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Let $T_1$ and $T_2$ be two theories having the same set of symbols. Assume that any interpretation of $T_1$ is a model of $T_1$ if and only if it is not a model of $T_2$. Then:

$T_1$ and $T_2$ are finitely axiomatizable.

(i.e. there are finite sets of sentences $A_1$ and $A_2$ such that, for any sentence $S$: $T_1$ proves $S$ if and only if $A_1$ proves $S$, and $T_2$ proves $S$ if and only if $A_2$ proves $S$).

/The proof will be by contradiction; assume $T_1$ or $T_2$ are not finitely axiomatizable, then .....?/

Any one have any idea of how to prove this argument?

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    @Sara: By symmetry, we may assume $T_2$ is not finitely axiomatizable; then every for every finite set $S$ of sentences of $T_2$, there is a model of $T_1$ in which $S$ is true (because there is some model of $S$ that is not a model for all of $T_2$, and hence is a model for $T_1$). Seems like this might lead somewhere...2010-12-04
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    @Taroccoesbrocco - it is not necessary to edit very old posts to fix minor formatting issues - it pops them to the front page needlessly.2018-02-24

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