- V. Gitman and T. A. Johnstone, “On ground model definability,” in Infinity, computability, and metamathematics: festschrift in honour of the 60th birthdays of peter koepke and philip welch, London, GB: College publications, 2014.
@INCOLLECTION{gitmanjohnstone:groundmodels, AUTHOR = {Victoria Gitman and Thomas A. Johnstone}, TITLE = {On ground model definability}, BOOKTITLE = {Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch}, Publisher = {College publications}, YEAR = {2014}, series = {Series:Tributes}, Address = {London, GB}, PDF={http://boolesrings.org/victoriagitman/files/2013/11/groundmodels.pdf}, EPRINT ={1311.6789} }
It took four decades since the invention of forcing for set theorists and to ask (and answer) what post factum seems as one of the most natural questions regarding forcing. Is the ground model a definable class of its set-forcing extensions? Laver published the positive answer in a paper mainly concerned with whether rank-into-rank cardinals can be created by small forcing [1]. Woodin obtained the same result independently and it appeared in the appendix of [2].
Theorem 1: [Laver, Woodin] Suppose $V$ is a model of ${\rm ZFC}$, $\mathbb P\in V$ is a forcing notion, and $G\subseteq\mathbb P$ is $V$-generic. Then in $V[G]$, the ground model $V$ is definable from the parameter $P(\gamma)^V$, where $\gamma=|\mathbb P|^V$.
Indeed, this definition of the ground model is uniform across all its set-forcing extensions. There is a first-order formula which, using a ground model parameter, defines the ground model in any set-forcing extension. Before Theorem 1, properties of the forcing extension in relation to the ground model could be expressed in the forcing language using the predicate $\check V$ for the ground model sets. But having a uniform definition of ground models in their set-forcing extensions was an immensely more powerful result that opened up rich new avenues of research. Hamkins and Reitz used it to introduce the Ground Axiom, a first-order assertion that a universe is not a nontrivial set-forcing extension [3]. Research on the Ground Axiom in turn grew into the set-theoretic geology project that reverses the forcing construction by studying what remains from a model of set theory once the layers created by forcing are removed [4]. Woodin made use of Theorem 1 in studying generic multiverses–collections of set-theoretic universes that are generated from a given universe by closing under generic extensions and ground models [2]. In addition, Theorem 1 proved crucial to Woodin’s pioneering work on suitable extender models, a potential approach to constructing the canonical inner model for a supercompact cardinal [5].
In this article we investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_\delta$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results.
Laver’s proof that ground models of ${\rm ZFC}$ are definable in their set-forcing extensions uses Hamkins’ techniques and results on pairs of models with the $\delta$-cover and $\delta$-approximation properties.
Definition: [Hamkins] Suppose $V\subseteq W$ are transitive models of (some fragment of) ${\rm ZFC}$ with the same ordinals and $\delta$ is a cardinal in $W$.
- The pair $V\subseteq W$ satisfies the $\delta$-cover property if for every $A\in W$ with $A\subseteq V$ and $|A|^W<\delta$, there is $B\in V$ with $A\subseteq B$ and $|B|^V<\delta$.
- The pair $V\subseteq W$ satisfies the $\delta$-approximation property if whenever $A\in W$ with $A\subseteq V$ and $A\cap a\in V$ for every $a$ of size less than $\delta$ in $V$, then $A\in V$.
Pairs of the form the ground model with its forcing extension, $V\subseteq V[G]$, satisfy the $\delta$-cover and $\delta$-approximation properties for any regular $\delta\geq\gamma^+$, where $\gamma$ is the size of the forcing poset. This result is a weak version of the following more general theorem [6].
Theorem 2: [Hamkins] Suppose $\delta$ is a cardinal and $\mathbb P$ is a poset which factors as $\mathbb R*\dot{\mathbb Q}$, where $\mathbb R$ is nontrivial (adds a new set) of size less than $\delta$ and $\vdash_{\mathbb R} \dot{\mathbb Q}$ is strategically $\lt\delta$-closed. Then the pair $V\subseteq V[G]$ satisfies the $\delta$-cover and $\delta$-approximation properties for any forcing extension $V[G]$ by $\mathbb P$.
Theorem 3 [Hamkins] Suppose $V$, $V’$ and $W$ are transitive models of ${\rm ZFC}$, $\delta$ is a regular cardinal in $W$, the pairs $V\subseteq W$ and $V’\subseteq W$ have the $\delta$-cover and $\delta$-approximation properties, $P(\delta)^V=P(\delta)^{V’}$, and $(\delta^+)^V=(\delta^+)^W$. Then $V=V’$.
