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.

Yip, it is what it is. Not an easy subject. Good that infinity just exists in math 😅.

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

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

@@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

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)

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

@@pikkuvarpunen-2.7182 Yeah , by now i have hit the limit in 5 different courses that I wanna complete , so I will get premium in the next few days In the mean time I am using the free platfrom Khan Academy Ideally I wanna use both to complement one another , since they are built differently

i unlocked brilliant completely for free by pretending to be an educator

Got me learning now thanks..

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

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)

You don't need naive set theory to assert the existence of first order models. Every model of a first order theory is a "set" in the sense that it's a bunch of objects which satisfy the axioms of that theory, but you don't need to assert that, for example, the power set of a model is a set, or the union of two models, etc. It is a theorem of first order logic that any consistent theory has a model.

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.

The details can be quite tricky, but roughly it can be thought of as a generalization of Cantor diagonalization. Or at least adding a generic real can be thought of this way. The main components of a forcing construction are a poset consisting of elements called forcing conditions and a class of special sets called names. These conditions are ordered in such a way that “stronger” conditions provide more information about the intended generic object you want to construct. For example, Cohen forcing uses partial functions with finite domains as conditions. Laver and Sacks forcing uses special types of trees as conditions. Grigorieff forcing generalizes lots of forcings by specifying certain types of Boolean ideals which can be used as domains for partial functions. The structure of the forcing typically is derived from the kind of question one wants to answer. The names, on the other hand, are a way of talking about objects that exist in the forcing extension, but that may not exist yet in the ground model. (The ground model is where you build the forcing poset and the names are kind of like a computer simulation of the generic extension.) Names can be quite tricky to handle due to their recursive construction. I like to conceptualize them as “cloudy” or “probabilistic” sets. We may not be able to say exactly which sets they are (Really I mean which sets they will become in the forcing extension.) given certain forcing conditions, but strong enough forcing conditions can give partial information about the structure of a name. Think of this as “resolving” some of the cloudiness. For example, I might have in my hand a name that contains the correct information to be the Prufer group or a Cohen-generic real, but will get only one or the other if certain conditions are met. If I travel to the left I get the group, if I travel to the right I get the real. The last critical thing to understand is density. Density in forcing roughly corresponds to the idea that a logical statement can be true “generically often”. A better way to think about this is that a property is generic if no matter how much information a condition gives you, you can always add a little bit more information to make the property true. This is the connection to diagonalization. For example, suppose you had a list of real numbers. Then you can generically extend any finite domain partial function to be different from the next real in your list and thus can obtain a “new real” generically by just making the right next digit choice at every step.

A beginner's guide to forcing: timothychow.net/forcing.pdf

Your videos are great, and I can guarantee you're a thousand times more competent mathematician than I am.

His pronunciation was the correct German interpretation. The correct anglicized version is your interpretation. Both are correct.

​@@firefox7857 In no world is "Geddle" correct lol.

@@firefox7857 lol, I see what you did there, nice.

@@firefox7857 Ich bin Deutscher, seit 26 Jahren sprech ich Deutsch als Muttersprache. Gödel wird nicht als Gedel ausgesprochen. And if you don't trust some random idiot, go with Prof Dr. Edmund Weitz of the HAW Hamburg in his video on Gödel incompleteness named "Gödel (miss)verstehen - Was sagt der Unvollständigkeitssatz wirklich aus?" around 1:15. Excellent channel btw, also has some english videos and his Christmas lectures are brilliant even when auto translated.

Well, this is just a beginning Edit: of set theory

this is true. but in any specific theory, after fixing what your axioms are, then there are some statements which "have to be" true in the theory if you want it to be consistent, some which "have to be" false, and some which can be either. so it's true that "truth" is a notion that doesn't exist outside of a theory, and for any axiom you could just as well have a different theory in which it is false instead