In On the Question of Absolute Undecidability, Peter Koellner investigates whether it is possible to prove or disprove $V = L$ using (EDIT: both first and second-order) reflection principles, ie. statements of the form*

$$V \vDash \varphi(A) \to \exists \alpha \ V_\alpha \vDash \varphi^\alpha(A^\alpha)$$

and shows that it cannot be done.

Starting from $ZF$, can these principles be used to prove or disprove the Axiom of choice or some of its weaker variants (eg. Dependent, Countable choice)?

* definition on page 13.

  • $\begingroup$ Although this is unrelated to the question on AC, perhaps the Robert's paper "A strong reflection principle" would be relevant. In this paper, there is a formulation of a reflection principle, of similar form, that implies the existence of 1-extendible cardinals (so it's incompatible with V=L). $\endgroup$
    – Yair Hayut
    Jan 12 '20 at 14:49
  • $\begingroup$ That seems to be an interesting paper. Bookmarked and saved in the Wayback Machine, thanks. $\endgroup$ Jan 12 '20 at 15:07

The question seems to me asking if sufficiently large cardinals defined by indescribability properties will prove the axiom of choice or its weak variants hold below such cardinals. (Disproving is moot since these are consistent with $V=L$.)

The answer is negative, but more complicated. First of all, we can violate any sort of choice "on a small set", well below our large cardinal, and that will preserve the reflection properties. So the question now is if we can make the failure in some sense large. And indeed we can. With Yair Hayut we developed a basic method for lifting elementary embeddings to symmetric extensions and we showed that it's consistent relative to large cardinals that there is a critical point whose successor is singular.

It's not hard to check that this reflects down, and that indeed this critical point satisfies any reflection principle we wanted (it was previously a supercompact cardinal, after all).

We continue research in this direction, and I hope we will have new results to announce soon.

  • $\begingroup$ This doesn't make sense to me. If reflection principles were a consequence of ZF, why would anybody bother trying to prove $V = L$ with them? Also, on page 14, Koellner says "This [reflection] principle yields inaccessible cardinals, Mahlo cardinals, weakly compact cardinals and more". Maybe, in my question, I mischaracterized the reflection principles? (But I've taken the definition from the paper.) Or maybe you only considered first-order reflection principles? Is that I also ask about second-order reflection something that I should have included in my question? $\endgroup$ Jan 12 '20 at 12:54
  • 1
    $\begingroup$ I will take a closer look. But just like Koellner says, you need to specify the language and logic in order to determine the extent of your Reflection principles. You specified nothing so I assumed you meant first order (which indeed is too weak to decide something like V=L). Ask, and ye shall receive an answer; ask a precise question, and ye shall receive a precise answer. $\endgroup$
    – Asaf Karagila
    Jan 12 '20 at 13:08
  • 1
    $\begingroup$ I've edited to reflect what I think you're asking. $\endgroup$
    – Asaf Karagila
    Jan 12 '20 at 13:20
  • $\begingroup$ And I've edited the question. Thank you. $\endgroup$ Jan 12 '20 at 13:23

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