Being HOD-of-a-set is invariant throughout the generic multiverse

Iowa State Capitol - Law Library _ Flickr - Photo Sharing!$\newcommand\HOD{\text{HOD}}$The axiom $V=\HOD$, introduced by Gödel, asserts that every set is ordinal definable. This axiom has a subtler foundational aspect than might at first be expected. The reason is that the general concept of “object $x$ is definable using parameter $p$” is not in general first-order expressible in set theory; it is of course a second-order property, which makes sense only relative to a truth predicate, and by Tarski’s theorem, we can have no first-order definable truth predicate. Thus, the phrase “definable using ordinal parameters” is not directly meaningful in the first-order language of set theory without further qualification or explanation. Fortunately, however, it is a remarkable fact that when we allow definitions to use arbitrary ordinal parameters, as we do with $\HOD$, then we can in fact make such qualifications in such a way that the axiom becomes first-order expressible in set theory. Specifically, we say officially that $V=\HOD$ holds, if for every set $x$, there is an ordinal $\theta$ with $x\in V_\theta$, for which which $x$ is definable by some formula $\psi(x)$ in the structure $\langle V_\theta,{\in}\rangle$ using ordinal parameters. Since $V_\theta$ is a set, we may freely make reference to first-order truth in $V_\theta$ without requiring any truth predicate in $V$. Certainly any such $x$ as this is also ordinal-definable in $V$, since we may use $\theta$ and the Gödel-code of $\psi$ also as parameters, and note that $x$ is the unique object such that it is in $V_\theta$ and satisfies $\psi$ in $V_\theta$. (Note that inside an $\omega$-nonstandard model of set theory, we may really need to use $\psi$ as a parameter, since it may be nonstandard, and $x$ may not be definable in $V_\theta$ using a meta-theoretically standard natural number; but fortunately, the Gödel code of a formula is an integer, which is still an ordinal, and this issue is the key to the issue.) Conversely, if $x$ is definable in $V$ using formula $\varphi(x,\vec\alpha)$ with ordinal parameters $\vec\alpha$, then it follows by the reflection theorem that $x$ is defined by $\varphi(x,\vec\alpha)$ inside some $V_\theta$. So this formulation of $V=HOD$ is expressible and exactly captures the desired second-order property that every set is ordinal-definable.

Consider next the axiom $V=\HOD(b)$, asserting that every set is definable from ordinal parameters and parameter $b$. Officially, as before, $V=\HOD(b)$ asserts that for every $x$, there is an ordinal $\theta$, formula $\psi$ and ordinals $\vec \alpha<\theta$, such that $x$ is the unique object in $V_\theta$ for which $\langle V_\theta,{\in}\rangle\models\psi(x,\vec\alpha,b)$, and the reflection argument shows again that this way of defining the axiom exactly captures the intended idea.

The axiom I actually want to focus on is $\exists b\,\left( V=\HOD(b)\right)$, asserting that the universe is $\HOD$ of a set. (I assume ZFC in the background theory.) It turns out that this axiom is constant throughout the generic multiverse.

Theorem. The assertion $\exists b\, (V=\HOD(b))$ is forcing invariant.

  • If it holds in $V$, then it continues to hold in every set forcing extension of $V$.
  • If it holds in $V$, then it holds in every ground of $V$.

Thus, the truth of this axiom is invariant throughout the generic multiverse.

Proof. Suppose that $\text{ZFC}+V=\HOD(b)$, and $V[G]$ is a forcing extension of $V$ by generic filter $G\subset\mathbb{P}\in V$. By the ground-model definability theorem, it follows that $V$ is definable in $V[G]$ from parameter $P(\mathbb{P})^V$. Thus, using this parameter, as well as $b$ and additional ordinal parameters, we can define in $V[G]$ any particular object in $V$. Since this includes all the $\mathbb{P}$-names used to form $V[G]$, it follows that $V[G]=\HOD(b,P(\mathbb{P})^V,G)$, and so $V[G]$ is $\HOD$ of a set, as desired.