Laver’s proof of Theorem 1 proceeds by combining the weak version of Theorem 2 with the uniqueness property of Theorem 3 as follows. A forcing extension $V[G]$ by a poset $\mathbb P$ of size $\gamma$ has the $\delta$-cover and $\delta$-approximation properties for $\delta=\gamma^+$, and moreover it holds that $(\delta^+)^V=(\delta^+)^{V[G]}$. It is not difficult to see that there is an unbounded definable class $C$ of ordinals such that for every $\lambda\in C$, the $\delta$-cover and $\delta$-approximation properties reflect down to the pair $V_\lambda\subseteq V[G]_\lambda$, and moreover both $V_\lambda$ and $V[G]_\lambda$ satisfy a large enough fragment of ${\rm ZFC}$, call it ${\rm ZFC}^*$, for the proof of Theorem 3 to go through. Letting $s=P(\delta)^V$, the sets $V_\lambda$, for $\lambda\in C$, are then defined in $V[G]$ as the unique transitive models $M\models {\rm ZFC}^*$ of height $\lambda$, having $P(\delta)^M=s$ such that the pair $M\subseteq V[G]_\lambda$ has the $\delta$-cover and $\delta$-approximation properties. Finally, we can replace the parameter $s=P(\delta)^V$ with $P(\gamma)^V$ by observing that $P(\gamma)^V$ is definable from $P(\delta)^V$ in $V[G]$ with the help of the $\delta$-approximation property.
Forcing constructions over models of ${\rm ZF}$ can be carried out in some overarching ${\rm ZFC}$ context because essential properties of forcing such as the definability of the forcing relation and the Truth Lemma do not require choice. Also, forcing over models of ${\rm ZF}$ preserves ${\rm ZF}$ to the forcing extension. Is every model of ${\rm ZF}$ definable in its set-forcing extensions? Although at the outset, it might appear that the $\delta$-cover and $\delta$-approximation properties machinery, used to prove definability ${\rm ZFC}$-ground models, isn’t applicable to models without full choice, we will show that much of it can be salvaged with only a small fragment of choice. We prove an analogue of Theorem 3 for models of ${\rm ZF}+{\rm DC}_\delta$ and derive from it a partial definability result for ground models of ${\rm ZF}+{\rm DC}_\delta$ and forcing extensions by posets admitting a gap at $\delta$. Posets admitting a gap at $\delta$ are particularly suited to forcing over models of ${\rm ZF}+{\rm DC}_\delta$ because they also preserve ${\rm DC}_\delta$ to the forcing extension.
Lévy introduced the dependent choice axiom variant ${\rm DC}_\delta$ asserting that for any nonempty set $S$ and any binary relation $R$, if for each sequence $s \in S^{\lt\delta}$ there is a $y\in S$ such that $s$ is $R$-related to $y$, then there is a function $f:\delta\to S$ such that $f\upharpoonright\alpha R f(\alpha)$ for each $\alpha<\delta$. It is easy to see that ${\rm DC}_\delta$ implies the choice principle ${\rm AC}_\delta$, the assertion that indexed families $\{A_\xi\mid \xi<\delta\}$ of nonempty sets have choice functions. The full ${\rm AC}$ is clearly equivalent to the assertion $\forall\delta \;{\rm DC}_\delta$, while ${\rm AC}_\delta$ is much weaker than ${\rm DC}_\delta$, as it provides choice functions only for already well-ordered families of nonempty sets. Indeed, for any fixed $\delta$, the principle ${\rm AC}_\delta$ does not imply ${\rm DC}_\omega$, while the assertion $\forall\delta {\rm AC}_\delta$ does imply ${\rm DC}_\omega$ but not ${\rm DC}_{\omega_1}$ [7]. Some of the natural models of ${\rm ZF}+{\rm DC}_\delta$ arise as symmetric inner models of forcing extensions and models of the form $L(V_{\delta+1})$. In [8], Hamkins defined that a poset $\mathbb P$ admits a gap at a cardinal $\delta$ if it factors as $\mathbb R*\dot{\mathbb Q}$, where $\mathbb R$ is nontrivial forcing of size less than $\delta$, and it is forced by $\mathbb R$ that $\dot{\mathbb Q}$ is $\leq\delta$-strategically closed. By Theorem 2, a ${\rm ZFC}$ ground model with a forcing extension by a poset admitting a gap at a $\delta$ satisfy the $\delta$-cover and $\delta$-approximation properties. Indeed the analogous result for ${\rm ZF}+{\rm DC}_\delta$ holds as well .
