Creating Models of $ZFC$

I know of only 2 main techniques to create a model of $ZFC$. The first one is creating a model which is an extension of $V$: this is forcing. The second technique is that of inner model theory and looking at subclasses of $V$. Do all methods to generate models of $ZFC$ fall in the these 2 categories or are there other radically different ways to generate of a model?

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Symmetric models (though AC usually fails there). What about iterates? And what about non-set theoretical? The stack semantics of a Boolean topos can give you a model of ZF. – David Roberts Nov 23 '11 at 4:34
But symmetric models are submodels of generic extensions and iterates are just transfinite extensions. However I don't know what a stack semantics of a Boolean topos is. Does it fall under the "extension" or "subclasses" kind of $V$? I am curious about a model generated neither by extension nor by inner model techniques. – Carlo Von Schnitzel Nov 23 '11 at 4:57
My first thought is the standard techniques for making nonstandard models: compactness and the incompleteness theorem. New models can also be made using ultrapowers of existing models of set theory. – Carl Mummert Nov 23 '11 at 5:01
No, Asaf, only when the ultrapower is wellfounded can you collapse it to an inner model. – François G. Dorais Nov 23 '11 at 14:25
No. Both forcing and inner models preserve $\omega$, and therefore the validity of arithmetical sentences (in fact, more generally, all $\Delta^1_3$-sentences, due to Shoenfield’s absoluteness theorem). In contrast, the incompleteness theorem produces models which differ in validity of $\Pi^0_1$ arithmetical sentences. Ultrapowers produce models that are (externally) not well-founded (unless the ultrafilter is measurable or something), so they cannot be realized as inner models either. – Emil Jeřábek Nov 23 '11 at 14:27

1 Answer

As Carl Mummert mentioned, the usual ways of constructing models — completeness and compactness — are also available for the construction of models of set theory. The models constructed in this way are usually not (externally) wellfounded but they have found some interesting uses.

For example, Joel David Hamkins proved the consistency of the Maximality Principle in this way. This principle asserts that any sentence $\phi$ that holds in some forcing extension and then in all further forcing extension — such a sentence is called forceably necessary — already holds in the ground model. He did this by showing that there is no sentence such that $\phi$ and $\lnot\phi$ are both forceably necessary. Therefore, the forceably necessary sentences form a consistent theory, which therefore has a model by completeness. [A simple maximality principle, JSL 68 (2003), 527–550. arXiv]

Another example (that I learned from Joel David Hamkins, but might be due to Sol Feferman) is the relative consistency with ZFC of the existence of some $\delta$ such that $V_\delta$ is an elementary submodel of $V$. The consistency of this follows from the reflection principle, which states that if $\phi(\bar{a})$ is any formula of set theory, then there is a closed unbounded class of $\delta$ such that $V_\delta \vDash \phi(\bar{a})$ iff $V \vDash \phi(\bar{a})$. It follows that every model of set theory satisfies every finite fragment of the said theory, therefore the whole theory is satisfiable by compactness.

End-extensions of models is another way by which new models can be constructed. This is an unusual method because it adds new ordinals and it can still form wellfounded models in some circumstances. This extends a well-known method of McDowell and Specker for models of arithmetic. Jack Silver, Jerome Keisler, and Ali Enayat have done significant work on this. In the opposite direction, there are various ways of cutting down models of set-theory to obtain new models. One such example is Cohen's minimal model of set theory, which is $L_\delta$ for the smallest ordinal $\delta$ (possibly equal to $Ord$) such that $L_\delta \vDash \mathrm{ZFC}$. [A minimal model for set theory, Bull. AMS 69 (1963), 537–540. MR0150036] (Note that these are not always inner-models since they often have fewer ordinals and they are generally not definable.)

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Thanks for mentioning my work, François! Although I have definitely used the $V_\delta\prec V$ idea, and I proved it is necessary for the maximality principle work, the equiconsistency of this particular hypothesis is not due to me, but was known much earlier. I think Feferman had a very strong early version of this. – Joel David Hamkins Nov 23 '11 at 14:36
Thanks, I'll correct that. – François G. Dorais Nov 23 '11 at 14:44