Conversely, suppose that $W$ is a ground of $V$, so that $V=W[G]$ for some $W$-generic filter $G\subset\mathbb{P}\in W$, and $V=\HOD(b)$ for some set $b$. Let $\dot b$ be a name for which $\dot b_G=b$. Every object $x\in W$ is definable in $W[G]$ from $b$ and ordinal parameters $\vec\alpha$, so there is some formula $\psi$ for which $x$ is unique such that $\psi(x,b,\vec\alpha)$. Thus, there is some condition $p\in\mathbb{P}$ such that $x$ is unique such that $p\Vdash\psi(\check x,\dot b,\check{\vec\alpha})$. If $\langle p_\beta\mid\beta<|\mathbb{P}|\rangle$ is a fixed enumeration of $\mathbb{P}$ in $W$, then $p=p_\beta$ for some ordinal $\beta$, and we may therefore define $x$ in $W$ using ordinal parameters, along with $\dot b$ and the fixed enumeration of $\mathbb{P}$. So $W$ thinks the universe is $\HOD$ of a set, as desired.

Since the generic multiverse is obtained by iteratively moving to forcing extensions to grounds, and each such movement preserves the axiom, it follows that $\exists b\, (V=\HOD(b))$ is constant throughout the generic multiverse. QED

Theorem. If $V=\HOD(b)$, then there is a forcing extension $V[G]$ in which $V=\HOD$ holds.

Proof. We are working in ZFC. Suppose that $V=\HOD(b)$. We may assume $b$ is a set of ordinals, since such sets can code any given set. Consider the following forcing iteration: first add a Cohen real $c$, and then perform forcing $G$ that codes $c$, $P(\omega)^V$ and $b$ into the GCH pattern at uncountable cardinals, and then perform self-encoding forcing $H$ above that coding, coding also $G$ (see my paper on Set-theoretic geology for further details on self-encoding forcing). In the final model $V[c][G][H]$, therefore, the objects $c$, $b$, $P(\omega)^V$, $G$ and $H$ are all definable without parameters. Since $V\subset V[c][G][H]$ has a closure point at $\omega$, it satisfies the $\omega_1$-approximation and cover properties, and therefore the class $V$ is definable in $V[c][G][H]$ using $P(\omega)^V$ as a parameter. Since this parameter is itself definable without parameters, it follows that $V$ is parameter-free definable in $V[c][G][H]$. Since $b$ is also definable there, it follows that every element of $\HOD(b)^V=V$ is ordinal-definable in $V[c][G][H]$. And since $c$, $G$ and $H$ are also definable without parameters, we have $V[c][G][H]\models V=\HOD$, as desired. QED

Corollary. The following are equivalent.

  1. The universe is $\HOD$ of a set: $\exists b\, (V=\HOD(b))$.
  2. Somewhere in the generic multiverse, the universe is $\HOD$ of a set.
  3. Somewhere in the generic multiverse, the axiom $V=\HOD$ holds.
  4. The axiom $V=\HOD$ is forceable.

Proof. This is an immediate consequence of the previous theorems. $1\to 4\to 3\to 2\to 1$. QED

Corollary. The axiom $V=\HOD$, if true, even if true anywhere in the generic multiverse, is a switch.

Proof. A switch is a statement such that both it and its negation are necessarily possible by forcing; that is, in every set forcing extension, one can force the statement to be true and also force it to be false. We can always force $V=\HOD$ to fail, simply by adding a Cohen real. If $V=\HOD$ is true, then by the first theorem, every forcing extension has $V=\HOD(b)$ for some $b$, in which case $V=\HOD$ remains forceable, by the second theorem. QED

Ehrenfeucht's lemma in set theory

  • G. Fuchs, V. Gitman, and J. D. Hamkins, “Ehrenfeucht’s lemma in set theory,” to appear in Notre Dame Journal of Formal Logic.  
    @ARTICLE{FuchsGitmanHamkins:EhrenfeuchtsLemmaInSetTheory,
    author = {Gunter Fuchs and Victoria Gitman and Joel David Hamkins},
    title = {Ehrenfeucht's lemma in set theory},
    journal = {to appear in Notre Dame Journal of Formal Logic},
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This is joint work with Gunter Fuchs and Victoria Gitman. $\newcommand\HOD{\text{HOD}}\newcommand\Ehrenfeucht{\text{EL}}$

Abstract. Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying $V=\HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq\HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form $\Ehrenfeucht(A,P,Q)$, which asserts that whenever an object $b$ is definable in $M$ from some $a\in A$ using parameters in $P$, with $b\neq a$, then the types of $a$ and $b$ over $Q$ in $M$ are different. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set $b$ is \emph{algebraic} in $a$ if it is a member of a finite set definable from $a$ (as in J. D. Hamkins and C. Leahy, Algebraicity and implicit definability in set theory). Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis, we answer two open questions posed in my paper with Leahy, by showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

When does every definable set have a definable member? CUNY Set Theory Seminar, October 2014

This will be a talk for the CUNY set theory seminar, October 10, 2014, 12pm  GC 6417.

