue is: 1.Take all subsets of ω. Easy, that's Beth1. 2.Take all subsets of Vα. Easy, that's Vα+1. 3.Take all subsets of the union of all Vαs. Here my issue is that Borcherds did not say whether α is a Natural or an Ordinal. Because if α is a Natural, then easy, our set is Vω+1. But if α is an Ordinal, then such a union cannot be created. But from what hell did Borcherds bring up Russell's Paradox here?? "If we allow any subset of the collection of all sets to be a set, then we can start looking for a set of all sets that aren't members of themselves." Umm yeah, I still don't see how this would make any sense. 7:00 "Questions that are really hard to answer" 1.How big is Pow(ω)? I mean, Beth1, right? Borcherds says it is Aleph1, and starts with the Continuum Hypothesis. OH, LET ME USE THIS PLACE TO EXPRESS MY DISGRUNTLEMENT WITH ALEPHS. I STILL HAVE NO IDEA WHAT THEY ARE, SO I WILL USE BETHS WHICH I AT LEAST KNOW ARE POWERSET-ITERABLE. I MAY HAVE INTERCHANGED ALEPHS AND BETHS INCORRECTLY BEFORE TODAY (2023-09-20). 9:15 Gödel's constructible universe is exactly like Neumann's, except that higher levels are not "all subsets of the lower level", but "all definable subsets of the lower level", whatever that means. And this method supposedly allows you to (in an easier way) count all the subsets of Aleph0, which allows to prove CH. 11:30 Umm, no! You canot just make a model of ZFC in which the powerset of Aleph0 doesn't contain some of its subsets! That would mean you powersetted it incorrectly! (Ugh! I hate models! How can you even make different ones on the same set of axioms!)
I'm really enjoying this series of lectures, as set theory has always fascinated me. One question (which is probably absurd): if P(omega) is so model dependent, is there a way in which some sort of "maximal model" makes sense? Some sort of "this is the biggest model that somehow makes sense considering", I guess.
Platonists accept a hypothetical "true" maximal model of math, while formalists accept that P(ω) isn't really a "concrete" concept in that you could always be missing some subset of ω (such as 0#)
The proper class of all sub*sets* of the proper class V makes sense. You just can't take the all the sub*classes* of V and expect them to form a class.
why should it be hard to talk about subsets? A is a subset of B if forall C we have C in A implies C in B. then the set of all subsets of B is the set of which a set A is an element iff it is a subset of B. where exactly is the issue in this statement?
This discussion is good in summarizing the standard position taught today, but this axiom is really the heart of what makes set theory controversial. The point of set theory is to manipulate completed infinite totalities, collections which are imagined to have been gathered up all together 'once and for all' at the beginning of time, and stripped of the ability to grow further, and now that they've been collected and pinned down, we are now enumerating their properties. This conception is unavoidable in mathematics past the renaissance, and I think it's mostly due to Simon Stevins' completed infinite decimal sequences. Equivalently, this is second order arithmetic, there are certain collections of integers/symbols which we feel are perfectly well enumerated in an infinite time limit, for example, computable reals, all the Gaussian primes, dense collections in function spaces like all continuous functions which take rational values at rational positions, and these countably infinite collections become definite well-defined infinite objects in the limit of infinite computing power and infinite time.
These countable structures are what you can (partially) represent on a computer, and they provide all the precise models of any first order axiomatic system. This is extremely important, because it means that this is 'as big as it gets' for purposes of precise (first order) logical reasoning. We can't define mathematical systems of reasoning any larger than what can be enumerated by these countable collections.
The proofs by Skolem and Godel clarify how this is so. For Skolem, starting for any model of a given first order language, we choose out of any model we are given a countable collection of witnesses for all the first order statements that are true or false, so a counterexample for every false statement, or an example for any predicate that is satisfied, and now we gather together all these witnesses into a (countable) set. By construction, every element of the original model is indistinguishable in its properties from one of the countably many witnesses we gathered, so all other members of the original set are philosophically extraneous and can be thrown away!
Similarly, in Godel's completeness theorem, we create witnesses for every existence proof, one by one, as they get proved, and since the deductions are countable, we end up with a countable model. What doesn't get constructed can be thrown away. This is also the philosophy behind Godel's L, except the deductions go on ordinally long, not just finitely long.
