Regula Krapf, Ph.D. 2017, University of Bonn

Regula Krapf successfully defended her PhD dissertation January 12, 2017 at the University of Bonn, with a dissertation entitled, “Class forcing and second-order arithmetic.”  I was a member of the dissertation examining committee. Peter Koepke was the dissertation supervisor.

Regula Krapf

Regula Krapf, Class forcing and second-order arithmetic, dissertation 2017, University of Bonn. (Slides)

Abstract. We provide a framework in a generalization of Gödel-Bernays set theory for performing class forcing. The forcing theorem states that the forcing relation is a (definable) class in the ground model (definability lemma) and that every statement that holds in a class-generic extension is forced by a condition in the generic filter (truth lemma). We prove both positive and negative results concerning the forcing theorem. On the one hand, we show that the definability lemma for one atomic formula implies the forcing theorem for all formulae in the language of set theory to hold. Furthermore, we introduce several properties which entail the forcing theorem. On the other hand, we give both counterexamples to the definability lemma and the truth lemma. In set forcing, the forcing theorem can be proved for all forcing notions by constructing a unique Boolean completion. We show that in class forcing the existence of a Boolean completion is essentially equivalent to the forcing theorem and, moreover, Boolean completions need not be unique.

The notion of pretameness was introduced to characterize those forcing notions which preserve the axiom scheme of replacement. We present several new characterizations of pretameness in terms of the forcing theorem, the preservation of separation, the existence of nice names for sets of ordinals and several other properties. Moreover, for each of the aforementioned properties we provide a corresponding characterization of the Ord-chain condition.

Finally, we prove two equiconsistency results which compare models of ZFC (with large cardinal properties) and models of second-order arithmetic with topological regularity properties (and determinacy hypotheses). We apply our previous results on class forcing to show that many important arboreal forcing notions preserve the $\Pi^1_1$-perfect set property over models of second-order arithmetic and also give an example of a forcing notion which implies the $\Pi^1_1$-perfect set property to fail in the generic extension.

Regula has now taken up a faculty position at the University of Koblenz.

The rearrangement number

  • A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, “The rearrangement number.” (manuscript under review)  
    @ARTICLE{BlassBrendleBrianHamkinsHardyLarson:TheRearrangementNumber,
    author = {Andreas Blass and J\"org Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
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    note = {manuscript under review},
    url = {http://jdh.hamkins.org/the-rearrangement-number},
    eprint = {1612.07830},
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Abstract.  How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We show that the minimum number of permutations needed for this purpose, which we call the rearrangement number, is uncountable, but whether it equals the cardinal of the continuum is independent of the usual axioms of
set theory. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied
cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally we deal briefly with some variants concerning rearrangements by a special sort of permutations and with rearranging some divergent series to become (conditionally) convergent.

This project started with Michael Hardy’s question on MathOverflow, How many rearrangements must fail to alter the value of a sum before you conclude that none do? I had proposed in my answer that we should think of the cardinal in question as a cardinal characteristic of the continuum, the rearrangement number, since we could prove that it was uncountable and that it was the continuum under MA, and had begun to separate it from other familiar cardinal characteristics. Eventually, the research effort grew into the collaboration of this paper. What a lot of fun!

Colloquium talk at Vassar | Lecture notes talk at CUNY | the original MathOverflow question

Here are the lecture notes for an introductory talk on the topic I had given at the Vassar College Mathematics Colloquium

Recent advances in set-theoretic geology, Harvard Logic Colloquium, October 2016

I will speak at the Harvard Logic Colloquium, October 20, 2016, 4-6 pm.

harvard

Abstract. Set-theoretic geology is the study of the set-theoretic universe $V$ in the context of all its ground models and those of its forcing extensions. For example, a bedrock of the universe is a minimal ground model of it and the mantle is the intersection of all grounds. In this talk, I shall explain some recent advances, including especially the breakthrough result of Toshimichi Usuba, who proved the strong downward directed grounds hypothesis: for any set-indexed family of grounds, there is a deeper common ground below them all. This settles a large number of formerly open questions in set-theoretic geology, while also leading to new questions. It follows, for example, that the mantle is a model of ZFC and provably the largest forcing-invariant definable class. Strong downward directedness has also led to an unexpected connection between large cardinals and forcing: if there is a hyper-huge cardinal $\kappa$, then the universe indeed has a bedrock and all grounds use only $\kappa$-small forcing.

