Reinhardt Cardinals without Choice

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.

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?

Research attempt

Scope check and current frontier (24 September 2026).

The source's §1.1 puts Separation and Collection in the language containing $j$, but requires elementarity only for the language of membership. Its full-elementarity theorem scheme does not make $j$ elementary for formulas that mention $j$ itself ([S1], §1.1). The unary-function presentation here is intertranslatable with the source's class-graph presentation by adding totality and functionality. Throughout this attempt, the nontriviality requirement in the supplied statement is retained.

Schlutzenberg's analysis of iterates is conditional on ZFR. In particular, individual sets being generic over an infinite iterate is not a construction of the whole universe as one set-forcing extension of that iterate ([S1], Theorems 3.14 and 3.19). His non-definability result also obstructs choosing the embedding by a formula from the indicated inner-model data ([S1], Theorem 3.21). A recent published obstruction is Mohammd's theorem: if $\delta>\kappa$ is regular and $j(\delta)=\delta$, no member of the range of $j$ that is a function $\delta\to\delta$ eventually dominates $j\!\upharpoonright\!\delta$. Its proof uses regularity, and its final inconsistency argument adds Choice ([E1], Theorem 2.2 and §5). These results supply useful tests, not a consistency or inconsistency proof for the exact choice-free target.

Attack map.

For a positive answer, one could try to inflate a rank-into-rank embedding to an entire model, or assemble increasingly accurate models of finite fragments. The respective missing assertions are closure under all expanded-language Replacement instances and a genuine finite-satisfiability construction from a separately specified assumption. Pure membership elementarity alone does not provide either assertion. For a negative answer, one could try to manufacture a dominating function prohibited by the recent theorem. This requires both a regular fixed domain and membership of the proposed dominator in $\operatorname{ran}(j)$, not just an arbitrary pointwise upper bound.

The attempt below pushes the rank-cut construction and the domination attack against the same object, the critical sequence. It proves an explicit obstruction to the first construction and gives a conditional example that breaks a tempting extension of the second. These are checks on possible arguments, not a claimed advance in consistency strength.

Attempt I: the critical sequence is a set, not an external list.

Work temporarily inside a hypothetical model of ZFR. Write $\kappa=\operatorname{crit}(j)$ and define \begin{equation}\label{wsref-0-72-61-31-33-35-37-3a-63-72-69-74-69-63-61-6c-2d-73-65-71-75-65-6e-63-65} c(n)=\kappa_n=j^n(\kappa),\qquad \lambda=\sup_{n<\omega}\kappa_n. \end{equation} Recursion followed by Replacement in the expanded language produces the function $c$ as a set. The ordinals $\kappa_n$ increase strictly, $j$ fixes $\omega$ pointwise, and pure-language elementarity applied to this set function gives \[ j(c)(n)=j(c(n))=\kappa_{n+1},\qquad j(\lambda)=\sup\operatorname{ran}(j(c))=\lambda. \] Thus $\operatorname{cf}(\lambda)=\omega$. No continuity of $j$ at arbitrary cofinalities has been used; the displayed equality is justified specifically by a set sequence of length $\omega<\kappa$.

A first proposed construction is to treat $V_\lambda$, with the restricted embedding, as a smaller universe. In fact, this gives a precise failure.

Rank-cut obstruction. Let $\theta$ be a limit ordinal in an ambient model of ZF, and let $h:V_\theta\to V_\theta$ be an elementary embedding of the membership structures with critical point $\kappa$. If \[ \sup_{n<\omega}h^n(\kappa)=\theta, \] then $(V_\theta,\in,h)$ fails an instance of Replacement in the expanded language, even if its membership reduct satisfies all of ZF.