The main point of Cohen's work is to explain how to produce sets/real-numbers that are FOR SURE not contained in Skolem reductions, or in Godel style step-by-step productions. that these organized countable collections, because you can create real number digits, which are specifically chosen to AVOID any definite ordered property like being equivalent to some Skolem witness.
For example, you could choose the binary digits of a real number completely 'at random', one by one. Such a real number, such a subset of the natural numbers, would still become precisely defined at the infinite time limit of infinitely many coin-tosses, all questions we can ask about this collection should get an answer at some finite time (and does get an answer, this is the fundamental theorem of forcing), but these random collections are fundamentally of a different kind than the ordered collections, so they aren't going to be inside a Skolem reduction. These randomly generated real numbers are defined to avoid all measure zero sets previously existing in a Skolem reduced model, which, incidentally, also avoids all one-point sets, you've created a new real number which isn't defined by any procedure, just by flipping coins.
The fact that you can always philosophically do this, you can always adjoin new real numbers, means that the previously existing collection of reals was never philosophically complete.
What Cohen actually did wasn't to use randomness, but to define reals which avoid previously definable meager sets, not measure zero sets, so he was imagining playing a game on the digits of the real number, where you specify countably many properties that a definite real number would have in your theory, and Cohen plays to avoid satisfying any of those properties, and because they all define meager sets, he can do it. One equivalent definition of a meager set is that if you are playing a digit game where you choose successive digits of the real number in turn, and player A tries to force the real into the set, and player B tries to force it out, then player B always wins. So Cohen is playing to generate a real number not in a Skolem reduction of Godel's L, so he's guaranteed to win.
This is why the concept of powerset is so model dependent, you can introduce as many Cohen reals as you like whenever you like. This isn't true in the same way for countable conceptions of completed infinity. This is why Cohen says the powerset axiom is so powerful. It's not because it's powerful logically, because it's not all that powerful logically, it's because it's philosophically ridiculous to create an axiom which steps you up completed cardinal totalities, when it is so easy to extend even just the real numbers to become larger than the entire universe.
The proper view of the real numbers is as a function type, which is never exhausted as a totality by any set. It's like a proper class. It's fundamentally bigger than anything in any set theory, ever. The sets are just various countable structures which are used to create definable collections inside the types, and they can only exhaust orderly methods of producing new objects, they are always surprised by disorderly methods, like Cohen's game, or choosing digits at random.
Fascinating comment! Did you get all this directly from reading Skolem, Godel, and Cohen, or is there a book you can recommend that covers this stuff?
Countability and Uncountability always confuse me from the beginning of my math journey. In advanced calculus I learned these concepts, and starting to get a little sense of it while learning Lebesgue’s theory. But it just never ends. Whenever we consider operation or collection, the extension to infinite always give those subtlety between countable and uncountable cases. After watching the lecture and reading this comment (which I do not understand at all ), I’m on one hand glad to hear that no one really know the answer, but on the other hand got terrified by how complex this matter is.
Very interesting. Where could I read further on this?
Beautifully put.
One note: what you call Gödels witnesses actually occur in Henkins method.
Happy birthday Professor Borcherds!
Happy one day late birthday Dr. Borcherds!
@@LillianRyanUhl
Happy birthday Richard, 2 months late!
God is great, but Richard is trying to be greater!
Cohen's book is great.
Godel's proof of Incompleteness is wrong, but Gentzen's counterstatement is right!
General recursive functions allow 1 = 0. This is the error of Godel's proof.
This is no more difficult than the set of natural numbers does not have cardinality a natural number.
Maybe we could say the sequence is complete when the greatest is equal to the greater!
*Hey, sonce TH-cam says it's been a year since you commented, the happy birthday Professor!*
Thank you for putting the time and effort into these lectures. I look forward to them.
I really appreciate all these video lectures! You're doing fantastic work :)
5:50 I'm not really a platonist, but I guess they wouldn't be particularly annoyed by this: they'd just say that V is not a set, so "all subsets" means something different for it than for V_alpha (for a particular alpha).
I.e. a set is a class which satisfies a first order logic predicate.
