The Continuum Hypothesis and the search for Mathematical Infinity, W. Hugh Woodin 

By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

Tamas Varhegyi Tamas Varhegyi isn’t this just what Cantor did in order to show that the set of natural numbers and the set of rational numbers have the same cardinality? What does this have to do with CH?

@@spacefertilizer He posted that comment as a reply to every other comment here.

@@whatno5090 He believes that he has mapped "floats" one-to-one with natural numbers, thus settling the question - because natural numbers and "floats" have the same cardinality, there is no cardinality strictly between the two. And he refuses to understand that he has only mapped one-to-one with natural numbers such real numbers which have a finite decimal expansion. (That this can be done is not a new result - all such numbers are rational, and that there are countably many rationals was proven by Cantor himself.) He also doesn't know, and refuses to learn, how are real numbers actually defined in mathematics. (But that doesn't prevent him from believing that he knows better than anybody else and spamming this nonsense all over RU-vid.)

@@MikeRosoftJH Sounds about right but I hope he eventually lets somebody talk to him one-on-one and sort this whole thing out

It might be beautiful but unfortunately wrong about CH. By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

"Their existence has falsifiable consequences on finite sets.... There's no way, that we know of, to explain why measurable cardinals are consistent that doesn't trace back to 'measurable cardinals exist' in some sense. The large cardinal hierarchy, which has emerged as a robust conception, is an increasingly stronger sequence of predictions about the universe. and the only way to account for this is to conceive of [them in] set theory... If we give up [the notion of large cardinals], we can't explain something that seems meaningful not only in number theory but in this universe." -- So far, up till 27:50 at least, the only support (as put forth by Woodin in this lecture to a non-professional audience) for the existence of large/measurable cardinals comprises (1) lack of proof of their nonexistence, (2) their falsifiability, and (3) the *attractiveness* of the idea of large cardinals.

We actually do accept that there's an interpretation where there are no large cardinals. What we reject is that we HAVE to stray from large cardinals just because we can't show there's a consistent interpretation where there are large cardinals. Woodin never claims that there is support for the EXISTENCE of them, rather that there is support for the inability to show they don't exist.

@@whatno5090The same can be said of God.

@@ZeroG I suppose so

It's really because they just don't want to let the idea of infinity go. It's too useful in mathematics(it seems) and to deny it's existence would create a rip between science and mathematics since all the mathematics based on infinity would have to be thrown away as just non-sense that just magically works because we really don't know what we are doing. In some sense the reward is to great. If there is nothing more(no infinity or no large cardinals or whatever) then the fantasy is over and it's all back to just basically mathematics. It's sorta like string theory turned out to be a complete flop. Essentially if there are large cardinals then it is as if "space and time warp back on themselves"(in the sense that the point of infinity can be understood by transforming by 1/z). After all, they are the "small cardinalities at the cardinal singularity".

It helped me a lot too. By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

@@tamasvarhegyi8813 bro what

+WriteRightMathNation I asked Prof. Woodin, and he told me that it was just a misspeak.

+Daniel Hill thank you.

By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

@@hilldaniel lul

@@tamasvarhegyi8813 no you didn’t. The continuum hypothesis hypothesises that the cardinality of the set of all subsets of the natural numbers is equal to the cardinality of the set of all countable ordinal numbers, omega one. All you have shown by doing this is that the number of rational numbers is equal to the number of natural numbers, which cantor already did. If you came up with it yourself good on you on though!

There is no such a thing as an infinite set. By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

@@tamasvarhegyi8813 This does not disprove the Continuum Hypothesis, since the floats are the rationals with terminating binary expansion, which are known to be countable. Also interesting take on "there is no such thing as an infinite set" when you claim that there is the set of all integers, which is clearly infinite.

You are simply regurgitating ideas without any proof. I actually have the algorithms which ends all arguments.

@@countingfloats No you don't. You have algorithms which merely show that countable sets are countable. You're incorrectly deducing from them that CH is false (and moreover that Cantor's diagonal argument is invalid),

Well, you absolutely have no idea about what the algorithms I published do, even though they are elementary. But let's change the subject. My guess is that you also believe that 0.9999...999 eventually turns into 1.0 if we go to infinity, am I right ? That is the prevailing view if I am correct. If so, tell me after how many 9's will this happen ? No, you cannot say infinity, because that is not a number. If it was a number, we could add another 9 to pass it and keep going. So we have to stop before it, right ? Where would you like to stop ? Give it a shot.

