TITLE: Classifying set theories whose standard models sharing the same ordinals are equal
QUESTION [14 upvotes]: 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.)

REPLY [21 votes]: 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.