The unsolvable problem that launched a revolution in set theory
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- เผยแพร่เมื่อ 29 พ.ค. 2024
- An introduction to the Continuum Hypothesis - a problem in set theory that cannot be proved correct or incorrect.
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A HUGE thank you to Luciano Salvetti, a graduate student at the University of Toronto in set theory, for helping me make this video!
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Intro: (0:00)
Continuum Hypothesis: (0:59)
What is Independence?: (2:04)
ZFC Axioms: (3:15)
Model of ZFC: (4:03)
Godel's Strategy: (5:23)
Cohen's Strategy: (5:35)
Showing Luciano some love in the comment section. ❤
While some authors present the Zermelo-Fraenkel axioms as including the axiom of the empty set, for the sake of simplicity, the standard axiomatization of set theory includes no such axiom. In fact, the existence of the empty set is a theorem, which is a consequence of the axiom of infinity, and the axiom schema of collection.
Also, I think that discussing the role of the axiom of choice here is important, since it allows us to say more about the continuum hypothesis. Without the axiom of choice, the continuum hypothesis is the statement that there exists no cardinal κ such that Aleph(0) < κ < 2^Aleph(0). Here, Aleph(0) is the cardinal of the set of natural numbers N, and 2^Aleph(0) is the cardinal of the power set of N. However, with the axiom of choice, in conjunction with the Zermelo-Fraenkel axioms, one can prove the well-ordering theorem. This means that for all sets X, there exists some =
Hello Angel! I find your comment important and relevant. During the realization of the video we decided to omit certain points for the sake of simplicity and accessibility to everyone. Some of these points are precisely the ones you are pointing out, so hopefully people will get to read this. In particular the clarification of the syntactic/semantic issue that became confusing at the end.
Also, thanks for taking the time to answer a lot of the questions in the comments!
Good comment, mate! Even with a non-math degree I could follow it fine, gives interesting and important context and corrections to the video. Thank you!
The incompleteness of consistency has 6 more orders of complexity, essentially not validly available in some sovereignty regimes, one of which is canada.
Beyond the criminals, satanists, and anti-christs, their laws are irrelevant, and not cured by the modeling of civility.
The idea is therefore alien to conformal existence, and, patently unlawful, to be even so cheaply discussed on a controlled bulletin board.
This was a very helpful comment. Thank you. People used to tell me that the CH is neither true nor false (it can be either you want). I always thought that sounded ridiculous.
Yes! When the world needed him most! He returned!
exactly my reaction!
He never really explains any math in his videos, though. It’s just symbols being drawn, and vague concepts talked about.
@@ophello Someone once said we should take the numbers out of math
@@chaotickreg7024 I agree arithmetic is not mathematics.
@@Alan-zf2tt Arithmetic is what happens after your math is done
This video felt more like an intro than an actual video. I hope there will be a dedicated series to this. This topic deserves it.
great video as usual!
small correction: at 1:47, "CH is not inconsistent" is the same as "is consistent", and thus not disprovable. I think you meant to say that the *negation* of CH is consistent w/ ZFC
You won’t get many likes with such a perfectionist comment, but I think Aleph0 would appreciate it more than anyone.
Thank you for clarifying this. The same correction occurred to me but I thought I was missing something.
@@charlesbrowne9590 Is there a problem with the idea of mathematical perfection?
@@charlesbrowne9590 He'll get likes from me. I was confused by this.
not ch is consistent
I vote for more about "forcing" as well. And, please, don't be shy about using abstruse mathematical notation! I'm sure your subscribers will love it!
Great video. I just want to add that "undecidable" should not be thought of as a "truth value" in the way "true" and "false" are. Rather, there is a class of truth values which are not "true" or "false", all of which are called undecidable. All true statements are equivalent, all false statements are equivalent, but not all undecidable statements are equivalent.
@@selfieelfie Wow, didn't expect to see you here.
If they are either 'true' of 'false' then they are equivalent to one or the other.
You might be right - I can't decide.
@MadAlly And then there is Tarski's undefinability theorem, which states, that there is no L-formula True(n). (That is, you can't prove arithmetically, that Arithmetics is true.)
so one might say "undecidable" acts similarly to the "undefined" one gets when dividing by zero
Great video. Seriously, huge props for including Luciano in the project and citing him. I'm a grad student in set theory, and math youtubers have, in recent years, constantly made videos about cardinality and CH with demonstrable errors. It's really frustrating seeing my subject consistently misunderstood. So, in all seriousness, taking that extra step to refer to somebody more knowledgeable is more than appreciated.