It doesn't Cohen showed the GCH is independent of ZFC and it's truth value has no impact on the system.

+Charles Hoskinson Right, I thought so. I've read the proof of "if ZFC is consistent then so is ZFC+ not CH (and at fortiori not GCH)" using forcing, but I think you can't prove it the other way around through forcing, right ?

CH or negation of CH has no effect on ZFC except as far as I understand the Axiom of choice only exists in a universe governed by GCH. If there are certain statements that do not require for proof GCH or CH as a matter of fact to hold or not to hold then the reader would mostly prefer a proof without either of the assumptions.

@@ujjalmajumdar618 That's not true. ZFC already includes the axiom of choice (C stands for "choice"). What is true is that ZF (set theory without axiom of choice) + generalized continuum hypothesis implies axiom of choice. (The reverse implication doesn't hold; neither ZF nor ZFC can prove continuum hypothesis, or the generalized continuum hypothesis, true or false - at least assuming that set theory itself is consistent.)

@MikeRosoftJH you are right sierpinski showed that Axiom of Choice follows from a universe governed by ZF and GCH

All subsets of the real numbers are contained within V_{\omega + 2}, and so are any bijections between these. Therefore, it could be settled by looking at the theory of V_{\omega+2}.

By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. If you want elegance, see the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

Hello, I completed the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. For further information you may contact me by email me at : secondcause@gmail.com

@@tamasvarhegyi8813 Spam, spam, spam, baked beans, spam, ...

@@tamasvarhegyi8813 go away troll

He might be a genius but he is wrong about CH. By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

@@tamasvarhegyi8813 sure you did

@@victormd1100 Sure, I did. Do I detect a condescending superior tint in your brief and empty response ? Try me, name any integer and I will produce its floating point equivalent or pick any floating point number (e.g. 374.117 or 0.000000004374 and I will send you its integer pair. Amazing, no ?

@@countingfloats okay, you are aware pi is a real number, right? Well, which integer corresponds to it?

@@victormd1100 Yes, I can even match Roman numeral integers with floating point numbers. floating_point_number_of_MMMCMXCIX := "2.443" floating_point_number_of_MMMM := "2.444" floating_point_number_of_MCCXXXIV := "37.1" floating_point_number_of_DCCCLXXXVIII := "8.86"

The "blueprints" are a simplified description of Cohen's method known as forcing, in which one starts with a model M of set theory and a partially ordered set P in M (which he calls a blueprint, although it's more commonly called a forcing notion), P "codes" information about some object G outside of M. By adjoining what is called a generic filter over P to M, you get an extension of the model in which the continuum hypothesis is now false or true, etc. en.wikipedia.org/wiki/Forcing_(mathematics) en.wikipedia.org/wiki/List_of_forcing_notions

timothychow.net/forcing.pdf

Fermat's last theorem. Luckily we have RU-vid so I had no excuse not to come up with the proof. I actually managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

@@tamasvarhegyi8813 Stop spreading misinformation.

@@countingfloats I have quite a bit of idea on this subject considering I wrote half of the entire wikipedia dedicated to this subject (cantor's attic).

@@countingfloats As I have repeatedly told you: you haven't proven anything that you claim. What you have proven is that natural numbers can be mapped one-to-one with real numbers with a finite decimal expansion. That's not a new result. All such numbers are rational, and that rational numbers are countably infinite has been known for about 100 years. And this has nothing to do with the continuum hypothesis. Continuum hypothesis is the proposition that there is no cardinality strictly between natural numbers and real numbers. (That natural numbers and real numbers can't be mapped one-to-one also has been known for about 100 years.) The problem is that you don't know how real numbers are defined (they *aren't* defined as decimal expansions), and refuse to learn. (To make things worse, you also use non-standard terminology; "floats" is short for "floating point numbers", which is not a term in calculus, but a specific representation of numbers in computer science. Likewise, "algorithm" means something else than how you use the word.) A saying goes: Who knows, and knows that he knows, is wise; follow him. Who does not know, and knows that he does not know, is simple; teach him. Who knows, and does not know that he knows, is asleep; awaken him. Who does not know, and does not know that he does not know, is a fool; avoid him. As I have said, you are completely ignorant about what real numbers and the continuum hypothesis are really about, refuse to learn, but believe yourself to know better than anybody else. By the way, continuum hypothesis has already been solved - the solution is that it can't be solved. The set theory axioms are insufficient to answer the question one way or other. It is true in some models of set theory, and false in others. In other words - assuming set theory itself is consistent - it is possible to add either the continuum hypothesis or its negation to the set theory axioms, and neither will make the theory inconsistent.