Proof. For every $n\in\omega$, the finite sequence recording the first $n$ iterates belongs to $V_\theta$: its entries have ranks bounded below the limit ordinal $\theta$, and finite coding adds only finitely many ranks. Consequently the formula saying that $y$ is the $n$th iterate of $\kappa$ is a functional expanded-language formula on $\omega$ in this structure. Replacement would place \[ a=\{h^n(\kappa):n<\omega\} \] in $V_\theta$. But this set of ordinals has rank $\sup_n(h^n(\kappa)+1)=\theta$, so it is not in $V_\theta$. This is the required failed instance.$\square$

This is subtler than absence of the restriction of the embedding to a set. For each $a\in V_\theta$, choose $\alpha<\theta$ with $a\subseteq V_\alpha$. Elementarity gives $h[V_\alpha]\subseteq V_{h(\alpha)}$. Hence the external graph $h\!\upharpoonright\!a$ has bounded rank below $\theta$, and lies in $V_\theta$. Thus this natural local amenability property can hold while expanded Replacement fails. Obtaining every one-step restriction of $h$ does not obtain the countable iteration function inside the rank cut.

In the hypothetical global ZFR model, $j\!\upharpoonright\!V_\lambda$ exists by expanded Replacement and is a pure elementary self-embedding of $V_\lambda$: relativize any fixed membership formula to $V_\lambda$ and apply $j$, using $j(V_\lambda)=V_\lambda$. The obstruction shows exactly why this rank cut is not itself a ZFR model. There is no inconsistency in the larger model: there $c$ and $\lambda$ are legitimate sets of higher rank.

Attempt II: can increasing the height repair the construction?

The next proposal is to choose a height above $\lambda$, close it under successively more $h$-definable operations, and take a limit. The exact test for this procedure can be stated without any appeal to an informal notion of "enough closure."

Bounded-witness criterion. In an ambient model of ZF, suppose $\theta$ is a limit ordinal, $V_\theta\models\mathrm{ZF}$, and $h:V_\theta\to V_\theta$ is a set function. Then $(V_\theta,\in,h)$ satisfies expanded Separation and Replacement if and only if the following condition holds for every expanded-language formula $\psi(x,y,\vec p)$: \begin{equation}\label{wsref-0-72-61-31-33-35-37-3a-62-6f-75-6e-64-65-64-2d-77-69-74-6e-65-73-73} \begin{gathered} a,\vec p\in V_\theta,\quad (V_\theta,\in,h)\models\forall x\in a\,\exists!y\,\psi(x,y,\vec p) \\[-2pt] \Longrightarrow\quad \exists\beta<\theta\quad (V_\theta,\in,h)\models \forall x\in a\,\exists y\in V_\beta\,\psi(x,y,\vec p). \end{gathered} \end{equation} Here the quantification over formulas is external; each instance is an ordinary assertion about the specified set structure.

Proof. Satisfaction for this set structure is available in the ambient ZF model. For $a\in V_\theta$, any subset of $a$ formed by ambient Separation has rank at most $\operatorname{rank}(a)$, and thus belongs to $V_\theta$. This proves expanded Separation. Under (ra1357:bounded-witness), ambient Separation on $V_\beta$ forms the range of the functional relation. This range is a subset of $V_\beta$ and belongs to $V_{\beta+1}\subseteq V_\theta$, proving the Replacement instance. Conversely, an internal range set has rank less than $\theta$ and therefore is contained in some $V_\beta$ with $\beta<\theta$.$\square$

Collection follows from these schemes without selecting one witness from each of arbitrarily many nonempty sets: associate to each input the least rank at which a witness exists, use Replacement for these uniquely determined ordinals, and collect a sufficiently large rank segment. Thus the criterion really tests the source's axiom convention, not a weaker "functional only" substitute.

Applying this test to the rank-cut proposal, the critical orbit supplies one failed bound. Increasing the height past $\lambda$ removes that particular failure, but says nothing about the next expanded formula or about arbitrary parameters at the new height. Moreover, an embedding on one rank segment does not come with a coherent elementary extension to a larger rank segment. Both the extension problem and all the bounds in (ra1357:bounded-witness) remain unproved. The criterion is an explicit audit of the desired construction, not an independent large-cardinal assumption from which a new relative consistency result has been obtained.

Attempt III: a dominator at the singular fixed point.

Return to the hypothetical global ZFR model and the set $c$ in (ra1357:critical-sequence). To try for a contradiction, remove the regularity hypothesis from Mohammd's domination theorem and use $\delta=\lambda$. The following computation disproves that proposed strengthening, conditionally on ZFR itself.

