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.)



Classifying set theories whose standard models sharing the same ordinals are equal

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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.


