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I've been looking up a little bit about the Continuum Hypothesis, I understand that it is independent of $ \textbf{ZFC} $, that is, cannot be proved or disproved using the current axioms in $ \textbf{ZFC} $.

Is there any mathematics that depend on it being true? Or equivalently not being true? As I understand it, we mostly assume it is true and namely $ 2^{\aleph_0} = \aleph_1 $. Would it be detrimental to mathematics that it were false, that there does exist a set $S$ such that its cardinality lies strictly between $ \aleph_0 $ and $ 2^{\aleph_0} $? Would there be a problem if it were found to be such that $ 2^{\aleph_0 } = \aleph_{27} $ or some other arbitrary number as long it did equal such a number.

Sorry if this has been asked before , couldn't find an answer really. I've only got a limited knowledge in set theory and models.

Mathew Duxbury
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    I think a possible answer could include small cardinals. – Renan Mezabarba Mar 22 '17 at 15:44
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    Several related questions were asked in the past. Have you searched the site for them? – Asaf Karagila Mar 22 '17 at 15:51
  • I've just followed the link in Martín-Blas Pérez Pinilla answer, sorry I didn't see them earlier. Maybe a better question - is there any ramifications to every day maths someone may do when initially studying? Or consequences in the maths relating to sciences? Most of the answers in those threads went a bit over my head. – Mathew Duxbury Mar 22 '17 at 18:28

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You clarify your question in a comment:

Is there any ramifications to every day maths someone may do when initially studying?

So let me address that. Since "everyday maths" is an informal term, it's impossible to prove something about it; but I can give a very good argument that the answer is "no." It's somewhat technical, but the gist of it is that if a sentence of everyday maths is provable from CH, or from "not CH", then it's just outright provable.

There are two key concepts at work here, each of them involving a heavy amount of set theory.

The first key concept is forcing. CH is a statement about sets, and as such is either true or false in any model $M$ of ZFC (the axioms of set theory). Now, there may be lots of models of ZFC (in fact, if ZFC is consistent, then there is at least one, and if there is one then there are infinitely many!) Models of ZFC are quite complicated objects, but over time we've learned how to understand them - and in particular, how to "tweak" them. Think about ways of constructing new groups (or rings, or fields) from old ones (e.g. by taking the algebraic closure); forcing is a way of building new models of ZFC from old ones. In particular, we can use forcing to do each of the following:

  • Given a model of ZFC in which CH is true, build a new model of ZFC in which CH is false.

  • Given a model of ZFC in which CH is false, build a new model of ZFC in which CH is true.

This is the proof that CH is independent from ZFC: the ZFC axioms alone aren't enough to prove or disprove it.

Now we come to the second key concept: absoluteness. An absoluteness result roughly is a statement of the form:

If $V, W$ are models of ZFC which are [related to each other somehow] and $\varphi$ is a [statement of some form], then $V$ and $W$ agree about whether $\varphi$ is true.

It turns out that a very strong absoluteness principle holds true for forcing extensions:

  • (Shoenfield absoluteness) Suppose $\varphi$ is a $\Pi^1_2$ sentence. If $V$ is gotten from $W$ by forcing, and $\varphi$ is true in either $V$ or $W$, then $\varphi$ is true in both $V$ and $W$.

(Actually Shoenfield absoluteness says even more than this.)

In other words, the truth/falsity of a $\Pi^1_2$ sentence can't be changed by forcing. I'm not going to explain what $\Pi^1_2$ is here, since it's rather technical, but it easily encompasses the vast majority of mathematics done today, even at the research level (for example, all Millenium Problems except the Hodge conjecture are $\Pi^1_2$; the Hodge conjecture is probably also $\Pi^1_2$, but I'm not sure since it's scary-looking).

Now combine the three bullet points above: CH can be "turned on" and "turned off" by forcing (Joel David Hamkins calls such statements "buttons"), and any sentence of everyday mathematics - being $\Pi^1_2$ - can't be changed from true to false or from false to true by forcing. So we get:

If $\varphi$ is a sentence of everyday mathematics provable from ZFC + CH, or provable from ZFC + "not CH", then $\varphi$ is just outright provable from ZFC.

Noah Schweber
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    An improvement in one direction: Any $\Pi^2_1$ sentence (in particular any sentence of second-order arithmetic), if provable in ZFC+CH, is provable in ZFC. The reason is that you can force CH without adding reals. – Andreas Blass Mar 23 '17 at 13:56
  • @AndreasBlass Indeed, that's true. And for the OP, the dual fails: "There is a nonconstructible real" is a $\Sigma^1_3$ sentence provable from ZFC + "not CH", but not from ZFC, and $\Sigma^1_3$ is vastly weaker than $\Pi^2_1$. – Noah Schweber Mar 23 '17 at 14:00
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    Nice post, Noah. One correction: the statements that can be turned on and then off repeatedly are the switches. A button, in contrast, is a statement that can be made true in such a way that it cannot subsequently be made false in a further extension. "You cannot unpush a button". – JDH Apr 03 '17 at 15:36
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In the theory of $C^*$-algebras CH is relevant. See for example Set theory and $C^*$-algebras.

EDIT: see also Continuum Hypothesis $\iff$ ??

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