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this is what has ignited a bit of a spark in me , trying to learn and UNDERSTAND math ... and thats how i got this video & ur channel in my recommended
All those "crazy" math teachers in school always talking about "the beauty of math" , which I never saw... I think I finally got a glimpse of what they see
How do you use the limited version? I've tried it once and it seemed to me like after one or two courses (or even exercises) I'd have to cough up the money to keep going
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The continuum hypothesis is my absolute favourite maths mystery. It boggles the mind.
It's not a mystery, we know the answer.
@@firefox7857 I find it hard to imagine what the universe in which it is false looks like. That is a mystery to me.
@@Kleithap Real numbers are fundamentally untenable in the physical world. We are incapable of thinking of uncountability (represent a state of thought by a quantized distribution of energy. There are countable states as it is quantized). It's purely a set theoretic concept.
@@ffc1a28c7 Nah a continuum is very much imaginable.
@@IsomerSoma A representation of a continuum is imaginable. You can not simultaneously imagine uncountable numbers of things (for that matter, aleph nought in untenable)
Outstanding, and many thanks to Luciano for joining forces with you.
Did you end this with the Axiom of Choice?
“Life exists on the fulcrum.” - JG.
Luciano - take some love. ❤
This is an excellent introduction. Perhaps a more advanced version of this video might explore the concept of "Forcing" in greater detail. Paul Cohen's proof is not easy to follow. Thank you.
This is a difficult topic (at least to me)
It's not difficult at all, the only thing that makes it difficult is set theory language, which makes it a pain in the ass to explain how it works. It works like this--- a truly uncountable set is a branching structure, there are at least two paths you can take with any finite amount of information you are given. An example is the real numbers--- you imagine you know some digits and then you can always ask Vanna White to uncover another digit, and there are at least two choices for that digit. This makes the real numbers into a "tree".
Now given any structure of ZFC, like some enormous aleph, you can map it to a collection of partially-known real numbers, think of a bunch of partially specified points in the tree, only finitely many digits are known of all these numbers, but you keep asking Vanna White to uncover more digits, one by one. As Vanna uncovers more digits, you prove more and more statements, for example, if one number called "x" has "3.141...." showing and nothing else, and Vanna uncovers the next digit, and it's "7", you immediately will prove "x is not equal to pi" from this information. But if you imagine going on forever, even when Vanna uncovers EVERY digit of every number, there will still be a ton of things you didn't prove either way, meaning the union of all statements you prove from partial data is never going to be complete.
So you define a different notion of 'true' and 'false' which is designed to end up complete in the limit, this is Cohen's notion of "forcing". A statement is "forced true" when you can prove it is true from the digits you already know, like "x is not equal to pi" was forced true in the example earlier. A statement is forced false if NO FURTHER EXTENSION is ever going to force it true, meaning, from this point on, no matter what Vanna uncovers, you are never going to prove it true (notice that isn't saying you will prove it false, only that you will never prove it true). You extend this definition in the obvious way by logical deduction, so if you can deduce something from forced true and forced false statements, they are also forced appropriately. If you think about it a second, given this new notion, EVERY statement is either going to be forced true or forced false in the limit of revealing every digit. That's by definition, we designed it to work that way.
So now you know that every statement is going to be decided in the sense of forcing, no matter what path you pick for your numbers. now pick a path through the set that is "generic". In the case of real numbers, you're picking digits, and making the real number generic just means that you avoid every single one of the countably many ways to name a property in set theory so specific that it restricts the real number digits to something more precise than any interval. For example "Less than 18 digits of my real number are not equal to 3" is too specific, because once I know 18 digits that aren't 3, the rest are all 3. A "too specific property" is formalized by the notion of a meager set. One way to define a meager set is that if you are playing a digit game, where your adversary gets to define a bunch of digits of a real number, then you get to name one digit, then your adversary gets a bunch more digits, then you get to name one digit, and so on, and your goal in naming digits is to avoid landing in the set eventually, then a set is meager iff you can always win this game, no matter what your opponent does. This means Vanna can always uncover new digits to avoid any given meager set, again, by definition.