Main Theorem 1: Suppose $V$ is a model of ${\rm ZF}+{\rm DC}_\delta$, $\mathbb P\in V$ is a forcing notion admitting a gap at $\delta$, and $G\subseteq\mathbb P$ is $V$-generic. Then in $V[G]$, the ground model $V$ is definable from the parameter $P(\delta)^V$.
Models of the theory ${\rm ZFC}^-$, known as set theory without powerset, are used widely throughout set theory. Typically, but not necessarily, these have a largest cardinal $\kappa$. The canonical ones are models $H_{\kappa^+}$, which are collections of all sets of hereditary size at most $\kappa$ for some cardinal $\kappa$. Models of ${\rm ZFC}^-$ also play a prominent role in the theory of smaller large cardinals, many of which, such as weakly compact, remarkable, unfoldable, and Ramsey cardinals, are characterized by the existence of elementary embeddings of ${\rm ZFC}^-$ models. Forcing over models of ${\rm ZFC}^-$ preserves ${\rm ZFC}^-$ to the forcing extension and the rest of standard forcing machinery carries over as well. However, we show that Laver’s ground model definability result cannot be generalized to ${\rm ZFC}^-$ ground models. Using forcing, we produce a ${\rm ZFC}$ universe with a cardinal $\kappa$ such that ground model definability fails for $H_{\kappa^+}$. In this case, ground model definability is violated in the strongest possible sense because $H_{\kappa^+}$ has a set-forcing extension in which it is not definable even using a parameter from the extension. We can set up the preparatory forcing so that $\kappa$ is any ground model cardinal and so that the forcing extension violating ground model definability is by a poset of the form ${\rm Add}(\delta,1)$ for some regular cardinal $\delta<\!<\kappa$. It will follow from our arguments that there is always a countable transitive model of ${\rm ZFC}^-$ violating ground model definability.
Main Theorem 2: Assume that $\delta, \kappa$ are cardinals such that $\delta$ is regular and either $2^{\lt\delta}\!<\!\kappa$ or $\delta\!=\!\kappa\!=\!2^{\lt\kappa}$ holds. If $V[G]$ is a forcing extension by ${\rm Add}(\delta,\kappa^+)$, then $H_{\kappa^+}^{V[G]}$ is not definable in its forcing extension by ${\rm Add}(\delta,1)$. It follows that there is a countable transitive model of ${\rm ZFC}^-$ that is not definable in its Cohen forcing extension.
[Bibtex]
@article {laver:groundmodel,
AUTHOR = {Laver, Richard},
TITLE = {Certain very large cardinals are not created in small forcing
extensions},
JOURNAL = {Ann. Pure Appl. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {149},
YEAR = {2007},
NUMBER = {1-3},
PAGES = {1--6},
ISSN = {0168-0072},
CODEN = {APALD7},
MRCLASS = {03E55 (03E35)},
MRNUMBER = {2364192 (2009e:03099)},
MRREVIEWER = {Paul Bradley Larson},
DOI = {10.1016/j.apal.2007.07.002},
URL = {http://dx.doi.org/10.1016/j.apal.2007.07.002},
}
[Bibtex]
@inproceedings {woodin:groundmodel,
AUTHOR = {Woodin, Hugh},
TITLE = {Recent developments on {C}antor's {C}ontinuum {H}ypothesis},
BOOKTITLE = {Proceedings of the continuum in {P}hilosophy and {M}athematics},
PUBLISHER = {Carlsberg Academy, {C}oppenhagen},
YEAR = {2004},
}
@article {reitz:groundaxiom,
AUTHOR = {Reitz, Jonas},
TITLE = {The ground axiom},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {Journal of Symbolic Logic},
VOLUME = {72},
YEAR = {2007},
NUMBER = {4},
PAGES = {1299--1317},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E35 (03E45)},
MRNUMBER = {2371206 (2009g:03079)},
DOI = {10.2178/jsl/1203350787},
URL = {http://dx.doi.org/10.2178/jsl/1203350787},
}
[Bibtex]
@ARTICLE{FuchsHamkinsReitz:Set-theoreticGeology,
AUTHOR = "Gunter Fuchs and Joel David Hamkins and Jonas Reitz",
TITLE = "Set-theoretic geology",
JOURNAL = "",
YEAR = "",
volume = "",
number = "",
pages = "",
month = "",
note = "submitted",
url = "http://arxiv.org/abs/1107.4776",
eprint = "1107.4776",
abstract = "",
keywords = "",
source = "",
file = F
}
[Bibtex]
@article {woodin:suitableextendermodels,
AUTHOR = {Woodin, W. Hugh},
TITLE = {Suitable extender models {I}},
JOURNAL = {J. Math. Log.},
FJOURNAL = {Journal of Mathematical Logic},
VOLUME = {10},
YEAR = {2010},
NUMBER = {1-2},
PAGES = {101--339},
ISSN = {0219-0613},
MRCLASS = {03E35 (03E25 03E45 03E55 03E60 03E65)},
MRNUMBER = {2802084 (2012g:03135)},
