I've recently read about Vopenka's Theorem (Jech 15.46), which states that if A is a set of ordinals, then $L[A]=HOD^{L[A]}[G]$, meaning that if $HOD^{L[A]}\subsetneq L[A]$ then $L[A]$ is a generic extension of a smaller model.

Are there any other theorems which state conclusively that a particular inner model is a generic extension of some smaller model? Even better - is there some sort of general criterion that we can apply that tells us if a model is a generic extension?


Yes. Lev Bukovský proved a very general theorem that deals precisely with this problem:

MR0332477 (48 #10804). Characterization of generic extensions of models of set theory, Fundamenta Mathematica 83 (1973), pp. 35–46.

Bukovský characterizes when, for a given regular cardinal $\lambda$, $V$ is a $\lambda$-cc generic extension of a given inner model $W$. For a modern write-up of Bukovský's theorem, see

Ralf D. Schidler. The long extender algebra, preprint. To appear in a special issue of Archive for Mathematical Logic.

Bukovský proved that $V$ is a $\lambda$-cc extension of $W$ if and only if $W$ uniformly $\lambda$-covers $V$, meaning that for all functions $f\in V$ whose domain is in $W$ and whose range is contained in $W$ there is some function $g\in W$ with the same domain and such that $f(x) \in g(x)$ and $|g(x)| < \lambda$ for all $x \in \operatorname{dom}(f)$. (Of course, relativizing to an inner model, we get a characterization of when that inner model is itself a generic extension of another one, which was your explicit question.)

The one drawback is that the result is so general that in specific situations its applicability may not be clear. In many cases, one uses ad hoc methods.

There is active work in what is now called set-theoretic geology, the study of the class of inner models $W$ of which $V$ is a generic extension. Reading the literature in this area may provide specific examples (and, anyway, it is a nice area). Moreover, the terminology it has introduced (including mantles, grounds, bedrocks, and other terms) is spreading, so it is a good idea to catch up on the basics early on. A good place to get started is

MR3304634. Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas. Set-theoretic geology. Ann. Pure Appl. Logic 166 (2015), no. 4, 464–501.

There are also some results involving large cardinals; this is actually a common situation. Typically, you start with an elementary embedding $j\!:V\to M$ with critical point $\kappa$, where $M$ has reasonable closure properties. Say you are in a situation where you define some forcing iteration $({\mathbb{P}}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha\le\kappa)$ and want to argue that it preserves, say, measurability of $\kappa$. The easiest way to do this would be to show that if $G$ is $\mathbb{P}_\kappa$-generic over $V$, then in $V[G]$ the embedding $j$ lifts to an embedding $\hat j\!:V[G]\to N$. Letting $$j({\mathbb{P}}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha\le\kappa)=({\mathbb{P}'}_\alpha,\dot{\mathbb{Q}'}_\alpha:\alpha\le j(\kappa)),$$ one common way of achieving this is to show that $\mathbb{P}'_\alpha=\mathbb{P}_\alpha$ for $\alpha\le\kappa$ and that, in $V[G]$, there is an $H$ generic over $M[G]$ for the tail of the iteration. If this is the case, then $j$ lifts straightforwardly: You simply map the $G$-interpretation of a $\mathbb{P}_\kappa$-name $\tau$ to the $G*H$-interpretation of the ${\mathbb{P}'}_{j(\kappa)}$-name $j(\tau)$. Effectively, what you are doing is identifying generic extensions of inner models in $V[G]$. A good place to read about this setting is

MR2768691. Cummings, James. Iterated forcing and elementary embeddings. In Handbook of set theory. Vols. 1, 2, 3, 775–883, Springer, Dordrecht, 2010.

Finally, there are scattered results that show cases where some inner models are generic extensions of other models and that are detected by techniques independent of Bukovský's approach. For instance, there is a nice characterization (due to Mathias) of when a sequence is generic for Prikry forcing. It leads to the following result: Suppose $\kappa$ is measurable and $U$ is a witnessing normal ultrafilter. You have an ultrapower embedding $j_{0,1}=j\!: V=M_0\to M_1$ mapping $\kappa_0=\kappa$ to $\kappa_1=j(\kappa)$ and $U_0=U$ to $U_1=j(U)$. You can iterate this situation in the obvious way to obtain embeddings $j_{n,n+1}\!:M_n\to M_{n+1}$. The direct limit $M_\omega$ is well-founded. Its measurable cardinal $\kappa_\omega$ is actually the supremum of the $\kappa_n$ so it is singular in $V$ (and $M_\omega$ does not see the sequence of the $\kappa_n$). Let $M_\omega^+=\bigcap_n M_n$. Solovay proved that $G=\{\kappa_n:n<\omega\}$ is generic over $M_\omega$ for the Prikry forcing corresponding to $U_\omega$ and that $M_\omega^+=M_\omega[G]$. Patrick Dehornoy has generalizations of this result, see for instance

MR0514228 (80k:03057). Dehornoy, Patrick. Iterated ultrapowers and Prikry forcing. Ann. Math. Logic 15 (1978), no. 2, 109–160.


Not quite an answer, but too long for a comment:

This is one of the questions studied in set-theoretic geology. Specifically, a model $V$ satisfies the ground axiom GA if it is not a (set) forcing extension of any smaller model. (Surprisingly, GA is in fact first-order expressible.)

In certain aspects, GA is weak: it is consistent with both CH (via $L$) and $\neg CH$, and is (relatively) consistent with GCH + the existence of a supercompact cardinal. Put another way: there is good evidence that there is not a simple combinatorial or large cardinal based property which implies that a model is a forcing extension.

An obvious direction to go at this point is to try to formalize the previous sentence via the Levy hierarchy. At the first level, this seems likely to me, though I don't immediately see how to prove it:

Guess: there is no $\Pi_1$ or $\Sigma_1$ sentence which implies $\neg$GA.

One level up, it seems to me (although I could be wrong) that Reitz' expression of the ground axiom is $\Pi_2$, so its negation is $\Sigma_2$. As for the $\Pi_2$ side, I think that breaks for silly reasons: unless I'm missing something, "$V=L[\{n: 2^{\aleph_{2n}}=\aleph_{2n+1}\}]$ and $\{n: 2^{\aleph_{2n}}=\aleph_{2n+1}\}$ is Cohen over $L$" is $\Pi^1_2$, and it clearly implies $\neg$GA.

Maybe this latter problem can be fixed by looking at "$\Pi_2$ and compatible with [---]," instead of merely $\Pi_2$, for some natural sentence [---] (e.g. a strong large cardinal property - although the inner model theory of such cardinals would have to be provably "bad" ...); however, my instinct is that looking for such a sentence is ultimately just pushing the question back a level.


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