Slides

The rearrangement number: how many rearrangements of a series suffice to verify absolute convergence? Mathematics Colloquium at Penn, September 2016

This will be a talk for the Mathematics Colloquium at the University of Pennsylvania, Wednesday, September 14, 2016, 3:30 pm, tea at 3 pm, in the mathematics department.

UPenn Campus
Abstract. The well-known Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. But how many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and, time permitting, an account of Freiling’s axiom of symmetry.

This talk is based in part on current joint work with Jörg Brendle, Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

Related MathOverflow post: How many rearrangements must fail to alter the value of a sum before you conclude that none do?

Set-theoretic geology and the downward-directed grounds hypothesis, CUNY Set Theory seminar, September 2016

This will be a talk for the CUNY Set Theory Seminar, September 2 and 9, 2016.

Blender3D EarthQuarterCut.jpgIn two talks, I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis.  This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC.

Although forcing is often viewed as a method of constructing larger models extending a given model of set theory, the topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model.  Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb{Q}\in W$.  It is a consequence of the ground-model definability theorem that every such $W$ is definable from parameters, and from this it follows that many second-order-seeming questions about the structure of grounds turn out to be first-order expressible in the language of set theory.

For example, Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis DDG and the strong DDG, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

Last year, Usuba proved the strong DDG, and I shall give a complete account of the proof, with some simplifications I had noticed. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

Among the consequences of Usuba’s result I shall prove are:

  1. Bedrock models are unique when they exist.
  2. The mantle is absolute by forcing.
  3. The mantle is a model of ZFC.
  4. The mantle is the same as the generic mantle.
  5. The mantle is the largest forcing-invariant class, and equal to the intersection of the generic multiverse.
  6. The inclusion relation agrees with the ground-of relation in the generic multiverse. That is, if $N\subset M$ are in the same generic multiverse, then $N$ is a ground of $M$.
  7. If ZFC is consistent, then the ZFC-provably valid downward principles of forcing are exactly S4.2.
  8. (Usuba) If there is a hyper-huge cardinal, then there is a bedrock for the universe.

Related topics in set-theoretic geology:

CUNY Set theory seminar abstract I | abstract II

The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme

  • J. D. Hamkins, “The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme.” (manuscript under review)  
    @ARTICLE{Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme,
    author = {Joel David Hamkins},
    title = {The {Vop\v{e}nka} principle is inequivalent to but conservative over the {Vop\v{e}nka} scheme},
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Abstract. The Vopěnka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopěnka scheme, which makes the Vopěnka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopěnka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for first-order assertions in the language of set theory.

Indras Net-03

The Vopěnka principle is the assertion that for every proper class $\mathcal{M}$ of first-order $\mathcal{L}$-structures, for a set-sized language $\mathcal{L}$, there are distinct members of the class $M,N\in\mathcal{M}$ with an elementary embedding $j:M\to N$ between them. In quantifying over classes, this principle is a single assertion in the language of second-order set theory, and it makes sense to consider the Vopěnka principle in the context of a second-order set theory, such as Godel-Bernays set theory GBC, whose language allows one to quantify over classes. In this article, GBC includes the global axiom of choice.

In contrast, the first-order Vopěnka scheme makes the Vopěnka assertion only for the first-order definable classes $\mathcal{M}$ (allowing parameters). This theory can be expressed as a scheme of first-order statements, one for each possible definition of a class, and it makes sense to consider the Vopěnka scheme in Zermelo-Frankael ZFC set theory with the axiom of choice.

Because the Vopěnka principle is a second-order assertion, it does not make sense to refer to it in the context of ZFC set theory, whose first-order language does not allow quantification over classes; one typically retreats to the Vopěnka scheme in that context. The theme of this article is to investigate the precise meta-mathematical interactions between these two treatments of Vopěnka’s idea.

Main Theorems.

  1. If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka scheme holds, but the Vopěnka principle fails.
  2. If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka principle holds.

It follows that the Vopěnka principle VP and the Vopěnka scheme VS are not equivalent, but they are equiconsistent and indeed, they have the same first-order consequences.

Corollaries.

  1. Over GBC, the Vopěnka principle and the Vopěnka scheme, if consistent, are not equivalent.
  2. Nevertheless, the two Vopěnka axioms are equiconsistent over GBC.
  3. Indeed, the two Vopěnka axioms have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for assertions in the first-order language of set theory. $$\text{GBC}+\text{VP}\vdash\phi\qquad\text{if and only if}\qquad\text{ZFC}+\text{VS}\vdash\phi$$

These results grew out of my my answer to a MathOverflow question of Mike Shulman, Can Vopěnka’s principle be violated definably?, inquiring whether there would always be a definable counterexample to the Vopěnka principle, whenever it should happen to fail. I interpret the question as asking whether the Vopěnka scheme is necessarily equivalent to the Vopěnka principle, and the answer is negative.