Because Vanna can avoid any meager set, you can make Vanna avoid ALL the meager sets inside a given countable model of set theory, because the union of countably many meager sets is still meager. These numbers obtained by Vanna are now 'generic' relative to your model of set theory, they have absolutely no specific property which you can write down in set theory which can specify a number in some way significantly better than saying "it's somewhere in that interval over there". In other words, you have freely specified a bunch of real numbers with no specific logically writable-down properties beyond the list of digits that you are uncovering.
With these new numbers added to your countable model, you can easily prove they are all forced to be different (meaning you're never going to prove x is equal to y no matter how many equal digits are uncovered, so 'x=y' is forced false), so if you have one such number for each of the countable many elements of some enormous aleph (in a countable model of ZFC), like aleph omega, then the continuum will now become bigger than aleph omega. That's the end of the continuum hypothesis proof.
There's one issue here, which is that you have to prove that adding these numbers doesn't "collapse cardinals", this means, you have to prove that aleph omega is still aleph omega at the end of the process of adjoining. This is proved using the 'countable chain condition', which states that any collection of disjoint already for sure unequal partially specified digit-sequences is countable (in any model of set theory, just countable). Then at no stage are you ever going to produce a map between two different cardinalities.
The best book on this is probably Nik Weaver's recent book, but Cohen's original book is definitely second best.
Paul Cohen's proof is a very bad place to start with learning forcing.
There are books like Kunen's introduction to independent proofs, or Bell's boolean valued models. One can try Jech's set theory 3rd millenium edition, but this book is harder.
Forcing is also explained in Kanamori's higher infinite.
Nice exposition can be also found in handbook of mathematical logic.
But in order to understand forcing well, i would suggest to get basic background in mathematical logic before (formulas, proofs, models, filters, boolean algebras).
Thank you to both Anna Clara Fenyo, and Elizabeth Harper, for your insights and recommendations. Your remarks may contain the germ of a script for the more advanced tutorial that I believe this subject deserves.
Some of the references offered (including one addition of my own) follow:
"Forcing for Mathematicians" by Nik Weaver, World Scientific Publishing, January 2014
"Set Theory and the Continuum Hypothesis", Paul J. Cohen (W.A. Benjamin, 1966), Dover Books on Mathematics, Illustrated Edition: Reprinted December 2008
"Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics, Vol. 102)" by Kenneth Kunen (Elsevier, 1980), North Holland, Reprinted December 1983
"What is Mathematical Logic?" by C. J. Ash, J. N. Crossley, C. J. Brickhill, J. C. Stillwell, and N. H. Williams (Oxford University Press, 1972) Dover Books on Mathematics, Reprinted October 2010
I found the chapter on Set Theory in Crossley, 1972, to offer a clear though incomplete overview. I have been looking for something more thorough, but which does not make quite the same demands as the original proof by Cohen, 1966.
@@christopherhume1631
If i were to choose only one, i would suggest Kunen. It provides not only forcing, but also rudiments of philosophy of set theory and it introduces everything basically from scratch
This is so awesome. Thanks Luciano for the collaboration!!
I think you are one of the channels that I wait the videos of the most eagerly. Great job !!!!
I'm always looking forward to seeing your videos! Thanks for highlighting this topic! Also, thanks to Luciano!
After 10 months he returned with 7 minute video. Absolute legend.
Good work Aleph 0 and Luciano Salvetti! I just started watching a 12 hour philosophy of mathematics series by Joel David Hamkins and really appreciate your short video's approachability in discussing important concepts and background in the foundations of mathematics. Thanks!
66 was a strange year where much happened with everything in a one +volume CRC. Thanks, Luciano for your hard work and thanks to your HONEST cohort. Refreshing to see this amongst the brainiacs. Good thing we're leading the way.
Outstanding 🤯 thanks Luciano! And the video itself it’s amazing and so easy to understand and learn.
Glad you're back. Made my day.
I think people will be surprised by a lot of proof theory stuff, like intuitionist mathematics, constructive proofs, martin lof type theory and etc.
Thanks Luciano to participate on the video.
Welcome back! Keep up the great work!
