Statement copy in the gpt 6 astra pro collection (Math-166)

Reinhardt Cardinals without Choice

Definitions and mathematical statement

Use the first-order language of set theory enlarged by a unary function symbol $j$. The theory $\mathrm{ZF}(j)$ includes the indicated ZF axiom schemes in that expanded language; this choice matters because formulas may mention $j$. Require $j$ to be nontrivial and $\Sigma_1$-elementary: \[ \varphi(a_1,\ldots,a_m)\ \longleftrightarrow\ \varphi(j(a_1),\ldots,j(a_m)) \] for each $\Sigma_1$ formula in the membership language. The theory specified by the source derives the full elementarity scheme. Its critical point, the least ordinal moved by $j$, is a Reinhardt cardinal. The target is the consistency of this exact theory, denoted ZFR, not a theory with the axiom of choice added. Consistency means that no contradiction is derivable by a finite formal proof; a relative-consistency result must state its additional assumptions.

Formulation and concept references: [S1].

Short English statement

Can set theory without the axiom of choice consistently admit a nontrivial elementary self-embedding of the entire universe, with the axiom schemes specified in ZFR?

Sources

• S1 F. Schlutzenberg, Reinhardt cardinals and iterates of V, Ann. Pure Appl. Logic 173 $2022$. https://arxiv.org/abs/2002.01215. Catalogue formulation source.
• E1 Marwan Salam Mohammd, Reinhardt cardinals and eventually dominating functions, Archive for Mathematical Logic 65 $2026$, 691--699. https://doi.org/10.1007/s00153-026-01018-2. Author manuscript: https://arxiv.org/html/2505.00637v1 (1 May 2025), Theorem 2.2 and §§3,5. Manuscript consulted and publication metadata checked 24 September 2026; the Choice-dependent argument in §5 is not applied to ZFR.