The proof of the main theorem involves the concept of a stretchable set $g\subset\kappa$ for an $A$-extendible cardinal, which has the property that for every cardinal $\lambda>\kappa$ and every extension $h\subset\lambda$ with $h\cap\kappa=g$, there is an elementary embedding $j:\langle V_\lambda,\in,A\cap V_\lambda\rangle\to\langle V_\theta,\in,A\cap V_\theta\rangle$ such that $j(g)\cap\lambda=h$. Thus, the set $g$ can be stretched by an $A$-extendibility embedding so as to agree with any given $h$.

Jacob Davis, PhD 2016, Carnegie Mellon University

Jacob Davis successfully defended his dissertation, “Universal Graphs at $\aleph_{\omega_1+1}$ and Set-theoretic Geology,” at Carnegie Mellon University on April 29, 2016, under the supervision of James Cummings. I was on the dissertation committee (participating via Google Hangouts), along with Ernest Schimmerling and Clinton Conley.

Jacob Davis

CMU web pageGoogle+ profile | ar$\chi$iv

The thesis consisted of two main parts. In the first half, starting from a model of ZFC with a supercompact cardinal, Jacob constructed a model in which $2^{\aleph_{\omega_1}} = 2^{\aleph_{\omega_1+1}} = \aleph_{\omega_1+3}$ and in which there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.  The same technique works with any uncountable cardinal in place of $\omega_1$.  In the second half, Jacob proved a variety of results in the area of set-theoretic geology, including several instances of the downward directed grounds hypothesis, including an analysis of the chain condition of the resulting ground models.

Giorgio Audrito, PhD 2016, University of Torino

Dr. Giorgio Audrito has successfully defended his dissertation, “Generic large cardinals and absoluteness,” at the University of Torino under the supervision of Matteo Viale.

The dissertation Examing Board consisted of myself (serving as Presidente), Alessandro Andretta and Sean Cox.  The defense took place March 2, 2016.

Giorgio Audrito defense (small)

The dissertation was impressive, introducing (in joint work with Matteo Viale) the iterated resurrection axioms $\text{RA}_\alpha(\Gamma)$ for a forcing class $\Gamma$, which extend the idea of the resurrection axioms from my work with Thomas Johnstone, The resurrection axioms and uplifting cardinals, by making successive extensions of the same type, forming the resurrection game, and insisting that that the resurrection player have a winning strategy with game value $\alpha$. A similar iterative game idea underlies the $(\alpha)$-uplifting cardinals, from which the consistency of the iterated resurrection axioms can be proved. A final chapter of the dissertation (joint with Silvia Steila), develops the notion of $C$-systems of filters, generalizing the more familiar concepts of extenders and towers.

Upward closure and amalgamation in the generic multiverse of a countable model of set theory

  • J. D. Hamkins, “Upward closure and amalgamation in the generic multiverse of a countable model of set theory,” RIMS Kyôkyûroku, pp. 17-31, 2016. (also available as Newton Institute preprint ni15066)  
    @ARTICLE{Hamkins2016:UpwardClosureAndAmalgamationInTheGenericMultiverse,
    author = {Joel David Hamkins},
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    year = {2016},
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    pages = {17--31},
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    note = {also available as Newton Institute preprint ni15066},
    url = {http://jdh.hamkins.org/upward-closure-and-amalgamation-in-the-generic-multiverse},
    eprint = {1511.01074},
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Abstract. I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.

This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.

The rearrangement number: how many rearrangements of a series suffice to verify absolute convergence? Vassar Math Colloquium, November 2015

This will be a talk for the Mathematics Colloquium at Vassar College, November 10, 2015, tea at 4:00 pm, talk at 4:15 pm, Rockefeller Hall 310

Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum introduced just recently, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on current joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

My notes are available here:

The rearrangement number, CUNY set theory seminar, November 2015

This will be a talk for the CUNY Set Theory Seminar on November 6, 2015.

The Riemann rearrangement theorem states that a convergent real series $\sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $\sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if the series is merely conditionally convergent, then one may find rearrangements for which the new sum $\sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent.  In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.