MRREVIEWER = {A. Kanamori},
DOI = {10.1142/S021906131000095X},
URL = {http://dx.doi.org/10.1142/S021906131000095X},
}
[Bibtex]
@article {hamkins:coverandapproximations,
AUTHOR = {Hamkins, Joel David},
TITLE = {Extensions with the approximation and cover properties have no
new large cardinals},
JOURNAL = {Fund. Math.},
FJOURNAL = {Fundamenta Mathematicae},
VOLUME = {180},
YEAR = {2003},
NUMBER = {3},
PAGES = {257--277},
ISSN = {0016-2736},
MRCLASS = {03E55 (03E40)},
MRNUMBER = {2063629 (2005m:03100)},
DOI = {10.4064/fm180-3-4},
URL = {http://dx.doi.org/10.4064/fm180-3-4},
}
@book {Jech1973:AxiomChoice,
AUTHOR = {Jech, Thomas J.},
TITLE = {The axiom of choice},
NOTE = {Studies in Logic and the Foundations of Mathematics, Vol. 75},
PUBLISHER = {North-Holland Publishing Co.},
ADDRESS = {Amsterdam},
YEAR = {1973},
PAGES = {xi+202},
MRCLASS = {04A25 (02K05)},
MRNUMBER = {0396271 (53 \#139)},
MRREVIEWER = {D. Pincus},
}
@article {hamkins:gapforcing,
AUTHOR = {Hamkins, Joel David},
TITLE = {Gap forcing},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {125},
YEAR = {2001},
PAGES = {237--252},
ISSN = {0021-2172},
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MRCLASS = {03E40 (03E55)},
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MRREVIEWER = {Renling Jin},
DOI = {10.1007/BF02773382},
URL = {http://dx.doi.org/10.1007/BF02773382},
}
When is there going to be a readable preprint on arXiv?
Working away at it :)!
Very nice, Vika! I really like what you and Tom have done.
Thanks Joel!
Incidentally, after theorem 3 you mention some things about Laver’s proof of theorem 1. But actually, Laver’s proof to my knowledge did not make any reference to the approximation and cover properties (he had a more complex argument that accomplished the task). Rather, it is my proof of theorem 1 that proceeds as you explain. When I told him my proof of the theorem, upon first hearing about it, he remarked that “clearly that is the right way to do it,” and he adopted my argument in his paper. He had told me at the time that he found the attribution problem of the theorem problematic. I guess the right way to say it is that he proved theorem 1 as you state it, whereas the strengthening of it to claim that every pseudo ground $W$, meaning an inner model with the $\delta$-approximation and cover properties and the right $\delta^+$, is definable in $V$ from parameter $P(\delta)^W$, is due to me. (And as you know, we can use just parameter $({}^{\lt\delta 2)^W$, which amounts to $P(\gamma)^W$ when $\delta=\gamma^+$.)
$({}^{\lt\delta} 2)^W$
Ah, I see! So Laver ended up using your proof of the ground model definability theorem. This will have to be a footnote in the paper.
Yes, that’s right. If you look at his paper, he credits me for the proof that he uses there. (His original proof is not published anywhere that I know.) He had contacted me because he noticed that his proof used a back-and-forth cover argument that was similar to the argument that Woodin and I had used in our paper on small forcing and Woodin cardinals, and it is this same method that arises in the approximation and cover property arguments.
I looked at his paper again, and unfortunately it is not quite so clear there as I had described. Basically, the situation is that Laver had proved his theorem 1 originally without any mention of approximation and cover, referring just to the forcing poset and its cardinality. He contacted me because of the similarity in his argument and my proof with Woodin in the back-and-forth covering process. At that point, I directed him to my paper on the approximation and cover, and sent him a proof of what you call theorem 3 (he had had a formulation stated in terms of the forcing notion and V_delta+1 with P inside V_delta), which implies theorem 1 in the way you describe. He liked that approach much better, and adopted it in his paper.
Actually, the original form of Laver’s theorem had a much larger parameter—he just used some enormous $V_\theta$ above the forcing notion—and it was my argument that brought it down to $P(\delta)^W$, where $\delta=|\mathbb{P}|^+$, because of theorem 2. This has now been brought down further, as you know, to $P(\mathbb{P})^W$, which aligns with Woodin’s version of the theorem.