What is known about the theory

($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"?

By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of reals is called $\kappa$-Suslin if it is the projection of a closed subset of $\omega^\omega \times \kappa^\omega$, and Suslin if it is $\kappa$-Suslin for some well-ordered cardinal $\kappa$.

Shoenfield proved the converse: assuming ZF, every ${\bf \Sigma}^1_2$ set of reals is Suslin (and in particular is $\aleph_1$-Suslin.) Every wellordered set of reals $A$ is Suslin (and in particular is $\left|A\right|$-Suslin) so the theory ($\ast$) is inconsistent with AC.

I was able to show that ($\ast$) is consistent relative to ZFC + "there is a generic Vopěnka cardinal," where the definition of "generic Vopěnka cardinal" is obtained by weakening the definition of "Vopěnka cardinal" to allow the elementary embeddings between structures to exist in a generic extension of V, rather than in V itself. (See Bagaria, Gitman, and Schindler [1] for the analogous variant of Vopěnka's principle.) As is typical of such "virtual large cardinal" properties, generic Vopěnka cardinals are between weakly compact cardinals and $0^\sharp$ in terms of consistency strength.

Moreover, I showed that ($\ast$) + "$\Theta = \aleph_2$" is equiconsistent with ZFC + "there is a generic Vopěnka cardinal", where $\Theta$ is the least ordinal that is not a surjective image of the reals. (So in ZFC, the statement $\Theta = \aleph_2$ would be equivalent to CH.)

I have two historical questions and one mathematical question:

  1. Has the theory ($\ast$) been considered before?

  2. Was any consistency strength upper bound for ($\ast$) already known?

  3. Can we obtain any nontrivial consistency strength lower bound (i.e., better than ZF itself) for ($\ast$)?

A couple of facts are worth mentioning here:

(A). Martin and Solovay proved that ZFC + MA + $\neg$CH + "$\aleph_1^L = \aleph_1$" implies that every $\aleph_1$-Suslin set of reals is ${\bf \Sigma}^1_2$.

(B). ZF + "$\Theta = \aleph_2$" implies that every Suslin set of reals is $\aleph_1$-Suslin, but this hypothesis is inconsistent with the hypothesis of Martin and Solovay.

[1] J. Bagaria, V. Gitman, and R. Schindler. Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Archive for Mathematical Logic, 56(1-2):1-20, 2017.

  • $\begingroup$ I am guessing that you also want DC there, or at least DC(R), right? $\endgroup$ – Asaf Karagila Jun 19 '18 at 7:14
  • $\begingroup$ @Asaf I would be interested in the answer either way (as written, or with DC). The equiconsistency that I mentioned holds with or without DC added. The model of ($\ast$) + "$\Theta = \aleph_2$" that I obtain from a model of ZFC + "there is a generic Vopěnka cardinal" satisfies DC, but the reverse direction does not require any kind of choice principle. $\endgroup$ – Trevor Wilson Jun 19 '18 at 15:31
  • $\begingroup$ Well. If $\omega_1$ is regular, the lower bound would be at least an inaccessible, almost automatically. My guess you can get probably get something like some large cardinal using some absoluteness. $\endgroup$ – Asaf Karagila Jun 19 '18 at 21:59
  • $\begingroup$ @Asaf Do you mean that ZF + ($\ast$) implies $\aleph_1^V$ is strong limit in $L$? I don't even see why it would imply that $\aleph_1^L < \aleph_1$. In light of the result of Martin and Solovay I mentioned, we would need to do something with $\kappa$-Suslin sets for some $\kappa > \aleph_1$, but I don't know what. $\endgroup$ – Trevor Wilson Jun 19 '18 at 23:52

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