Define the set function $f:\lambda\to\lambda$ by \[ m(\alpha)=\min\{n<\omega:\alpha<\kappa_n\},\qquad f(\alpha)=\kappa_{m(\alpha)+1}. \] The minimum exists by cofinality of the critical sequence. This definition is in the pure membership language with the set parameter $c$; Replacement produces $f$. Set $g=j(f)$. Since $j(\lambda)=\lambda$ and $j(c)$ is the shifted sequence, applying pure elementarity to the defining formula gives \begin{equation}\label{wsref-0-72-61-31-33-35-37-3a-73-69-6e-67-75-6c-61-72-2d-64-6f-6d-69-6e-61-74-6f-72} g(\alpha)=\kappa_{n(\alpha)+2},\qquad n(\alpha)=\min\{n<\omega:\alpha<\kappa_{n+1}\}. \end{equation} For every $\alpha<\lambda$, strict order preservation yields \[ \alpha<\kappa_{n(\alpha)+1} \quad\Longrightarrow\quad j(\alpha)<j(\kappa_{n(\alpha)+1}) =\kappa_{n(\alpha)+2}=g(\alpha). \] Thus $g\in\operatorname{ran}(j)$ is a function $\lambda\to\lambda$ that strictly dominates $j\!\upharpoonright\!\lambda$ everywhere. The formula also handles endpoints $\alpha=\kappa_n$, because the defining inequality is strict. This rules out the proposed regularity-free contradiction. It is not a model of ZFR and not evidence that ZFR is consistent; it is a derivation about what any such model would contain.

At a regular fixed cardinal $\delta$, the analogous attempt to make an upper envelope from many embeddings has the opposite obstruction. For example, let $\mathcal E_\delta$ be the set of elementary self-embeddings of $V_\delta$, including the identity. It is definable from $V_\delta$ by the satisfaction relation for set structures, so $j(\mathcal E_\delta)=\mathcal E_\delta$. If all the suprema \[ F(\alpha)=\sup\{k(\alpha):k\in\mathcal E_\delta\},\qquad \alpha<\delta, \] were less than $\delta$, then $F:\delta\to\delta$ would satisfy $j(F)=F$ and would dominate $j\!\upharpoonright\!\delta$, because the restriction of $j$ is one of these embeddings. Theorem 2.2 would contradict this. Therefore this route cannot obtain the required bounded envelope. This is a special case of the more refined small-family obstruction already proved in [E1], §3, not a new inconsistency argument. Regularity bounds suprema indexed by ordinals below $\delta$; it does not bound a family of arbitrarily many embeddings without a justified size bound.

Stress test.

There are four essential logical safeguards. First, the proof that a parameter-free definable ordinal is fixed cannot be applied to the definition "the least ordinal moved by $j$": that formula contains $j$. If full expanded-language elementarity were added, the critical point would indeed be fixed, contradicting nontriviality, but that is a different theory. Second, the rank-cut argument uses an internally definable cofinal sequence. External countability or external singularity of a model's ordinal height is not by itself a violation of its Replacement scheme.

Third, compactness would give a consistency proof from actual models of every finite fragment, built in a stated metatheory. No such independent family has been constructed here. Assuming global ZFR to obtain its rank-reflection fragments would assume the desired consistency rather than establish a new lower assumption. Lastly, (ra1357:singular-dominator) uses only pure elementarity with the set parameter $c$; it does not accidentally assume that $j$ is elementary for its own defining predicate.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The rank-cut construction fails at a displayed expanded Replacement instance, even though all one-step restrictions of the embedding may exist internally. The bounded-witness criterion identifies the additional closure a repaired construction would have to provide. In the other direction, the explicit function (ra1357:singular-dominator) defeats the attempt to remove regularity from the recent domination obstruction. These elementary deductions are proved here; no priority claim is made for them.

A positive resolution still requires a model construction or a proved relative-consistency implication from a stated assumption. A negative resolution requires a finite contradiction in the exact choice-free theory. Neither has been obtained.

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.