Suppose is a ctm and . Must have an outer model of which is cardinal preserving and not a set forcing extension? Number Theory is full of problems which seem completely out of reach. And one can easily generate such problems in a foundational guise.
I have a couple of questions about Hugh’s question.
1. The question as formulated involves both forcing extensions and general (outer) extensions. Are there appropriate formulations that
a. Do not refer to forcing at all. b. Given any ctm , refer only to forcing extensions.
2. What are the known natural conditions on that are known to be sufficient? Also for variants in 1 above.
With regard to Hugh’s second statement, I think he is proposing an informal family of questions that are at present all intractable. I and I think others would like to get a better sense of just what this informal family of questions is or is like.
Concerning Hugh’s questions involving big numbers. I remember raising on the FOM email list this kind of question many years ago:
Is the -th digit in the binary expansion of zero?
Here say is some ridiculously large Ackerman number, or whatever. This gives us intractable $\Delta_0$ sentences which can be completely solved with a computer of absurd size. I conjectured that there is no proof or refutation of this question in ZFC + large cardinals in any remotely reasonable sized proof.
Continuing from my earlier messages that try to get an understanding of maybe how AxC follows from some foundational important sense of “set theoretic maximality”.
I need to present more ideas to make further progress, but before moving on to this, let’s pause and see what we have discovered already.
It appears that we have stumbled on a very fundamental class K of set theoretic statements with the following properties.
PRESENTATION OF K. K consists of all sentences of set theory of the form “a given sentence in first order predicate calculus with equality has a model on any infinite set domain”.
- Every set can be linearly ordered.
- Every infinite set has a one-to-one function from into .
- For every infinite A, there is a one-to-one function from into .
It is well known that 2 is equivalent to AxC over ZF. (1 and 3) does not imply 2 over ZF. 1 implies 3 and 3 implies 1 are not provable in ZF.
Also notice that these examples are in K for UNIVERSAL SENTENCES. Also note that 1 is in K’ for UNIVERSAL SENTENCES WITH NO FUNCTION SYMBOLS. Thus we have two important fragments of K here.
We have the following main results for the original full K.
- In a weak fragment of ZFC we can prove that the set of true elements of K is arithmetical – in fact, complete .
- Some elements of K are provably equivalent, in a weak fragment of ZF, to AxC.
- Some elements of K are provable in a weak fragment of ZFC, not provable in ZF, and do not imply AxC over ZF.
- In fact, in a weak fragment of ZFC, every element of K is provably equivalent to a Pi01 sentence.
But note that satisfiability in infinite models of universal sentences with no function symbols in infinite models is decidable (due to Ramsey). So for this fragment, we get the sharper
1′. Every sentence in the fragment of K is provable or refutable in a weak fragment of ZFC. 2. Same. 3. Same. 4′. Same as 1′.
A foundational program is to gain a complete understanding of K and its interesting fragments. What does this entail?
It would appear that there are two orthogonal components to truth of elements of K. One is the purely”combinatorial” component. The other is the purely “set theoretic” component. For this reason, it is probably best to work over ZF* = ZF + the true sentences. Of course, we can define important fragments of K, where every element is provable or refutable in a weak fragment of ZFC – in which case the purely “combinatorial” component is trivial. But generally, we want to work over ZF*. We seek
i. Interesting necessary or sufficient conditions on elements of K to be provable in ZF*.ii. Interesting necessary or sufficient conditions on elements of K to imply AxC over ZF*. iii. Interesting necessary or sufficient conditions for determining whether one element of K implies another over ZF*.
For “innocent” enough fragments of K, we can realistically hope to get precise necessary and sufficient conditions, and associated decision procedures. Also, in general, robustness would be very good to have – that for present purposes, ZF behaves exactly like weak fragments of ZF.
There is an important variant of K which we call K’.
PRESENTATION OF K’. K’ consists of all sentences of set theory of the form “a given sentence in first order predicate calculus with equality has a model on any nonempty set domain”.
We have the same 1-4 above, and also 1′-4′. BUT, there is a problem with Examples 2,3. IN FACT, is there an instance of K’ which is provably equivalent to AxC over ZF?
It appears that the answer is yes.
*) Every nonempty set D is in one-one correspondence with disjoint union B disjoint union C, where B,C are subsets of A.
*) is equivalent to AxC, and lies in the purely universal part of K’.
QUESTIONS. Are the elements of K,K’ provably equivalent over ZF? Are the purely universal elements of K,K’ provably equivalent over ZF?
In the next message I hope to bring maximality back into the picture, while maintaining the fundamental character of the investigation.