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Same disclaimer as v64: my background is also mathematics, but not in this field.

As he says, what was proved so far is that: a_0 < a_1 <= p <= t <= 2^(a_0) The first inequality is by definition (a_1 is "the smallest cardinal greater than a_0), the second one is a theorem by Hausdorff, the third one is supposedly easy, the last one is trivial. Also, if the first one is an equality, then the second one is, too.

The continuum hypothesis claims that they are all equal; that's been proved by Cohen to be independent of what you call "A" (and is generally called ZFC), by "Forcing" which is not easy, technical, and has won Cohen a Fields medal... And you're right in saying that what is proved in an axiom set is true in any larger axiom set.

However, the main point is that "we cannot have p<t, and hence p=t" is false; that is the whole point of indecidability. A statement can be "true", "false" or "not decidable". Many people apparently expected the p=t question to fall into this last group (with some enlarged axiom set "A+" permitting to prove the strict inequality).

And no, the statement "The continuum hypothesis is undecidable in ZFC" is proved within ZFC, otherwise it wouldn't have been proved at all... To give a trivial counter-example, as a corollary of Cohen's proof, the axiom choice "ZFC+CH" (ZFC + continuum hypothesis) is as valid an axiom set as ZFC is. Within it, you can certainly prove that "ZFC implies CH" since... CH is simply true.

Thanks.

I think the idea I had in my head was this: If p=t or p!=t is decidable in ZFC+AoC, then it must be that p=t because otherwise that contradicts that the continuum hypothesis is undecidable.

So a better way to sum up this result is: "The question of whether p=t or p!=t is decidable in ZFC+AoC. And oh yeah, they happen to be equal."

How does that sound?

I hate to be that guy but a friendly correction: ZF + AoC is what ZFC refers to.

I'm not sure I follow your line of thinking. If it's decidable it could either be p=t or p!=t, no? Ie. If the contium hypothesis can be proved in ZFC, it could be either.

Afaik, Cohen proved, by using Gödel, that in fact p=t or p!=t is not provable in ZFC. Ie contium hypothesis is not provable in ZFC.

As I understand, Malliaras and Shelah have proved p=t ie contium hypothesis is true by using set and model theory. As I understand they are not relying solely on ZFC to prove this since that would disprove Cohen and my understanding is they do not.

Further, I understand this result to mean there can only be two infinities which can be said to be different, the cardinal of the countable rational numbers and the cardinal of the uncountable real numbers. But I don't really understand any of this shit so it is likely I've got it wrong.

Disclaimer: I have a math background, but this isn't my field.

The continuum hypothesis was proved to be independent of ZFC. From glancing at the first few paragraphs of the actual paper [2], it's been established for decades that the following inequality holds:

aleph_1 <= p <= t,

and that aleph_1 = p can be shown to imply aleph_1 = t. Note that the inequalities aren't strict inequalities, and this proof is essentially saying that they were equivalent all along.

[1] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

[2] https://arxiv.org/pdf/1208.5424.pdf

The article is misleading on this point. The Continuum Hypothesis says that it is undecidable in the standard axioms whether there is a cardinality between countable and the continuum (i.e., the cardinality of the real). This work shows that p and t are different, which also means that t is of greater cardinality than the reals.

Oh, thank you. So the article just threw in Continuum Hypothesis because different infinities 'n stuff.

But what is confusing to me is that the article suggested p and t are defined as collections of subsets of the natural numbers. The power set of the natural numbers has the same cardinality as the reals, right? Since p and t are both subsets of this power set, they must have cardinalities below the cardinality of the reals.

Wait I'm confused. When you say p and t are different, you mean they are cardibalities of different sets but still equal in "size", right? Because I've understood they've proved p to be of equal "size" of t, which I guess means they have a one to one relation? Fuck this is confusing

What is p and t? You say that this work shows that p and t are different. What two entities are proven to be equal then in this work?

p and t are implicitly defined cardinalities (i.e., the smallest cardinalities with given properties). See Definition 1.1 of the source article (https://arxiv.org/pdf/1208.5424.pdf)