Woot!! He's back! Thanks for another AMAZING video! Love it! Please keep them coming! Shout out to Luciano Salvetti for his insight! Brilliant stuff
Dear Aleph 0! Your videos are all superb! They are packed with so many wonderful and complex ideas in mathematics and physics and their unifying inter-relationship. I have also enjoyed your other presentations on Galois Theory, Stokes Theorem and DeRahm's Theorem. Thank you for creating and and uploading them. Kudos to Luciano also for this Continuum Hypothesis material. I would like to suggest if you can give names of accessible books to further explore these topics in detail if viewers choose to do so. It would greatly help in the dissemination of scientific knowledge among the general public. Cheers!
Thanks Luciano! And good luck with your studies.
Greetings from a colleague from Germany. (But with focus not in set theory, though. I wanted to keep my sanity, lol.)
What is your focus on?
@@bengal_tiger1984 Differential geometry, cohomology, and scattering theory (in the periphery of the "can you hear the shape of a drum" thing)
YOURE BACK :DDDD I remember reading an article about this stuff before, it’s super interesting! Cant wait to take the set theory
course at my Uni
Just want to say: I love your work. You have a unique presentation style, and you fill a niche in TH-cam math for covering very high-level topics in an accessible way. Keep it up!
Wow that was awesome! Thanks to both of you.
Good thing I have notifications on!
Great video, thanks for returning
This is interesting. Thanks Aleph 0 and Luciano!
Amazing video!
Thank you for making interesting concepts easily accessable to the world.
You always teach me something new Luciano! You're a legend!
These are wonderful videos! Thank you so much!
Thnak you Aleph 0 and Luciano Salvetti!
Glad to see you back at it, friend.
Finally a comprehensive explanation of continuum hypothesis I been trying to find for years. Thanks
As always, very informative video. You sir, are making an impact on the world. !!
Another great video, I hope you keep making them.
Thank you so much Luciano and Professor 🙂🙏🏻🙏🏻🌼🌼...
Brilliant!!!! Thank you Aleph 0 and Luciano!!!! Was looking for a great explanation to this very interesting topic. I remember this lesson during my time as math major (22 years ago). Unfortunately, I changed majors to physics. But I still have tons of love for math.
Happy to see that you haven't abbandoned the channel!
thank you for the great intuition on undecidability. had real trouble getting my head around it.
Thanks so much for uploading this, just when I needed it. This topic was one of the topics I suggested you upload next. And you did it. I am writing an important paper on the foundations of math and the future of math and physics, and this is something I am addressing. Topos theory and Grothendiecks Algebraic Geometry would be an awesome next topic. Or possibly abc conjecture and the purported proof by Mochizuki (although this is pushing it)...
Great video! Thanks to Luciano.
Thanks to you and Luciano. I now feel like I understand the place of the Continuum hypothesis in mathematics better. I also understand a little bit more about zermelo Frankel axiom's
Thank you Luciano Salvetti !👍👏
Thank you very much Luciano!
Luciano, we love you.
Absolutely fantastic quality
Thanku u Luciano selvetti for delivering such immense knowledge to us💓
Fantastic video! Loved it
Love your videos. Keep it up buddy.
Thank you Luciano!
Awesome video! I kind of want to learn more about model theory now
I know this is a science video but that song is incredible. Taking Flight. Holy moly bro. That song is way to good.
Thanks Luciano!
Great video Luciano😊
Awesome work!
For those who wish to get a flavor for Cohen forcing, this is roughly how one proceeds: Think of an object. If there is no obvious objection to its existence, then it *does* exist in the universe of sets, for that universe is vast. Use the notion of a *poset* to create increasingly consistent approximations to this object, and we obtain a forcing axiom for that object.
In other words, think of it this way: The Continuum Hypothesis is like asking whether unicorns exist according to the laws of physics (unicorns being a set of cardinality between the natural numbers and the real numbers). Cantor claims these unicorns *do not exist.* Gödel answers the question using his constructible universes, saying that "yes, Cantor claiming that unicorns do not exist is consistent with the laws of physics." Cohen, however, answered it by saying "Well, we do not observe these unicorns in our (constructible) universe, however, there is no good reason why unicorns should not be allowed to exist, provided we can model our universe in a way that does not violate reasonable laws of physics (such as gravity, quantum mechanics, etc.), but may take liberties with the planets or molecular makeup of these unicorns. Who is to say these unicorns don't feed off silicon, or methane gases, etc.? So, let us create some reasonable guesses, as to the details of these unicorns: They look and run like horses, but they have horns, and they may run through walls (these are not violations of the laws of quantum mechanics, for example). Thus, we can approximate the biological nature and habitats of these unicorns to arbitrarily closer degrees of exactness, by running our arguments to their increasingly *filtered* conclusions, and eventually we obtain a perfectly reasonable model of a universe that works just like ours, except unicorns *do exist!*
Thus, the statement "unicorns exist" is in fact independent of our current knowledge of the laws of physics, because consistent arguments can be made for and against such an existence. However, if we extend those laws of physics, (such as adding the law of *constructibility*), we can indeed demonstrate universes in which unicorns are not only plausible, but inevitable.