Abstract. Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ZF in which every definable nonempty set has a definable element are precisely the models of V=HOD.  Indeed, V=HOD is equivalent to the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $\Sigma_2$-definability, because every model of ZFC has a forcing extension satisfying $V\neq\text{HOD}$ in which every $\Sigma_2$-definable set has an ordinal-definable element.

This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions and answers on MathOverflow, namely,

Definable collections without definable members
A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together.
Is it consistent that every definable set has a definable member?
A similar question asked last week by (anonymous) user38200
Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable member?
A question I had regarding the limits of an issue in my answer to the previous question.

In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students.

Large cardinals need not be large in HOD

  • Y. Cheng, S. Friedman, and J. D. Hamkins, “Large cardinals need not be large in HOD,” Annals of Pure and Applied Logic, vol. 166, iss. 11, pp. 1186-1198, 2015.  
    @ARTICLE{ChengFriedmanHamkins2015:LargeCardinalsNeedNotBeLargeInHOD,
    title = "Large cardinals need not be large in {HOD} ",
    journal = "Annals of Pure and Applied Logic ",
    volume = "166",
    number = "11",
    pages = "1186 - 1198",
    year = "2015",
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    issn = "0168-0072",
    doi = "10.1016/j.apal.2015.07.004",
    eprint = {1407.6335},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://jdh.hamkins.org/large-cardinals-need-not-be-large-in-hod},
    author = "Yong Cheng and Sy-David Friedman and Joel David Hamkins",
    keywords = "Large cardinals",
    keywords = "HOD",
    keywords = "Forcing",
    keywords = "Absoluteness ",
    abstract = "Abstract We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal κ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in V, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals. "
    }

Abstract. We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $\kappa$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

In this article, we prove that large cardinals need not generally exhibit their large cardinal nature in HOD, the inner model of hereditarily ordinal-definable sets, and there can be a divergence in strength between the large cardinals of the ambient set-theoretic universe $V$ and those of HOD. Our general theme concerns the questions:

Questions.

1. To what extent must a large cardinal in $V$ exhibit its large cardinal properties in HOD?

2. To what extent does the existence of large cardinals in $V$ imply the existence of large cardinals in HOD?

For large cardinal concepts beyond the weakest notions, we prove, the answers are generally negative. In Theorem 4, for example, we construct a model with a supercompact cardinal that is not weakly compact in HOD, and Theorem 9 extends this to a proper class of supercompact cardinals, none of which is weakly compact in HOD, thereby providing some strongly negative instances of (1). The same model has a proper class of supercompact cardinals, but no supercompact cardinals in HOD, providing a negative instance of (2). The natural common strengthening of these situations would be a model with a proper class of supercompact cardinals, but no weakly compact cardinals in HOD. We were not able to arrange that situation, however, and furthermore it would be ruled out by Conjecture 13, an intriguing positive instance of (2) recently proposed by W. Hugh Woodin, namely, that if there is a supercompact cardinal, then there is a measurable cardinal in HOD. Many other natural possibilities, such as a proper class of measurable cardinals with no weakly compact cardinals in HOD, remain as open questions.

CUNY talkRutgers talk | Luminy talk

Large cardinals need not be large in HOD, International Workshop on Set Theory, CIRM, Luminy, September 2014

I shall speak at the 13th International Workshop on Set Theory, held at the CIRM Centre International de Rencontres Mathématiques in Luminy near Marseille, France, September 29 to October 3, 2014. 

Abstract.  I shall prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals. There are many open questions.

This talk will include joint work with Cheng Yong and Sy-David Friedman.

Article | Participants | Slides

Large cardinal indestructibility: two slick new proofs of prior results

$\newcommand\HOD{\text{HOD}}$

I’ve recently found two slick new proofs of some of my prior results on indestructibility, using the idea of an observation of Arthur Apter’s.  What he had noted is:

Observation. (Apter [1])  If $\kappa$ is a Laver indestructible supercompact cardinal, then $V_\kappa\subset\HOD$.  Indeed, $V_\kappa$ satisfies the continuum coding axiom CCA.

