Address at the Dean's List Ceremony

Posted on October 15, 2013 by Joel David Hamkins

As the designated faculty speaker, selected after nominations from all the various departments at the college, I made the following remarks, in full academic regalia, at the Dean’s List ceremony this evening at the College of Staten Island.

Thank you very much to the Dean for the kind introduction.

I am pleased and honored to be here, amongst the elite of the College of Staten Island, students who have made the Dean’s list for academic accomplishment. You have proved yourselves in the challenges of academic life and you should be proud. I am proud of you.

You are our intellectual powers, engines of thought and reason. Probably your minds are running all the time—it does seem a little hot in here…

In the classic Wim Wenders film, “Wings of Desire,” some characters are able to hear and experience the thoughts of others. And although one ordinarily imagines a library as a place of quiet contemplation, in the film the library was a cacophy of voices, a hundred trains of thought running through, each audibly expressing the content of a person’s mind.

This is how I like to imagine the arena of the mind, lively and exciting.

But also playful. Perhaps I’m called on to offer some advice to you, and my advice is this: pursue an attitude of playful curiosity about your subject, whatever it is. Play with new ideas, explore all facets of them, going beyond whatever had been expected of you. You will be led to vast new lands of imagination.

A while back, my son Horatio, in fourth grade at the time, showed me his math homework. His teacher had said, “I am thinking of a number. It has a 3 in the ten’s place. The digits add to 10. It is prime. What is my number?” Can anyone solve it? Yes, 37 is a solution, and Horatio also had found it. He asked me, “are there other solutions?” Now perhaps an unimaginative person might think only of two-digit numbers, and in this case, 37 is the only solution. But with imagination, we can seek out larger possibilities. And indeed Horatio and I worked together at the cafe and realized that 433 is another solution, as well as 631 and 1531. I checked a table of prime numbers, and found that 10333 is also a solution, as is 100,333. I began to wonder: are there infinitely many solutions? I had no idea how to prove such a thing.

So I posted the question to one of the math Q & A sites, and it immediately rocketed to the top, getting thousands of views. Mathematicians all around the world were thinking about this playful version of my son’s homework problem. Some of these mathematicians wrote computer programs to find more and more
solutions; one young mathematician found all the solutions up to one hundred million, systematically. Another found enormous solutions, gigantic prime numbers, one with 416 digits (mostly zeros), which were also solutions of the problem. But with regard to whether there are infinitely many solutions or not, we still had no answer. It turns out to be an open mathematical question; nobody knows, and the question leads to deep number-theoretic waters.

Another instance of playfulness began when I was a graduate student. A prominent visiting mathematician (Lenore Blum) gave a talk about the theory of infinitely precise computation, concerning computational devices able to undertake perfectly accurate real number arithmetic. After the talk, inspired, a group of us speculated about other infinitary notions of computability: what could or should one do with a computational device able to undertake infinitely many steps of computation? We played with the idea, and over several years a theory gradually emerged. Although some of my research colleagues had discouraged me, I ignored them and continued to play with and develop my theory. In time, our ideas grew into a new theory of infinitary computation. We had invented the subject now known as infinite time Turing machines. Those playful ideas led to a new theory that is now studied around the world, with new masters theses written on it and Ph.D. dissertations written on it, and conferences focused on it; our original article now has hundreds of citations.

Play is not always easy. It took years of seriously hard work in the case I just mentioned; but essential to that work was play. So please play! Explore your ideas further, and see where they take you!

Thank you very much.

City University of New York, since 1995

Posted on October 6, 2013 by Joel David Hamkins

I am a professor at the City University of New York, where I have held a faculty position since 1995. (I have taken various leaves of absence for various appointments at other universities.) The City University is the nation’s largest urban university system, with over 250,000 students spread over 11 senior colleges and more, with the doctoral programs centered largely at the Graduate Center, which borrows much of its faculty from the colleges.

The College of Staten Island is one of the senior colleges of the City University of New York, situated on an ample wooded campus surrounded by Willowbrook Park and the Greenbelt nature preserve. I am Professor of Mathematics at the college, and this is where I do all my undergraduate and masters-degree level teaching at CUNY. The mathematics department has research strengths in many areas, including probability, topology, logic and set theory and also applied mathematics, housing the CUNY High Performance Computing Center. Most of our undergraduate mathematics majors aim to become mathematics teachers, and almost all of our masters-level mathematics students are current high school teachers gaining their certifications. I was appointed to the mathematics faculty at the college in 1995, and served as Assistant Professor 1995-1998; Associate Professor 1999-2002; tenure granted 2000; and full Professor since 2003.

I am also on the doctoral faculty of the CUNY Graduate Center, which is home to most of the university’s doctoral programs and is in many ways the center of research life at CUNY. Located in midtown Manhattan just across the corner from the Empire State Building, the Graduate Center forms the main part of the Advanced Learning Superblock, joined by Oxford University Press and the NYPL Science Library. The Mathematics program has diverse research strengths, including a remarkably large faculty in mathematical logic, as does the program in Computer Science. The CUNY Graduate Center Philosophy program is one of the world’s top-rated universities in the area of mathematical logic (and a few years ago was rated number one in this category). We offer a vigorous schedule of logic seminars at the Graduate Center, with many distinguished visiting speakers and audiences filled with faculty and students from around the NYC metropolitan region.

I am a member of the Graduate Center doctoral faculty in three areas:

Professor of Mathematics, doctoral faculty, since 1997.
Professor of Philosophy, doctoral faculty, since 2013.
Professor of Computer Science, doctoral faculty, since 2002.

