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

Question

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.)

Answer 1

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.

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.

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 implicitly yields a negative answer to Friedman's question. See Taranovsky's answer below for a proposed striking generalization of Schultzenberg's construction.

Answer 2

$\newcommand\Ord{\text{Ord}}\newcommand\Z{\mathbb{Z}}\newcommand\Q{\mathbb{Q}}$This is a nice question, and I don't have much to say about it, except that I did want to mention that it is important in your account that you are talking about standard models, which I take to mean well-founded or transitive models. If you dropped that, the phenomenon would disappear, for the following reason.

Theorem. For any consistent theory $T$ extending ZF, there are two models of $T$ with the same ordinals, such that they are not isomorphic by any isomorphism fixing the ordinals.

Proof. Let $M$ be any countable computably saturated model of $T$. It follows that the natural numbers of the model $\omega^M$ are nonstandard and have order type $\omega+\Z\cdot\Q$. The ordinals of $M$ have type $\Ord^M$, which is the same as $\omega^M\cdot\Ord^M$, by an internal isomorphism, and this has order type $(\omega+\Z\cdot\Q)\cdot\Ord^M$. It follows that there is an order-automorphism $\pi:\Ord^M\cong\Ord^M$ that shifts the ordinals within the nonstandard parts of these blocks of $\omega^M$ by one (or one can make more complicated automorphisms). That is, we shift all the $\Z$-chains by one. Since $\Ord^M\subset M$, we may extend $\pi$ to an isomorphism $\pi:M\cong N$ to some model $N$, where $M$ and $N$ have exactly the same ordinals, but where many ordinals $\alpha$ that are even in $M$ are odd in $N$ and vice versa (this will be true exactly for the ordinals that are a nonstandard natural number successor of the largest limit ordinal below them). Thus, $M$ and $N$ have exactly the same ordinals, but are not isomorphic by any isomorphism fixing those ordinals. QED

In particular, if ZFC is consistent, then there can be models of ZFC+V=L that have exactly the same ordinals, but think different things are true of those ordinals.

Answer 3

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 (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.