go post constantly man i'm addicted
Thank you for the video.
dude i just got interested in pure maths and ran into your channel. please keep posting, love your content!
I'd love to hear more about Forcing. I haven't seen a lot of resources online that discuss the topic.
Also, I was wondering how other set theories like NBG and MK play into this question.
NBG is equiconsistent with ZFC, since it is a conservative extension of ZFC. MK is not a conservative extension of ZFC, and is stronger than NBG and ZFC, but the continuum hypothesis is undecidable in MK.
@@angelmendez-rivera351 Thank you
MK uses a stronger form of choice if I recall correctly and is not equivalent to ZFC.
Forcing is a hard topic. If you want to learn more about it, I’d suggest starting light with some classical set theory. The standard is Ken Kunen’s book, but I think that is like reading Rudin on a first run through real analysis. Start with Halmos, then find any book on introductory set theory. It should go up to at least the equivalence of AC with Well Ordering and Zorn as well as discuss the reflection principle. You’ll need to know some model theory and Gödel’s theorems, which themselves require some decent background.
Once you get that, Kunen should be tolerable. Forcing is essentially “outer model theory” and so you need to understand those things first to get it.
@@HPTopoG I have Kunen's book but I was never able to get deeply into it. I'll take your other suggestions. Thank you.
@@seanspartan2023Sure thing. Don’t skip the sections on cardinal arithmetic and infinite combinatorics. They are critical for understanding everything that comes later.
I love that you continue to make videos. What future video plans do you have?
Great video - thanks to Luciano
In fact, I was impatiently waiting for a new video from you, especially since there are topics that you did not complete explaining in previous videos, especially the last Fermat theorem. I jumped out of joy when I received a notification from your channel with a new video, but I was waiting for more from you.
Especially the explanation using mathematical equations, and not just impromptu speech.
Because most of the viewers of this channel are specialists in this field. In any case, I thank you and your colleague who helped you (Luciano). I am waiting for a new video from you soon. Good luck to you and your friend.
Good to see you again
You’re back!
Believe or not, TIL that Cantor lived into the 1900s and there are photographs of him! For some reason I've always thought he was more like Euler's contemporary.
Thanks Luciano Salvetti!
Luciano, thank you!
"In some sense" is a good way of describing it😌
thank you Luciano!
We need more of this videos with same quality
Thank you Luciano.
Thank you Luciano
Can’t wait for the next video
Welcome Back!!!!!!!
Vai Luciano, gran bel lavoro!!!
THANKS!! LUCIANO SALVETTI
A very Refreshing topic.
great explanation
Yesss the best math channel returns!
Cheers Luciano!
amazing video!
welcome back!!!
great video thank you. Please do one about the riemann hypothesis. the biggest problem in mathematics
I went to Hebrew school in Boro Park, a neighborhood of Brooklyn, NYC, where Paul Cohen also attended. Along with a third pre- bar mitvah boy, we sometimes skipped the class and chatted together. I don't mean to convey that Paul and I were friends, just acquaintances, but we, the other boy and I, knew that he was studying higher maths. I understood that he was close to his grandfather. I don't know if his grandfather was at all mathematical. We met again at Brooklyn College in 1950, where I struggled with physics. I believe that the maths chairman, a Prof Borofsky - a very likable guy - recognized Paul's potential and arranged for him to go to the U of Chicago to do graduate work in mathematics. I envied Paul's escape from Brooklyn College's rigid curriculum.
Love it!!!!!!!!!!!!!!!
Thanks Luciano
Great, this channel is perfect
kudos Luciano!
thank you Luciano
welcome back buddy
Ugh thank you. Everything else I saw on this topic was so unnecessarily thick. The bit about “True, False, or Choose” makes a lot of sense
Gracias por tan interesante tema.
Thank you!