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Do the following statements mean the same thing? If not, what's the difference? Assume that $T_0, T_1$ are first-order languages and that $T_1$ is a model-theoretic extension of $T_0$.

Every model for $T_0$ can be expanded into a model for $T_1$.

vs.

For any model $M$ of $T_0$ there is a model $N$ of $T_1$ such that $M \equiv N\upharpoonright L_0$.

Notes:

  • "$N\upharpoonright L_0$" means the reduct of $N$ to the language of $M$.

  • "$\equiv$" means elementary equivalence, not isomorphism, so that, for instance, $(\mathbb{N},<)\equiv(\mathbb{N}+\mathbb{Z},<)$, as can be proved e.g. via quantifier elimination, but, clearly, $(\mathbb{N},<)\not\cong(\mathbb{N}+\mathbb{Z},<)$.

My confusion may arise from an inadequate understanding of the meanings of one or more of the terms model-theoretic extension, expansion, reduct, elementary equivalence and isomorphism.

Please assume in your answer minimal knowledge of logic on my part.

Evan Aad
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1 Answers1

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This was explained in my answer to part 3b of your previous question: they are not equivalent (although the first trivially implies the second). Isomorphism (= existence of a structure-preserving bijection) is strictly stronger than elementary equivalence (= satisfy the same first-order sentences), and this drives the separation.

In more detail:

Let $L_0$ be the language of arithmetic, and $L_1$ be that language expanded with a new constant symbol $c$. Let $T_0$ be the theory of $(\mathbb{N}; +, \times, 0, 1)$, and $T_1$ be that theory together with the sentences $$\mbox{"$c>1+1+1+...+1$"}$$ ($n$ "1"s) for each $n\in\mathbb{N}$. Then:

  • Clearly $T_1$ is a model-theoretic (in fact, proof-theoretic) extension of $T_0$.

  • For any model $N$ of $T_1$ and any model $M$ of $T_0$, $M\equiv N$. (This is because $T_0$ is a complete theory in the language $L_0$.)

  • But the model $(\mathbb{N}; +, \times, 0, 1)$ has no expansion to a model of $T_1$: wherever we "put" $c$, we wind up violating one of the axioms of $T_1$.

(Note, to allay possible worries, that $T_1$ does indeed have models - this can be proved using the compactness theorem, and is a good exercise.)

You asked in a comment to bof's answer how to tell that two structures are elementarily equivalent. There are several ways to do this:

  • If (the deductive closure of) $S$ is a complete $L$-theory and $A, B\models S$ are $L$-structures, then $A\equiv B$ by definition: if $A\models \varphi$, then we can't have $\neg\varphi\in S$, so since $S$ is complete we have $\varphi\in S$, but then $B\models\varphi$ since $B\models S$.

  • We can use Ehrenfeucht-Fraisse games to show that two structures are elementarily equivalent.

  • Los' Theorem says that we can show that two structures are elementarily equivalent if they have isomorphic ultrapowers (the converse is the Keisler-Shelah isomorphism theorem).

  • Quantifier elimination, model completeness, or similar can be used to give a complete description of the theory of a structure, and so facilitate proofs of elementary equivalence.

  • And sometimes you can come up with really silly model chain constructions too.

And there are lots of other methods.

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Noah Schweber
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  • Thanks. How can it be shown that $T_0$ is a complete theory in $L_0$? – Evan Aad Feb 11 '17 at 06:39
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    @EvanAad Any theory of the form "$Th(M)$" for an $L$-structure $M$ is a complete theory in $L$ - for each $\varphi$ in $L$, either $M\models \varphi$ (in which case $\varphi\in Th(M)$) or $M\models\neg\varphi$ (in which case $\neg\varphi\in Th(M)$). And here, $T_0=Th(\mathbb{N}; +, \times, 0, 1)$, so is a complete $L_0$-theory. – Noah Schweber Feb 11 '17 at 15:13