Specifically, we define the rearrangement number $\newcommand\rr{\mathfrak{rr}}\rr$ (“double-r”), a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $\sum_n a_n$ remains convergent and with the same sum after all rearrangements $\sum_n a_{p(n)}$ by a permutation $p\in P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted $\newcommand\rrsum{\rr_{\scriptscriptstyle\Sigma}}
\rrsum$, is the smallest cardinality of a family $P$ of permutations, such that if a series $\sum_n a_n$ is conditionally convergent, then there is a rearrangement $\sum_n a_{p(n)}$, by some permutation $p \in P$, which converges to a different sum. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that $\mathfrak{b}\leq\rr\leq\mathop{\bf non}(\mathcal{M})$, that $\mathfrak{d}\leq \rrsum$, and that $\mathfrak{b}<\rr$ is relatively consistent.

MathOverflow question | CUNY Set Theory Seminar

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

Upward countable closure in the generic multiverse of forcing to add a Cohen real

I’d like to discuss my theorem that the collection of models $M[c]$ obtained by adding an $M$-generic Cohen real $c$ over a fixed countable transitive model of set theory $M$ is upwardly countably closed, in the sense that every increasing countable chain has an upper bound.

I proved this theorem back in 2011, while at the Young Set Theory Workshop in Bonn and continuing at the London summer school on set theory, in a series of conversations with Giorgio Venturi. The argument has recently come up again in various discussions, and so let me give an account of it.

We consider the collection of all forcing extensions of a fixed countable transitive model $M$ of ZFC by the forcing to add a Cohen real, models of the form $M[c]$, and consider the question of whether every countable increasing chain of these models has an upper bound. The answer is yes!  (Actually, Giorgio wants to undertake forcing constructions by forcing over this collection of models to add a generic upward directed system of models; it follows from this theorem that this forcing is countably closed.) This theorem fits into the theme of my earlier post, Upward closure in the toy multiverse of all countable models of set theory, where similar theorems are proved, but not this one exactly.

Theorem. For any countable transitive model $M\models\text{ZFC}$, the collection of all forcing extensions $M[c]$ by adding an $M$-generic Cohen real is upward-countably closed. That is, for any countable tower of such forcing extensions
$$M[c_0]\subset M[c_1]\subset\cdots\subset M[c_n]\subset\cdots,$$
we may find an $M$-generic Cohen real $d$ such that $M[c_n]\subset M[d]$ for every natural number $n$.

Proof. $\newcommand\Add{\text{Add}}$Suppose that we have such a tower of forcing extensions $M[c_0]\subset M[c_1]\subset\cdots$, and so on. Note that if $M[b]\subset M[c]$ for $M$-generic Cohen reals $b$ and $c$, then $M[c]$ is a forcing extension of $M[b]$ by a quotient of the Cohen-real forcing. But since the Cohen forcing itself has a countable dense set, it follows that all such quotients also have a countable dense set, and so $M[c]$ is actually $M[b][b_1]$ for some $M[b]$-generic Cohen real $b_1$. Thus, we may view the tower as having the form:
$$M[b_0]\subset M[b_0\times b_1]\subset\cdots\subset M[b_0\times b_1\times\cdots\times b_n]\subset\cdots,$$
where now it follows that any finite collection of the reals $b_i$ are mutually $M$-generic.

Of course, we cannot expect in general that the real $\langle b_n\mid n<\omega\rangle$ is $M$-generic for $\Add(\omega,\omega)$, since this real may be very badly behaved. For example, the sequence of first-bits of the $b_n$’s may code a very naughty real $z$, which cannot be added by forcing over $M$ at all. So in general, we cannot allow that this sequence is added to the limit model $M[d]$. (See further discussion in my post Upward closure in the toy multiverse of all countable models of set theory.)

We shall instead undertake a construction by making finitely many changes to each real $b_n$, resulting in a real $d_n$, in such a way that the resulting combined real $d=\oplus_n d_n$ is $M$-generic for the forcing to add $\omega$-many Cohen reals, which is of course isomorphic to adding just one. To do this, let’s get a little more clear with our notation. We regard each $b_n$ as an element of Cantor space $2^\omega$, that is, an infinite binary sequence, and the corresponding filter associated with this real is the collection of finite initial segments of $b_n$, which will be an $M$-generic filter through the partial order of finite binary sequences $2^{<\omega}$, which is one of the standard isomorphic copies of Cohen forcing. We will think of $d$ as a binary function on the plane $d:\omega\times\omega\to 2$, where the $n^{th}$ slice $d_n$ is the corresponding function $\omega\to 2$ obtained by fixing the first coordinate to be $n$.