Proof. The continuum coding axiom asserts that every set of ordinals is coded into the GCH pattern (it follows that they are each coded unboundedly often). If $x\subset\kappa$ is any bounded set of ordinals, then let $\mathbb{Q}$ be the forcing to code $x$ into the GCH pattern at regular cardinals directly above $\kappa$. This forcing is ${\lt}\kappa$-directed closed, and so by our assumption, $\kappa$ remains supercompact and in particular $\Sigma_2$-reflecting in the extension $V[G]$. Since $x$ is coded into the GCH pattern of $V[G]$, it follows by reflection that $V_\kappa=V[G]_\kappa$ must also think that $x$ is coded, and so $V_\kappa\models\text{CCA}$. QED

First, what I noticed is that this immediately implies that small forcing ruins indestructibility:

Theorem. (Hamkins, Shelah [2], Hamkins [3]) After any nontrivial forcing of size less than $\kappa$, the cardinal $\kappa$ is no longer indestructibly supercompact, nor even indestructibly $\Sigma_2$-reflecting.

Proof.  Nontrivial small forcing $V[g]$ will add a new set of ordinals below $\kappa$, which will not be coded unboundedly often into the continuum function of $V[g]$, and so $V[g]_\kappa$ will not satisfy the CCA.  Hence, $\kappa$ will not be indestructibly $\Sigma_2$-reflecting there. QED

This argument can be seen as essentially related to Shelah’s 1998 argument, given in [2].

Second, I also noticed that a similar idea can be used to prove:

Theorem. (Bagaria, Hamkins, Tsaprounis, Usuba [4])  Superstrong and other large cardinals are never Laver indestructible.

Proof.  Suppose the superstrongness of $\kappa$ is indestructible. It follows by the observation that $V_\kappa$ satisfies the continuum coding axiom. Now force to add a $V$-generic Cohen subset $G\subset\kappa$.  If $\kappa$ were superstrong in $V[G]$, then there would be $j:V[G]\to M$ with $V[G]_{j(\kappa)}=M_{j(\kappa)}$. Since $G$ is not coded into the continuum function, $M_{j(\kappa)}$ does not satisfy the CCA.  This contradicts the elementarity $V_\kappa=V[G]_\kappa\prec M_{j(\kappa)}$. QED

The argument shows that even the $\Sigma_3$-extendibility of $\kappa$ is never Laver indestructible.

I would note, however, that the slick proof does not achieve the stronger result of [4], which is that superstrongness is never indestructible even by $\text{Add}(\kappa,1)$, and that after forcing to add a Cohen subset to $\kappa$ (among any of many other common forcing notions), the cardinal $\kappa$ is never $\Sigma_3$-extendible (and hence not superstrong, not weakly superstrong, and so on).  The slick proof above uses indestructibility by the coding forcing to get the CCA in $V_\kappa$, and it is not clear how one would argue that way to get these stronger results of [4].

[1] Arthur W. Apter and Shoshana Friedman. HOD-supercompactness, inestructibility, and level-by-level equivalence, to appear in Bulletin of the Polish Academy of Sciences (Mathematics).

[2] Joel David Hamkins, Saharon Shelah, Superdestructibility: A Dual to Laver’s Indestructibility,  J. Symbolic Logic, Volume 63, Issue 2 (1998), 549-554.

[3] Joel David Hamkins, Small forcing makes any cardinal superdestructible, J. Symbolic Logic, 63 (1998).

[4] Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba, Superstrong and other large cardinals are never Laver indestructible, to appear in the Archive of Math Logic (special issue in memory of Richard Laver).

Large cardinals need not be large in HOD, Rutgers logic seminar, April 2014

 

I shall speak at the Rutgers Logic Seminar on April 21, 2014, 5:00-6:20 pm, Room 705, Hill Center, Busch Campus, Rutgers University.

Abstract. I will show that large cardinals, such as measurable, strong and supercompact cardinals, need not exhibit their large cardinal nature in HOD.  Specifically, it is relatively consistent that a supercompact cardinal is not weakly compact in HOD, and one may construct models with a proper class of supercompact cardinals, none of them weakly compact in HOD.  This is current joint work with Cheng Yong.

Article

Large cardinals need not be large in HOD, CUNY Set Theory Seminar, January 2014

This will be a talk for the CUNY Set Theory Seminar, January 31, 2014, 10:00 am.

