The Alephs aren't fields. Finite fields have absolutely nothing at all to do with this. They're apples and rutabagas. — fishfry
The cardinality of an infinite set ("aleph") is indeed formally not a Galois field. What I was saying, isn't a formal argument. I was just pointing out an uncanny similarity, i.e. just some kind of pattern:
[1] arithmetic is allowed in finite fields of prime powers only. This rule creates gaps between the field sizes.
At the same time, arithmetic is allowed in the integers (or rationals), i.e. a set with cardinality of aleph-0, and is also allowed in the reals (or computable numbers or similar), i.e. a set with cardinality of aleph-1.
[2] Continuum hypothesis. There is an insurmountable gap between aleph-0 and aleph-1. There are no infinite cardinalities in between.
As far as I am concerned, there is an uncanny similarity between the gaps in between finite calculation fields and the gaps between infinite calculation fields. Furthermore, with aleph[k]=2^aleph[k-1], the aleph[k] cardinalities are also prime powers; which is another uncanny similarity.
Of course, formally proving the link between both phenomena, is something else altogether. That is obviously another exercise.
One can Google around. A lot of work has been done by Woodin, Hamkins, and other contemporary set theorists. Work on CH has been ongoing for decades. It just doesn't make the mainstream news. — fishfry
Well, of course, people are still working on CH. I was just pointing out how "rigged" the game is. The context to work in, is really difficult. The proof that any axiomatization of CH is independent of ZFC, makes doing something meaningful, really hard. The structure required is incredibly annoying. You need other axioms, but which must still have a real provability distance from CH, because otherwise you would just be axiomatizing the final result itself ... Well ...
One big new idea is Woodin's Ultimate-L. It's so new and so technical it doesn't have a Wikipedia entry. This MathOverflow thread has some references. Nothing in this topic is comprehensible to laymen, just mentioning it since it's the state of the CH art. — fishfry
Apparently, the current status is:
How Woodin changed his mind: new thoughts on the Continuum Hypothesis.
This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true.
Well, yeah. My own intuition is in favour of CH. So, I obviously like Woodin's new approach better.
This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.
That explains why there is no Wikipedia entry for Woodin's work yet. They are not supposed to report on proofs that still have gaps in them. A proof with existing remaining gaps is a research hypothesis and not something to be propagated as "finished work" in places like Wikipedia.
If you look at the subtleties he needs in order to make his point:
[1] A Reinhardt cardinal, [2] An n-huge cardinal, [3] A huge cardinal, [4] An extendible cardinal, [5] A supercompact cardinal, [6] A superstrong cardinal, [7] A Woodin cardinal, [8] A measurable cardinal, [9] A (strongly) inaccessible cardinal.
The complexity proposed is substantially beyond what is done in existing, established theories. Therefore, what he is doing, looks quite ... ambitious.
Then, he writes:
As mentioned in the article, too much would have to be said to give an argument as to why the Axiom of Projective Determinacy, PD, should be accepted.
There, he is hitting the core of the problem. In fact, there should be no argument as to why axiom PD is accepted. If he does that, it is not an axiom. So no, all he needs to do, is to prove that there is a real distance, a serious series of transformative derivation steps from PD to CH.
There is no such thing as the quality of an axiom, because that implies "justification" of an axiom, which is exactly what is not allowed. All we need to see, is that CH is not trivial from PD. That would be enough.
I think that Woodin's work is interesting, but before further digging into the nitty-gritty details of the pyramid of vocabulary on which his work rests, I will just let him first complete the gaps in his argument!
;-)
Another idea is Hamkins's set-theoretic multiverse.
http://jdh.hamkins.org/themultiverse/ — fishfry
The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer.
In my opinion, Hamkins' work sounds less promising than Woodin's work, with which I can somehow identify. Hamkins does not talk about juxtaposing a set of axioms provably independent from ZFC from which there exists a noticeable derivation distance from CH, but from which he successfully derives CH. Therefore, I cannot identify with Hamkins work.
Here's an accessible article that surveys the modern developments.
https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis — fishfry
He writes:
How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say.
Well, CH is not unsolvable. It is just unsolvable from ZFC. That does not mean that it would be unsolvable from anything else. For example, it is trivially solvable from itself. However, that is not what we want. We want some larger distance between CH and the X someone would solve it from. This X needs to be independent from ZFC, and that is a problem, because ZFC is extremely powerful. So, finding such X is not easy.
Gödel, however, [...] taking the view that his incompleteness theorems, though they show that some provably undecidable statements do exist, have nothing to do with whether the continuum hypothesis is solvable or not.
Yes, of course!
what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.
Yes, X needs to be independent from ZFC (not easy to achieve) and still have a real distance from CH. Next, CH needs to be proven from X. That is what needs to happen, but as I argued before, that is clearly not easy. Furthermore, it is also a very counter-intuitive procedure to follow.