Now, we enumerate the countably many open dense subsets for the forcing to add a Cohen real $\omega\times\omega\to 2$ as $D_0$, $D_1$, and so on. There are only countably many such dense sets, because $M$ is countable. Now, we construct $d$ in stages. Before stage $n$, we will have completely specified $d_k$ for $k<n$, and we also may be committed to a finite condition $p_{n-1}$ in the forcing to add $\omega$ many Cohen reals. We consider the dense set $D_n$. We may factor $\Add(\omega,\omega)$ as $\Add(\omega,n)\times\Add(\omega,[n,\omega))$. Since $d_0\times\cdots\times d_{n-1}$ is actually $M$-generic (since these are finite modifications of the corresponding $b_k$’s, which are mutually $M$-generic, it follows that there is some finite extension of our condition $p_{n-1}$ to a condition $p_n\in D_n$, which is compatible with $d_0\times\cdots\times d_{n-1}$. Let $d_n$ be the same as $b_n$, except finitely modified to be compatible with $p_n$. In this way, our final real $\oplus_n d_n$ will contain all the conditions $p_n$, and therefore be $M$-generic for $\Add(\omega,\omega)$, yet every $b_n$ will differ only finitely from $d_n$ and hence be an element of $M[d]$. So we have $M[b_0]\cdots[b_n]\subset M[d]$, and we have found our upper bound. QED

Notice that the real $d$ we construct is not only $M$-generic, but also $M[c_n]$-generic for every $n$.

My related post, Upward closure in the toy multiverse of all countable models of set theory, which is based on material in my paper Set-theoretic geology, discusses some similar results.

Upward closure in the generic multiverse of a countable model of set theory, RIMS 2015, Kyoto, Japan

Philosophers Walk Kyoto Japan (summer)This will be a talk for the conference Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) in Kyoto, Japan, September 16-18, 2015.

Abstract. Consider a countable model of set theory amongst its forcing extensions, the ground models of those extensions, the extensions of those models and so on, closing under the operations of forcing extension and ground model.  This collection is known as the generic multiverse of the original model.  I shall present a number of upward-oriented closure results in this context. For example, for a long-known negative result, it is a fun exercise to construct forcing extensions $M[c]$ and $M[d]$ of a given countable model of set theory $M$, each by adding an $M$-generic Cohen real, which cannot be amalgamated, in the sense that there is no common extension model $N$ that contains both $M[c]$ and $M[d]$ and has the same ordinals as $M$. On the positive side, however, any increasing sequence of extensions $M[G_0]\subset M[G_1]\subset M[G_2]\subset\cdots$, by forcing of uniformly bounded size in $M$, has an upper bound in a single forcing extension $M[G]$. (Note that one cannot generally have the sequence $\langle G_n\mid n<\omega\rangle$ in $M[G]$, so a naive approach to this will fail.)  I shall discuss these and related results, many of which appear in the “brief upward glance” section of my recent paper:  G. Fuchs, J. D. Hamkins and J. Reitz, Set-theoretic geology.


Erin Carmody

Erin Carmody successfully defended her dissertation under my supervision at the CUNY Graduate Center on April 24, 2015, and she earned her Ph.D. degree in May, 2015. Her dissertation follows the theme of killing them softly, proving many theorems of the form: given $\kappa$ with large cardinal property $A$, there is a forcing extension in which $\kappa$ no longer has property $A$, but still has large cardinal property $B$, which is very slightly weaker than $A$. Thus, she aims to enact very precise reductions in large cardinal strength of a given cardinal or class of large cardinals. In addition, as a part of the project, she developed transfinite meta-ordinal extensions of the degrees of hyper-inaccessibility and hyper-Mahloness, giving notions such as $(\Omega^{\omega^2+5}+\Omega^3\cdot\omega_1^2+\Omega+2)$-inaccessible among others.

Erin Carmody

G+ profile | math genealogy | MathOverflow profileNY Logic profilear$\chi$iv

Erin Carmody, “Forcing to change large cardinal strength,”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2015.  ar$\chi$iv | PDF

Erin has accepted a professorship at Nebreska Wesleyan University for.the 2015-16 academic year.


 

Erin is also an accomplished artist, who has had art shows of her work in New York, and she has pieces for sale. Much of her work has an abstract or mathematical aspect, while some pieces exhibit a more emotional or personal nature. My wife and I have two of Erin’s paintings in our collection:
OceanIMG_0597