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.)

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    $\begingroup$ +1. Very interesting question. But: every extension of a "canonical" theory is also canonical, according to your definition, and I wouldn't expect this to be a feature of canonicity. So is this the right terminology? How about: $T$ is ordinal-categorical, or models of $T$ are determined by their ordinals, or...? $\endgroup$ – Joel David Hamkins Dec 17 '14 at 0:03
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    $\begingroup$ I thought of that. I couldn't think of a good term, but I like ordinal-categorical. I will edit the post. $\endgroup$ – Jesse Elliott Dec 17 '14 at 0:04

An old theorem of Harvey Friedman answers the question:

Theorem. If $T$ is an r.e. extension of KP (Kripke-Platek set theory) and $T$ is $\alpha$-categorical for all countable ordinals $\alpha$, then $T$ proves $V = L$.

In the above, "$T$ is $\alpha$-categorical" means that $T$ has at most one standard model of ordinal height $\alpha$. Friedman's result appears as Theorem 6.3 of his 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:

1. The theory $ZF + V = L(0 ^{\#})$ need not be ordinal categorical since by Friedman's theorem, two standard 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 $\Delta^1_3$.

2. It is open whether the conclusion of Friedman's theorem continues to hold if its hypothesis is weakened to "$T$ is $\delta$-categorical", where $\delta$ is the ordinal height of the Shepherdson-Cohen minimal model of set theory.

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    $\begingroup$ This is my way of giving another plus one: thanks Ali! $\endgroup$ – François G. Dorais Dec 17 '14 at 21:17
  • $\begingroup$ Thanks! I don't have immediate free access to the book. Can you state the definition of "$\alpha$-categorical"? $\endgroup$ – Jesse Elliott Dec 18 '14 at 6:41
  • $\begingroup$ @Jesse Elliot: I have included the def. in my new edit. $\endgroup$ – Ali Enayat Dec 18 '14 at 7:15

$\newcommand\Ord{\text{Ord}}\newcommand\Z{\mathbb{Z}}\newcommand\Q{\mathbb{Q}}$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.


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