@@countingfloats As I have repeatedly said, your function only covers numbers with a finite decimal expansion. So here's a couple of numbers that your function does not cover: 1/3, square root of 2, pi. Yes, your function covers every truncation of these numbers to a finite number of decimal places. That's not a matter of dispute. But it does not cover the numbers themselves. (And don't tell me that 1/3, square root of 2, or pi are not numbers. That would have shown even more ignorance about what real numbers are.) Continuum hypothesis is a statement about real numbers as conventionally defined in mathematics. It is not a statement about "floats" under your definition. So in order to say anything about the continuum hypothesis, you first need to know and understand how real numbers are defined. Another equivalent way to state the continuum hypothesis is: there is no set whose cardinality is strictly between natural numbers and the set of all sets of natural numbers. (Note: there are countably many finite sets of natural numbers, but uncountably many sets of natural numbers overall - just like there are countably many real numbers with a finite decimal expansion, but uncountably many real numbers overall.)

Yes but those axioms are so fundamental to the rest of mathematics - mathematics which has been successful across the years - that it would be extremely counterproductive to assume they are false when they really help give ground to the rest of mathematics (not like that would matter much). Axiom of choice is a little more contested...but many people see the fruits in what it proposes, and I generally tend to agree, even though you have to swallow some uncomfortable decisions later on with regards to measure theory etc.

Would a more restricted mathematics (e.g., one without choice or ramified infinity) have any effect on claims by other fields like physics or engineering? To what extent do practical disciplines even rely on the subset of assertions in classical mathematics + choice + infinity + ... that aren't found in more conservatively defined paradigms?

The short answer is, I don't think so no. But this is coming from someone who believes that Mathematics is invented, rather than discovered, which I can assume is your stance based on your question. This is the long answer: The Mathematical discipline aims to answer questions within its own framework and seeks to probe deeper within that framework to answer even deeper questions. Mathematics never is able to answer questions outside what it is capable of. Tools such as Calculus are only useful in the real world because they have been set up in the Mathematical framework and studied to give what answers appear to be correct within that framework. What goes on in the real world does not affect Mathematics in the slightest, it is independent. Mathematics though is influenced by the real world because we are creatures of the real world. We want results about the real world and we want tools to be able to make predictions about the real world. This does not inhibit Mathematics though, as it may probe within that artificial framework as much as it pleases to gain even deeper results which would have been previously invisible to us.

People still arguing over the axiom of choice smh

@declup "Would a more restricted mathematics (e.g., one without choice or ramified infinity) have any effect on claims by other fields like physics or engineering?" You realize that without the axiom of infinity, you can't do calculus right? You realize that ZFC - the axiom of infinity is equivalent as a theory to just PA and PA can't prove anything about analysis. Would physics and engineering be weaker disciplines if they weren't allowed to use real numbers, let alone complex numbers, and calculus? Yes, of course they would. I feel like you might not have been exposed to the history of these ideas but you're not the first person to object to choice or the axiom of infinity, all of these objections appeared during the foundational crisis when these formal systems were being made and the fact that we use the axioms now is a testament to the philosophical and mathematical arguments that have been made for them, as well as the fruit of their labor. There are people who reject choice, obviously there are people who reject infinity (finitists), and there are people who reject the law of the excluded middle. The fact remains that if you choose to reject those axioms you're stuck with a very weak form of mathematics that is not able to prove a lot of things that you, honestly, believe to be true. A perfect example is the work that Harvey Friedman has done on independence. arxiv.org/abs/math/9811187 He proved that there are statements in number theory, statements about finite things, that cannot be proven in PA or even in ZFC. To prove them, you need to assume a large cardinal exists, so you need a strengthened form of the axiom of infinity. Again, you probably believe these statements, they're just regular combinatorial statements about finite objects, but in order to prove them formally you need to assume a large cardinal axiom. So, do you actually not believe in the axiom of infinity? It seems like you don't believe in it, but you believe in statements that it's required to prove. In your other comment above you misinterpreted what Woodin said about the falsification of the large cardinals. He's talking about what I just linked to above. "Their existence has falsifiable consequences on finite sets," and "If we give up [the notion of large cardinals], we can't explain something that seems meaningful not only in number theory but in this universe." He's saying that if we give up the idea of large cardinals, we will give up the finite combinatorial statements that Friedman showed require an axiom of strong infinity. It's not just that "the axioms are falsifiable," it's that they're falsifiable by things that even the strongest skeptic of infinity believes to be true, because the statements are about finite things.

