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The fundamental theorem of model theory says that:

Theorem: A first order theory is consistent if and only if it has a model.

In the above theorem we assume that the domain of any model is a non-empty "set". But in set theory sometimes we use proper class models of a theory. (For example $\langle L,\in\rangle \models ZFC$). Now a simple question is:

Question: Does existence of a proper class model for a first order theory (which is a weaker assumption than existence of a set model) imply its consistency? How can we formalize a similar theorem for proper class models?

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    $\begingroup$ Is there a source for naming this theorem the "fundamental theorem of model theory"? It sounds like the Completeness Theorem to me. $\endgroup$ – Ari Brodsky Oct 7 '13 at 5:26
  • $\begingroup$ Dear Ari, you are right but I just used the phrase "the fundamental theorem" as a description to emphasis the importance of this theorem for model theory. $\endgroup$ – user36136 Oct 7 '13 at 5:57
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$\newcommand{\ZFC}{\text{ZFC}}\newcommand{\KM}{\text{KM}}$

The answer must of course be negative, since every model of a theory $T$ is a proper class model of $T$, from its own perspective, but this cannot imply $\text{Con}(T)$ because of the incompleteness theorem.

But the question is actually more problematic than this, since we cannot generally even express that a proper class model is a model of a given theory as a single statement; rather, it is generally a scheme. For example, in the case of the constructible universe, one often hears it said that the constructible universe $L$ is a model of $\ZFC+V=L$, but this is not a single assertion in the language of set theory. Rather, what one means is that one can prove in $\ZFC$ that any given axiom of $\ZFC$ holds also in $L$. Furthermore, because of Tarski's theorem on the non-definability of truth, we aren't really able to formulate the assertion "$L$ is a model of $\ZFC$" as a single assertion in the first-order language of set theory. In this sense, the answer to your question is no.

Meanwhile, however, if you work in a stronger theory, such as Kelley-Morse set theory, then you can recover an affirmative answer. This is because $\KM$ proves the existence of truth predicates for first-order truth relative to any given class. Thus, in $\KM$, if you have a proper class model of a theory $T$, then $\KM$ can build the truth predicate for first-order satisfaction in this model and see that it is consistent, so the answer turns to Yes. The assertion that $L\models\ZFC$, for example, can be formalized as a single assertion in Kelley-Morse set theory, and furthermore, this assertion will imply $\text{Con}(\ZFC)$, essentially by induction on proofs.

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    $\begingroup$ The key point here is the difference between "$T$ is consistent" as a metatheory statement that is implied by the existence of a proper class model, versus Con($T$), which is an object-theory statement. For example, Con(ZFC) can fail inside a proper class model of ZFC, but if there is any proper class model of ZFC then ZFC is consistent, from the perspective of the metatheory. JDH is correct that "we cannot generally even express that a proper class model is a model of a given theory as a single statement" of the object theory but it is a single statement of the metatheory (by inspection...) $\endgroup$ – Carl Mummert Oct 7 '13 at 13:24
  • $\begingroup$ @CarlMummert I am confused as I thought that given a class $M$ (a unary predicate in the language of set theory) and a relational class $E$ (a binary predicate), saying that '$(M,E)$ is a proper class model of ZFC' amounts to saying that a certain scheme of sentences are provable in ZFC. The fact that we are able to prove a set of sentences in ZFC does not imply that ZFC is consistent (from the perspective of the metatheory). What am I missing? $\endgroup$ – Noel Vaillant Oct 7 '13 at 14:04
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    $\begingroup$ Noel, that is one way to interpret what that assertion means. Another way to interpret it, in Kelley-Morse theory, is as the assertion that there is a truth predicate for which all the axioms of ZFC come out true. This way of saying it is not a scheme, and makes reference to the internal consistency statement of the model, as Carl mentions for the object theory. Thus, assertions of class models of a theory in KM do imply the object theory consistency statement, but meta-theoretic statements of class models (as schemes) do not. $\endgroup$ – Joel David Hamkins Oct 7 '13 at 14:08
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    $\begingroup$ @Noel Valient: We don't have to read "a proper class $M$ is a model of ZFC" as a statement within ZFC. For example, there is a well known argument that the cumulative hierarchy is a model of ZFC minus replacement. That argument is given entirely in an informal metatheory, and it does not claim anything about provability in ZFC. Now, when we want to study the proper classes of a model $M$ of ZFC from within that model, we look at $M$-definable classes; but if we study them from outside the model there is no reason to think a proper class over $M$ would be definable over $M$ by a formula. $\endgroup$ – Carl Mummert Oct 7 '13 at 14:52
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    $\begingroup$ Yes, that is exactly right. When undertaken in ZFC itself, the assertion that ZFC holds in L is a theorem scheme, a separate theorem for each axiom of ZFC. When undertaken in KM, however, we can make the stronger statement that the class L has a satisfaction predicate for which every ZFC axioms comes out true, and this is a single (second-order) statement in KM. $\endgroup$ – Joel David Hamkins Oct 7 '13 at 16:22

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