Abstract. I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I will begin with the example of a measurable cardinal that is not measurable in HOD. After this, I will describe how to force a more extreme divergence.  For example, among other possibilities, it is relatively consistent that there is a supercompact cardinal that is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

Article

Algebraicity and implicit definability in set theory

  • J. D. Hamkins and C. Leahy, “Algebraicity and Implicit Definability in Set Theory,” Notre Dame J. Formal Logic, vol. 57, iss. 3, pp. 431-439, 2016.  
    @article{HamkinsLeahy2016:AlgebraicityAndImplicitDefinabilityInSetTheory,
    author = "Hamkins, Joel David and Leahy, Cole",
    doi = "10.1215/00294527-3542326",
    fjournal = "Notre Dame Journal of Formal Logic",
    journal = "Notre Dame J. Formal Logic",
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    pages = "431--439",
    publisher = "Duke University Press",
    title = "Algebraicity and Implicit Definability in Set Theory",
    volume = "57",
    year = "2016",
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We aim in this article to analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity and its companion concept of implicit definability. In place of the class HOD of hereditarily ordinal definable sets, for example, we consider the class HOA of hereditarily ordinal-algebraic sets. In place of the pointwise definable models of set theory, we examine its (pointwise) algebraic models. And in place of G&ouml;del’s constructible universe L, obtained by iterating the definable power set operation, we introduce the implicitly constructible universe Imp, obtained by iterating the algebraic or implicitly definable power set operation. In each case we investigate how the change from definability to algebraicity affects the nature of the resulting concept. We are especially intrigued by Imp, for it is a new canonical inner model of ZF whose subtler properties are just now coming to light. Open questions about Imp abound.

Before proceeding further, let us review the basic definability definitions. In the model theory of first-order logic, an element $a$ is definable in a structure $M$ if it is the unique object in $M$ satisfying some first-order property $\varphi$ there, that is, if $M\models\varphi[b]$ just in case $b=a$. More generally, an element $a$ is algebraic in $M$ if it has a property $\varphi$ exhibited by only finitely many objects in $M$, so that $\{b\in M \mid M\models\varphi[b]\}$ is a finite set containing $a$. For each class $P\subset M$ we can similarly define what it means for an element to be $P$-definable or $P$-algebraic by allowing the formula $\varphi$ to have parameters from $P$.

In the second-order context, a subset or class $A\subset M^n$ is said to be definable in $M$, if $A=\{\vec a\in M\mid M\models\varphi[\vec a]\}$ for some first-order formula $\varphi$. In particular, $A$ is the unique class in $M^n$ with $\langle M,A\rangle\models\forall \vec x\, [\varphi(\vec x)\iff A(\vec x)]$, in the language where we have added a predicate symbol for $A$. Generalizing this condition, we say that a class $A\subset M^n$ is implicitly definable in $M$ if there is a first-order formula $\psi(A)$ in the expanded language, not necessarily of the form $\forall \vec x\, [\varphi(\vec x)\iff A(\vec x)]$, such that $A$ is unique such that $\langle M,A\rangle\models\psi(A)$. Thus, every (explicitly) definable class is also implicitly definable, but the converse can fail. Even more generally, we say that a class $A\subset M^n$ is algebraic in $M$ if there is a first-order formula $\psi(A)$ in the expanded language such that $\langle M,A\rangle\models\psi(A)$ and there are only finitely many $B\subset M^n$ for which $\langle M,B\rangle\models\psi(B)$. Allowing parameters from a fixed class $P\subset M$ to appear in $\psi$ yields the notions of $P$-definability, implicit $P$-definability, and $P$-algebraicity in $M$. Simplifying the terminology, we say that $A$ is definable, implicitly definable, or algebraic over (rather than in) $M$ if it is $M$-definable, implicitly $M$-definable, or $M$-algebraic in $M$, respectively. A natural generalization of these concepts arises by allowing second-order quantifiers to appear in $\psi$. Thus we may speak of a class $A$ as second-order definable, implicitly second-order definable, or second-order algebraic. Further generalizations are of course possible by allowing $\psi$ to use resources from other strong logics.

The main theorems of the paper are:

Theorem. The class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets: $$\text{HOA}=\text{HOD}.$$

Theorem. Every pointwise algebraic model of ZF is a pointwise definable model of ZFC+V=HOD.

In the latter part of the paper, we introduce what we view as the natural algebraic analogue of the constructible universe, namely, the implicitly constructible universe, denoted Imp, and built as follows:

$$\text{Imp}_0 = \emptyset$$

$$\text{Imp}_{\alpha + 1} = P_{imp}(\text{Imp}_\alpha)$$

$$\text{Imp}_\lambda = \bigcup_{\alpha < \lambda} \text{Imp}_\alpha, \text{ for limit }\lambda$$

$$\text{Imp} = \bigcup_\alpha \text{Imp}_\alpha.$$

Theorem.  Imp is an inner model of ZF with $L\subset\text{Imp}\subset\text{HOD}$.