lol same

Vw is not the set of all finite sets. It's the set of all hereditarily finite sets. There's a difference. Also, CH is not undefined, CH is well-defined and its a quite common exercise to write its definition in purely FOST, the base language of all of set theory, so its essentially as defined as you possibly can get.

+LogicalBelief Numbers are not concepts?

+LogicalBelief infinity means infinite cardinal numbers. they are "numbers" not simply a concept, they just dont behave like finite cardinals

+carelessbear What else should they be ?

Nomen Nominandum Uhm, exactly my point, you tell me? lol I know what he means though. You cant assign a number to infinity because its simply a contradiction, also infinities are vry very problematic and unrealistic to work with if not simply false.

+carelessbear Sure. I have to read the comments a bit slower next time. Sorry !

Yeah I noticed it as well

Gurdle?

When you're on Woodin's level, you can pronounce it however you want

@@drewduncan5774 Haha! High level in mathematics; less so in linguistics.

You're not!

When people don't understand something they ascribe theological implications to it.

Cantor also ascribed theological importance to set theory.

You are absolutely right, that is one gets when trying to do arithmetic with infinity. By the way I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

Are you taking the union or the Cartesian product? Since you claimed you'd get back N, I'm assuming you actually meant union. You wouldn't actually get back the entirety of N anyways, you're missing elements made up of more than the powers of just 1 prime, like 6 for example. Also, in that case, the cardinality of your set would be NxN = N^2 which is easily proved with the diagonalization method. Should you actually have meant Cartesian product, you'd have gotten a subset of N^N, not N, so you proved a subset of N^N = N^N, whoopdy-doo :).

The answer is that you haven't proven that N^N has the same cardinality as N. The reason is: every natural number is finite. So when you separate any natural number into a product of primes, the result will also have finitely many elements. If you take a sequence with infinitely many non-zero elements, such a sequence does not represent a natural number (any natural number is divisible by finitely many primes). What is true is: the set of all finite sequences of natural numbers has the same cardinality as natural numbers themselves. (On the other hand, N^N - set of all functions from natural numbers to natural numbers - is uncountable; it has the same cardinality as the continuum.)

Now take the surreal numbers. You have more surreal numbers than you do real numbers. What is the cardinal of surreals? What about of the powerset of surreals?

Set theory says that everything you just said is wrong. Infinity came in different sizes. And the cardinality of N is smaller than R. In fact it is infinitely smaller, i.e, if you randomly pick any number on the real like, the probability that you hit an integer is zero. These things are not new, and they have been proven long time ago.

@@vuongbinhan Your reply is precluded by what I have pointed out. Again, a cardinal is just an alias for an ordinal. All infinities are equal.

@@vuongbinhan forgive him, he probably has never heard about diagonal argument

You either did not watch the video or watched the video and understood nothing of it.

You are right, I say a lot more .I managed to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

Wrong. Tells you at the very beginning.

4:22

I made it simple for everyone and proved the CH.By managing to prove the Continuum Hypothesis in spite of everything said in this video by showing that the floats and integers can be put into 1:1 correspondence with each other. See the recently published YTube video : "Proving the Continuum Hypothesis" Tamas Varhegyi

"Humble" opinion.

Sorry but I don't think you're qualified to criticize professional mathematicians? There is no possibility for a numerical example, because this is abstract, but it doesn't make it any less correct - it's like criticizing philosophy because there's no proof of the ideas involved. But at least here, we know the theories we're working in are logically sound. And excuse me, do you see any numerical examples in "actual mathematics" either - when does a journal published article in, say topology or algebraic geometry, going to include numerical examples either?

You know, in math we also use quantifiers to talk about things. Not only numerical values.