941: New Stable Maximality/1

[Dmytro Taranovsky]

Dear Harvey Friedman,

Can you provide some intuition about the connection with the stationary reflection principle (SRP)?

Here is my attempt; perhaps you can make it more precise:
Order invariant relations (or properties) -- a set up to do set theory
E to be emulated -- a fragment of a model of set theory
N tail successor (NTS) -- corresponds to a symmetry of k-subtle cardinals
stable S -- a fragment of a model with k-subtle cardinals
maximality of S -- ensures that key structures in E are present in S

It would also be interesting to have a version with a strength between Con(k-subtle) and Con(k+1-subtle) for a given k.  One possibility may be to restrict NTS to change only up to k' largest values in the sequence (counting duplicates as a single value; k' is a new parameter).

Definition 6 (in your post) -- I think you meant GNTS rather than NTS.  (I am also interested about the intuition on the connection with k-Mahlo cardinals.)

A number of different schemas are equiconsistent with SRP.  Which forms of SRP do you find most useful/enlightening for the arithmetic incompleteness?

We have:

- k-subtle, k-almost ineffable, k-ineffable cardinals

- A basic axiomatization of finite iterations/tuples of reflective cardinals (in my paper "Reflective Cardinals" https://arxiv.org/abs/1203.2270 ) is equiconsistent with SRP.  Two reflective k-tuples S and S' satisfy the same statements with parameters in V(min(S union S')), which resembles your invariance under NTS.

- Existence of a nontrivial elementary embedding j:V->V (in ZFC with symbol j but without separation and replacement for j-formulas) is conservative over ZFC.  With the critical point axiom (existence of the least ordinal moved by j), it is conservative over SRP.  A weakening of the critical point axiom gives conservation over ZFC + {n-Mahlo: n<omega}.  These and other results are in my paper "Elementarily self-embeddable models of ZFC" https://web.mit.edu/dmytro/www/BTEE.htm .

Your NTS (and stability) reminds me of elementary embeddings, and your NTS -> GNTS may correspond to weakening the critical point axiom.

Sorry for the late reply.  I do wish there was more engagement by others with your postings and their interesting results.

Sincerely,
Dmytro Taranovsky
https://web.mit.edu/dmytro/www/main.htm

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20220805/5cafb3ff/attachment-0001.html>