Theorem.  It is relatively consistent with ZFC that $\text{Imp}\neq L$.

Theorem. In any set-forcing extension $L[G]$ of $L$, there is a further extension $L[G][H]$ with $\text{gImp}^{L[G][H]}=\text{Imp}^{L[G][H]}=L$.

Open questions about Imp abound. Can $\text{Imp}^{\text{Imp}}$ differ from $\text{Imp}$? Does $\text{Imp}$ satisfy the axiom of choice? Can $\text{Imp}$ have measurable cardinals? Must $0^\sharp$ be in $\text{Imp}$ when it exists? (An affirmative answer arose in conversation with Menachem Magidor and Gunter Fuchs, and we hope that $\text{Imp}$ will subsume further large cardinal features. We anticipate a future article on the implicitly constructible universe.)  Which large cardinals are absolute to $\text{Imp}$? Does $\text{Imp}$ have fine structure? Should we hope for any condensation-like principle? Can CH or GCH fail in $\text{Imp}$? Can reals be added at uncountable construction stages of $\text{Imp}$? Can we separate $\text{Imp}$ from HOD? How much can we control $\text{Imp}$ by forcing? Can we put arbitrary sets into the $\text{Imp}$ of a suitable forcing extension? What can be said about the universe $\text{Imp}(\mathbb{R})$ of sets implicitly constructible relative to $\mathbb{R}$ and, more generally, about $\text{Imp}(X)$ for other sets $X$? Here we hope at least to have aroused interest in these questions.

This article arose from a question posed on MathOverflow by my co-author Cole Leahy and our subsequent engagement with it.

Algebraicity and implicit definability in set theory, CUNY, May 2013

This is a talk May 10, 2013 for the CUNY Set Theory Seminar.

Abstract.  An element a is definable in a model M if it is the unique object in M satisfying some first-order property. It is algebraic, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing a. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interesting new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.

This is joint work with Cole Leahy (MIT).

NYlogic abstract | MathOverflow post

Jonas Reitz

Jonas Reitz earned his Ph.D under my supervision in June, 2006 at the CUNY Graduate Center.  He was truly a pleasure to supervise. From the earliest days of his dissertation research, he had his own plan for the topic of the work: he wanted to “undo” forcing, to somehow force backwards, from the extension to the ground model. At first I was skeptical, but in time, ideas crystalized around the ground axiom (now with its own Wikipedia entry), formulated using a recent-at-the-time result of Richard Laver.  Along with Laver’s theorem, Jonas’s dissertation was the beginning of the body of work now known as set-theoretic geology.  Jonas holds a tenured position at the New York City College of Technology of CUNY.

Jonas Reitz


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Jonas Reitz, “The ground axiom,” Ph.D. dissertation, CUNY Graduate Center, June, 2006.  ar$\chi$iv

Abstract.  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.

Must there be numbers we cannot describe or define? Pointwise definability and the Math Tea argument, Bristol, April 2012

This is a talk I plan to give to the set theory seminar at the University of Bristol on April 18, 2012.

An old argument, heard at a good math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

slides | article

Must there be non-definable numbers? Pointwise definability and the math-tea argument, KGRC, Vienna 2011

This talk will be a part of the “Advanced Introduction” series for graduate students at the the Kurt Gödel Research Center, November 4, 2011.

An old argument, heard perhaps at math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

Slides | Article

Generalizations of the Kunen inconsistency, KGRC, Vienna 2011

This is a talk at the research seminar of the Kurt Gödel Research Center, November 3, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

Slides | Article

Set-theoretic geology

  • G. Fuchs, J. D. Hamkins, and J. Reitz, “Set-theoretic geology,” Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015.  
    @article{FuchsHamkinsReitz2015:Set-theoreticGeology,
    author = "Gunter Fuchs and Joel David Hamkins and Jonas Reitz",
    title = "Set-theoretic geology",
    journal = "Annals of Pure and Applied Logic",
    volume = "166",
    number = "4",
    pages = "464--501",
    year = "2015",
    note = "",
    MRCLASS = {03E55 (03E40 03E45 03E47)},
    MRNUMBER = {3304634},
    issn = "0168-0072",
    doi = "10.1016/j.apal.2014.11.004",
    eprint = "1107.4776",
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = "http://jdh.hamkins.org/set-theoreticgeology",
    }

The Inner Core

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.