Transitive ZFC model

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A transitive model of ZFC is a transitive set $M$ such that the structure $\langle M,\in\rangle$ satisfies all of the ZFC axioms of set theory. The existence of such a model is strictly stronger than Con(ZFC) and stronger than an iterated consistency hierarchy, but weaker than the existence of an worldly cardinal, a cardinal $\kappa$ for which $V_\kappa$ is a model of ZFC, and consequently also weaker than the existence of an inaccessible cardinal. Not all transitive models of ZFC have the $V_\kappa$ form, for if there is any transitive model of ZFC, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.

Nevertheless, every transitive model $M$ of ZFC provides a set-theoretic forum inside of which one can view nearly all classical mathematics taking place. In this sense, such models are inaccessible to or out of reach of ordinary set-theoretic constructions. As a result, the existence of a transitive model of ZFC can be viewed as a large cardinal axiom: it expresses a notion of largeness, and the existence of such a model is not provable in ZFC and has consistency strength strictly exceeding ZFC.

Minimal transitive model of ZFC

If there is any transitive model $M$ of ZFC, then $L^M$, the constructible universe as computed in $M$, is also a transitive model of ZFC and indeed, has the form $L_\eta$, where $\eta=\text{ht}(M)$ is the height of $M$. The minimal transitive model of ZFC is the model $L_\eta$, where $\eta$ is smallest such that this is a model of ZFC. The argument just given shows that the minimal transitive model is a subset of all other transitive models of ZFC.

Omega model of ZFC

An $\omega$-model of ZFC is a model of ZFC whose collection of natural numbers is isomorphic to the actual natural numbers. In other words, an $\omega$-model is a model having no nonstandard natural numbers, although it may have nonstandard ordinals. (More generally, for any ordinal $\alpha$, an $\alpha$-model has well-founded part at least $\alpha$.) Every transitive model of ZFC is an $\omega$-model, but the latter concept is strictly weaker.

Consistency hierarchy

The existence of an $\omega$-model of ZFC and implies Con(ZFC), of course, and also Con(ZFC+Con(ZFC)) and a large part of the iterated consistency hierarchy. This is simply because if $M\models\text{ZFC}$ and has the standard natural numbers, then $M$ agrees that Con(ZFC) holds, since it has the same proofs as we do in the ambient background. Thus, we believe that $M$ satisfies ZFC+Con(ZFC) and consequently we believe Con(ZFC+Con(ZFC)). It follows again that $M$ agrees with this consistency assertion, and so we now believe $\text{Con}^3(\text{ZFC})$. The model $M$ therefore agrees and so we believe $\text{Con}^4(\text{ZFC})$ and so on transfinitely, as long as we are able to describe the ordinal iterates in a way that $M$ interprets them correctly.

Transitive models of ZFC fragments

Every finite fragment of ZFC admits numerous transitive models, as a consequence of the reflection theorem.

Transitive models and forcing

Countable transitive models of set theory were used historically as a convenient way to formalize forcing. Such models $M$ make the theory of forcing convenient, since one can easily prove that for every partial order $\mathbb{P}$ in $M$, there is an $M$-generic filter $G\subset\mathbb{P}$, simply by enumerating the dense subsets of $\mathbb{P}$ in $M$ in a countable sequence $\langle D_n\mid n\lt\omega\rangle$, and building a descending sequence $p_0\geq p_1\geq p_2\geq\cdots$, with $p_n\in D_n$. The filter $G$ generated by the sequence is $M$-generic.

For the purposes of consistency proofs, this manner of formalization worked quite well. To show $\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\varphi)$, one fixes a finite fragment of ZFC and works with a countable transitive model of a suitably large fragment, producing $\varphi$ with the desired fragment in a forcing extension of it.

Transitive model universe axiom

The transitive model universe axiom is the assertion that every set is an element of a transitive model of ZFC. This axiom makes a stronger claim than the Feferman theory, since it is asserted as a single first-order claim, but weaker than the universe axiom, which asserts that the universes have the form $V_\kappa$ for inaccessible cardinals $\kappa$.

The transitive model universe axiom is sometimes studied in the background theory not of ZFC, but of ZFC-P, omitting the power set axiom, together with the axiom asserting that every set is countable. Such an enterprise amounts to adopting the latter theory, not as the fundamental axioms of mathematics, but rather as a background meta-theory for studying the multiverse perspective, investigating how the various actual set-theoretic universe, transitive models of full ZFC, relate to one another.