How does it not contradict the Cohen's theorem?

[Demystifier]

Can someone explain this article?
https://www.quantamagazine.org/math...l&utm_source=facebook.com&utm_campaign=buffer
How is it related to the Cohen's theorem that continuum hypothesis is undecidable?
Is it an alternative way to compare infinities, or what?

 

[yossell]

Not an expert on this but...

(a) I see no reason why the paper should contradict or be in tension with Cohen's theorem. The main result of the paper is that t and p have the same cardinality. That's compatible with both the continuum hypothesis and its negation. The definitions of t and p is technical -- the paper's sidebar has an explanation which I couldn't improve!

(b) there's no new way to compare infinities -- it's only that the infinities of direct interest are not the continuum and aleph 1.

(c) If the continuum hypothesis were true, t = p would follow quickly. From the article: `mathematicians figured out that both sets are larger than the natural numbers.' From the definitions of t and p they are no bigger than the continuum. So if the continuum were the next size up t and p and the continuum would all be equal. Therefore: you were never going to be able to show that t and p were different sizes in set-theory. For that result would show that the continuum hypothesis was false in set-theory, contradicting Cohen's result. But they did not refute it.

(d) Still, t = p had never been shown to be *independent* of set-theory -- even with all the methods of forcing that had been developed. To everyone's surprise, the result has turned out to follow from set-theory.

(e) The very last part of the article I did find badly put and a little confusing.

"Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis. Still, the overwhelming feeling among experts is that this apparently unresolvable proposition is false: While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found."

I don't believe that mathematicians have thought this problem would settle the continuum hypothesis since Cohen's result.

The proposition is not apparently unresolvable: it is definitely unresolvable in standard set-theory. But standard set-theory may not be the final word on sets. Insofar as mathematicians venture this far, there are many who think that the continuum hypothesis is not true. The situation is like the axiom of choice -- it once seemed contentious and is now part of standard set-theory and thought of as correctly characterising the sets. In the future, perhaps some other axiom will become part of standard set theory, an axiom which negates CH. Moreover, it's not that there are other sets waiting to be found -- we know that aleph zero has a unique successor -- the cardinality of all well-orderings of the natural numbers. All that we don't know, and all that is unprovable in standard set-theory, is whether this set has the same cardinality as the continuum.

 

[Zafa Pi]

(b) there's no new way to compare infinities -- it's only that the infinities of direct interest are not the continuum and aleph 1.

The infinities of interest are those of t and p (which are =), and you say they are not Aleph1 and(?) c. Assuming the CH (which is consistent) they in fact are.
Maybe I'm not reading your sentence correctly.

we know that aleph zero has a unique successor -- the cardinality of all well-orderings of the natural numbers. All that we don't know, and all that is unprovable in standard set-theory, is whether this set has the same cardinality as the continuum.

Wow. I find this hard to believe since I think I can prove the cardinality is c.
For an infinite subset of N, order it with [1.ω) and order it's complement with [ω,α) where ω ≤ α ≤ 2ω. There are c infinite subsets of N. Many of these will have the same order type.
Perhaps you are thinking of the number of countable well order types.

 

[yossell]

I take your points Zafa,

I guess I should have said `it is not the case that the infinities of direct interest are aleph 1 and c.' As put, it looks as though I am making the too strong assertion that the cardinalites of t and p are not aleph 1 and c -- I hope that in the rest of the post it's clear that it's very much open whether the cardinalities of t and p are c.

Yes, you're right, I was confusingly thinking of the well orderings of N as types.

"and order it's complement with..."
Wow. I think you meant its. ;)