Because proper classes usually have higher order predicates, with 2nd order being the coarsest to formalize the incompleteness theorems.
Does that make sense?
Would you consider making an introduction to forcing as an addition to this series?
Powerset, obviously necessary for the Von Neumann hierarchy.
1:30 "What if we remove the Powerset Axiom?"
Consider the model Hk = all sets of cardinality < k. Borcherds says that if you take k to be a Regular Cardinal > ω, then this will be a model of ZFC-P.
Oh, like Beth5 is Regular! I thought that after taking the union of all Hk's sets we could create a set with cardinality ≥ k, but now I see that when k is Regular, that is impossible! (Irregular Cardinals like Bethω can be created by unionizing Beth0,Beth1,Beth2...) You cannot progress from Beth5 to Beth6 without Powerset!
(Though idk how in my example we would even reach Beth5, will we not get stuck at Aleph/Beth0/1?)
[Jesus Christ how much I hate the unexplained distinction between Beths and Alephs!]
2:30 ZFC→Con(ZFC-P), the way of showing that Powersets strictly strengthens ZFC that already has been used multiple times in this series. And Borcherds gives no proof yet again.
2:45 "But what is a subset of A?" - "It's a set of some elements of A."
"But what is a set?" - And we cannot use Von Neumann this time, because we are yet to cumulate it.
"Maybe a subset of A is the set of all elements of A defined by some formula." - It must be a formula in some language then, and a language will have only a countable number of statements, and we want Powerset to be able to give us uncountable subsets.
4:15 ZF bypasses these problems by ignoring them, assuming we know what a subset is.
Borcherds says the issue is:
1.Take all subsets of ω. Easy, that's Beth1.
2.Take all subsets of Vα. Easy, that's Vα+1.
3.Take all subsets of the union of all Vαs. Here my issue is that Borcherds did not say whether α is a Natural or an Ordinal. Because if α is a Natural, then easy, our set is Vω+1. But if α is an Ordinal, then such a union cannot be created.
But from what hell did Borcherds bring up Russell's Paradox here??
"If we allow any subset of the collection of all sets to be a set, then we can start looking for a set of all sets that aren't members of themselves."
Umm yeah, I still don't see how this would make any sense.
7:00 "Questions that are really hard to answer"
1.How big is Pow(ω)? I mean, Beth1, right? Borcherds says it is Aleph1, and starts with the Continuum Hypothesis.
OH, LET ME USE THIS PLACE TO EXPRESS MY DISGRUNTLEMENT WITH ALEPHS. I STILL HAVE NO IDEA WHAT THEY ARE, SO I WILL USE BETHS WHICH I AT LEAST KNOW ARE POWERSET-ITERABLE. I MAY HAVE INTERCHANGED ALEPHS AND BETHS INCORRECTLY BEFORE TODAY (2023-09-20).
9:15 Gödel's constructible universe is exactly like Neumann's, except that higher levels are not "all subsets of the lower level", but "all definable subsets of the lower level", whatever that means. And this method supposedly allows you to (in an easier way) count all the subsets of Aleph0, which allows to prove CH.
11:30 Umm, no! You canot just make a model of ZFC in which the powerset of Aleph0 doesn't contain some of its subsets! That would mean you powersetted it incorrectly! (Ugh! I hate models! How can you even make different ones on the same set of axioms!)
Can anyone guide me to a good introduction to model theory? I seem to have a rough time with what I've come across. Something for dummies out there?
I'm really enjoying this series of lectures, as set theory has always fascinated me. One question (which is probably absurd): if P(omega) is so model dependent, is there a way in which some sort of "maximal model" makes sense? Some sort of "this is the biggest model that somehow makes sense considering", I guess.
Platonists accept a hypothetical "true" maximal model of math, while formalists accept that P(ω) isn't really a "concrete" concept in that you could always be missing some subset of ω (such as 0#)
The proper class of all sub*sets* of the proper class V makes sense. You just can't take the all the sub*classes* of V and expect them to form a class.
This is so great.
why should it be hard to talk about subsets? A is a subset of B if forall C we have C in A implies C in B. then the set of all subsets of B is the set of which a set A is an element iff it is a subset of B. where exactly is the issue in this statement?
BRAVO!!
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
I agree with you.