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Coming from here and the Wikipedia page on conservative extension, which defines a proof-theoretic conservative extension like this:

a theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if every theorem of $T_1$ is a theorem of $T_2$, and any theorem of $T_2$ in the language of $T_1$ is already a theorem of $T_1$.

And a model-theoretic conservative extension as this:

an extension $T_2$ of a theory $T_1$ is model-theoretically conservative if $T_{1}\subseteq T_{2}$ and every model of $T_{1}$ can be expanded to a model of $T_{2}$.

From the Wiki and the linked question, I gather that model theoretic conservative extensions are a subset of proof-theoretic conservative extensions. But the question that keeps bugging me is what is a (preferably constructible,) example of a proof-theoretic conservative extension that is not also model-theoretic conservative extension? What would that look like?

Pineapple Fish
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1 Answers1

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There are many examples! A convenient tool is that if $T\subseteq S$ are consistent theories and $T$ is complete, then $S$ is automatically a proof-theoretic extension of $T$. This lets us generate lots of examples by focusing on model-theoretic behavior. Here's a couple I like:

  • Let $X$ be the $\{<,c\}$-theory of dense linear order without endpoints (so $c$ is just some labelled point). Now consider the larger language $\{<,c,f\}$ with $f$ a unary function, and let $X'$ be the extension of $X$ in this language which says that $f$ gives an order-isomorphism between the part of the universe $<c$ and the part of the universe $>c$. $X'$ is not a model-theoretic conservative extension: consider the model of $X$ with underlying set $\mathbb{R}\setminus\mathbb{Q}_{>0}$ with the usual ordering and with $c$ interpreted as $0$.

  • Similarly, if $T$ is any complete first-order theory, we can whip up a theory $T^\star$ which describes two models of $T$ and the expansion $T^+$ of $T^\star$ which says that these two models are isomorphic (explicitly, via a new function symbol). In general, $T^+$ won't be a model-theoretically-conservative extension of $T^\star$ (and if we require $T$ to have no finite models and be non-countably-categorical, $T^+$ won't even be model-theoretically-conservative over $T^\star$ for countable models). However, $T^+$ will be proof-theoretically conservative over $T^\star$ (this is a good exercise).

  • Yet another example: take a structure $\mathcal{M}$ with an unrealized type $p$, and let $T$ be the complete theory of $\mathcal{M}$ together with axioms saying that a new constant symbol realizes the type $p$.

Noah Schweber
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