Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For example, if $T$ proves $V = L$, then $T$ is ordinal-categorical. I think the same is true if $T$ proves $V = L(0^\sharp)$.

Is there anything else that can be said about such theories $T$? In particular, is there a way to find an axiom or axiom schema A such that all ordinal-categorical theories $T$ are precisely the (recursively axiomatizable) extensions of ZF+A? If this is not known, is finding such an A a goal of the inner model program?

(Note: In the original post I used the word "canonical" instead of "ordinal-categorical.)
['https://mathoverflow.net/questions/190902', 'https://mathoverflow.net', 'https://mathoverflow.net/users/17218/']
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> In this edit, the statement of Friedman's theorem is reformulated (the previous formulation was incorrectly stated). Thanks to Dmytro Taranovsky and Farmer Schultzenberg for pointing out the blooper. *See also Remark 2 for recent progress (January 2023) on this topic by Schultzenberg and Taranovsky.*
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An old theorem of Harvey Friedman answers the question:

**Theorem.** *Under a mild set theoretical hypothesis* $\mathrm{H}$ (see Note 1 below), *there is a cofinal subset* $U$ *of* $\omega\_1$ (see Note 2 below) *such that if* $T$ *is any r.e. extension of* $\mathrm{ZF + V \neq L}$ *that has a countable transitive model* $M$ *of height* $\alpha$, *then* $T$ *has another model* $N \neq M$ *of height* $\alpha$.

**Note 1.** The mild set theoretical hypothesis $\mathrm{H}$ above asserts that there is an ordinal $\alpha$ of uncountable cofinality such that $V\_{\alpha} \models \mathrm{ZF}$. Thus the existence of a strongly inaccessible cardinal implies $\mathrm{H}$.

**Note 2.** $U$ consists of $\omega\_1 \cap G$, where $G$ consists of ordinals $\alpha$ such that $L\_{\mathrm{n}(\alpha)} \cap V\_{\alpha}=L\_{\alpha}$. Here $\mathrm{n}(\alpha)$ is Friedman's notation (in the paper below) for the next admissiable ordinal after $\alpha$, i.e., the least admissible ordinal greater than $\alpha$; this ordinal is denoted $\alpha^{+,\mathrm{CK}}$ [in this MO question of Taranovsky](https://mathoverflow.net/questions/438081/minimum-transitive-models-and-v-l).

**Note 3.**  The above Theorem follows from putting together the proof of the hard direction of Theorem 6.2 together with the proof of Lemma 6.3.1 of Friedman's paper below:

*Countable models of set theories*, In A. R. D. Mathias & H. Rogers (eds.), **Cambridge Summer School in Mathematical Logic**, Lecture Notes in Mathematics vol. 337, Springer-Verlag. pp. 539--573 (1973).

Two Remarks are in order.

**Remark 1.** It is known that if the theory $\mathrm{ZF + V = L(0 ^{\#}})$ has transitive model of height $\alpha < \omega\_1$, then $\alpha \in U$; coupled with Friedman's above theorem, this implies that $\mathrm{ZF + V = L(0 ^{\#}})$ has more than one transitive model of height $\alpha$. Thus two transitive models of set theory of the same height could both believe that $0 ^{\#}$ exists, and yet they might have distinct $0 ^{\#}$s. Recall that the complexity of $0 ^{\#}$ is $\Delta^1\_3$.

**Remark 2.** In the introduction to his aforementioned paper, Friedman posed the hitherto open question of whether the conclusion of the theorem above holds for $\alpha$ = the ordinal height of the Shepherdson-Cohen minimal model of set theory (it is known that $\alpha \notin U$). Schultzenberg's construction to [this MO question of Taranovsky](https://mathoverflow.net/questions/438081/minimum-transitive-models-and-v-l) implicitly yields a negative answer to Friedman's question. See Taranovsky's answer below for a proposed striking generalization of Schultzenberg's construction.
For every c.e theory $T$ extending KP (Kripke-Platek) with a model $M$ of height $α<ω\_1$, the intersection of all such $M$ is a subset of $L\_{α^{+,\mathrm{CK}}}$. This holds since the existence of such $M$ is $Σ^1\_1(α)$. Every model $\text{ZF}+0^\#$ (or just $\text{KP}+0^\#$) of height $α$ includes a set outside of $L\_{α^{+,\mathrm{CK}}}$, so such theories do not have unique models for countable $α$.

The interplay of uniqueness and non-uniqueness — and the conditions to guarantee uniqueness — is an important theme in inner model theory. Theories such as $\text{ACA}\_0$ or the primitive recursive set theory have unique minimal $ω$-models, but then for c.e. theories extending $\text{ATR}\_0$, the intersection of all $ω$-models (if there are any) equals HYP. (HYP is the set of hyperarithmetic sets; it corresponds to $L\_{ω\_1^{\mathrm{CK}}}$, the minimal transitive model of KPω.) With $0^\#$, uniqueness of minimal transitive models fails. And going further, $L[M\_1^\#]$ does not satisfy $V=HOD$: Despite being an inner model, $L[M\_1^\#]$ lacks sufficient closure to iterate and identify the true $M\_1^\#$ (the sharp for a Woodin cardinal).

Back to the question, for every ordinal-categorical (as defined in the question) c.e. theory $T$, all transitive models are constructible, and for a model $M⊨T$ of possibly uncountable height $α$, $M⊂L\_{α^{+,\mathrm{CK}}}$ (uncountable $α$ can handled using countable elementary submodels). Thus, if $On^M$ is a cardinal (and $T⊢\text{KP}$), then $M ⊨ V=L$.

Surprisingly, however, we have the following, which adapts Farmer Schlutzenberg's answer to my recent question [Minimum transitive models and V=L](https://mathoverflow.net/questions/438081/minimum-transitive-models-and-v-l?noredirect=1&lq=1) (the discussions in the two questions complement each other).

**Theorem:** There are ordinal-categorical c.e. theories extending $\text{ZFC} + V≠L$ that have arbitrarily large transitive models (assuming ZFC has arbitrarily large transitive models).

**Proof:** To get such models, we use existence of (non-trivial) $≤κ$-closed forcings with (as viewed from the generic extension) unique generics. Ordinarily, we might have $V[G]=V[G']$ (which interferes with defining our $G$ in $V[G]$), but we can get unique generics by encoding $G$ into choices for subsequent rounds of forcing and iterating $ω$ times (with full support). Next, for every ordinal $κ$ (including uncountable $κ$), the minimum transitive model $M$ of ZFC of height $>κ$ is pointwise definable with ordinals $<κ$ as constants. Fix a formula picking a $κ$ (as above) in $M$, and a parameter-free definable forcing as above. Using the pointwise definability of $M$ (with ordinals $<κ$ as constants) and the $≤κ$-closure, every dense open set has a parameter-free definable (in $M$) dense open subset. Thus, we can fix a choice of the generic $G$ using a computable schema as in the linked answer. Specifically, enumerate formulas; if a formula $φ\_0$ defines a dense open set here, then pick its first element (under some fixed parameter-free definable well-ordering), and then repeat with $φ\_1, φ\_2, ...$, picking the first element compatible with the previous ones. Our theory will be $\text{ZFC} + V=M[G]$ with $M$ and $G$ as above.

A remaining open question is whether there is an ordinal-categorical theory $\text{ZFC} + A + V≠L$ (with a transitive model) where $A$ is a single statement. A model of such a theory cannot be obtained by set forcing (unlike a schema, a statement in the generic extension would be forced by some condition in the poset), and so may require new (perhaps, still forcing-like) construction methods.