I regularly teach graduate courses at the Graduate Center and supervise the dissertation research of my Ph.D. graduate students there. I am also currently a member of the Executive Committee of the mathematics program.

Satisfaction is not absolute, CUNY Logic Workshop, September 2013

Posted on September 4, 2013 by Joel David Hamkins

This will be a talk for the CUNY Logic Workshop on September 27, 2013.

Abstract. I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory $𝑀 1$ and $𝑀 2$ , for example, can agree on their natural numbers $⟨ ℕ, +, \cdot, 0, 1, < ⟩ 𝑀 1 = ⟨ ℕ, +, \cdot, 0, 1, < ⟩ 𝑀 2$ , yet disagree on arithmetic truth: they have a sentence $𝜎$ in the language of arithmetic that $𝑀 1$ thinks is true in the natural numbers, yet $𝑀 2$ thinks $\neg 𝜎$ there. Two models of set theory can agree on the natural numbers $ℕ$ and on the reals $ℝ$ , yet disagree on projective truth. Two models of set theory can have the same natural numbers and have a computable linear order in common, yet disagree about whether this order is well-ordered. Two models of set theory can have a transitive rank initial segment $𝑉 𝛿$ in common, yet disagree about whether this $𝑉 𝛿$ is a model of ZFC. The theorems are proved with elementary classical methods.

This is joint work with Ruizhi Yang (Fudan University, Shanghai). We argue, on the basis of these mathematical results, that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

Article

A brief history of set theory…

Posted on August 28, 2013 by Joel David Hamkins

A few months ago, Peter Doyle sent me a cryptic email, containing only the following photo and a subject line containing the title of this post.

I was mystified, until François Dorais subsequently explained that he had given a short presentation on recent progress in foundations for prospective graduate students at Dartmouth.

I’m glad to know that the upcoming generation will have an accurate historical perspective on these things! 🙂

The role of the axiom of foundation in the Kunen inconsistency, CUNY September 2013

Posted on August 28, 2013 by Joel David Hamkins

This will be a talk for the CUNY Set Theory Seminar on September 20, 2013 (date tentative).

Abstract. The axiom of foundation plays an interesting role in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself, for the truth or falsity of the Kunen assertion depends on one’s specific anti-foundational stance. The fact of the matter is that different anti-foundational theories come to different conclusions about this assertion. On the one hand, it is relatively consistent with ZFC without foundation that the Kunen assertion fails, for there are models of ZFC-F in which there are definable nontrivial elementary embeddings $𝑗 : 𝑉 \to 𝑉$ . Indeed, in Boffa’s anti-foundational theory BAFA, the Kunen assertion is outright refutable, and in this theory there are numerous nontrivial elementary embeddings of the universe to itself. Meanwhile, on the other hand, Aczel’s anti-foundational theory GBC-F+AFA, as well as Scott’s theory GBC-F+SAFA and other anti-foundational theories, continue to prove the Kunen assertion, ruling out the existence of a nontrivial elementary embedding $𝑗 : 𝑉 \to 𝑉$ .

This talk covers very recent joint work with Emil Jeřábek, Ali Sadegh Daghighi and Mohammad Golshani, based on an interaction growing out of Ali’s question on MathOverflow, which lead to our recent article, The role of the axiom of foundation in the Kunen inconsistency.

Weak embedding phenomena in $𝜔 1$ -like models of set theory, Collaborative Incentive Research Grant award, 2013-2014

Posted on August 26, 2013 by Joel David Hamkins

V. Gitman, J. D. Hamkins and T. Johnstone, “Weak embedding phenomena in $𝜔 1$ -like models of set theory,” Collaborative Incentive Research Grant award program, CUNY, 2013-2014.

Summary. We propose to undertake research in the area of mathematical logic and foundations known as set theory, investigating a line of research involving an interaction of ideas and methods from several parts of mathematical logic, including set theory, model theory, models of arithmetic and computability theory. Specifically, the project will be to investigate the recently emerged weak embedding phenomenon of set theory, which occurs when there are embeddings between models of set theory (using the model-theoretic sense of embedding here) in situations where there can be no $Δ 0$ -elementary embedding. The existence of the phenomenon was established recently by Hamkins, who showed that every countable model of set theory, including every countable transitive model, is isomorphic to a submodel of its own constructible universe and thus has such a weak embedding into its constructive universe. In this project, we take the next logical step by investigating the weak embedding phenomena in $𝜔 1$ -like models of set theory. The study of $𝜔 1$ -like models of set theory is significant both because these models exhibit interesting second order properties and because their construction out of elementary chains of countable models directs us to create structurally rich countable models.

Exploring the Frontiers of Incompleteness, Harvard, August 2013

Posted on August 20, 2013 by Joel David Hamkins

I will be participating in the culminating workshop of the Exploring the Frontiers of Incompleteness conference series at Harvard University, to take place August 31-September 1, 2013. Rather than conference talks, the program will consist of extended discussion sessions by the participants of the year-long series, with the discussion framed by very brief summary presentations. Peter Koellner asked me to prepare such a presentation on the multiverse conception, and you can see the slides in The multiverse perspective in set theory (Slides).

My previous EFI talk was The multiverse perspective on determinateness in set theory, based in part on my paper The set-theoretical multiverse.

Satisfaction is not absolute, Connecticut, October 2013

Posted on August 18, 2013 by Joel David Hamkins

This will be a talk for the Logic Seminar in the Mathematics Department at the University of Connecticut in Storrs on October 25, 2013.

Abstract. The satisfaction relation $N ⊧ 𝜑 [⃗ 𝑎]$ of first-order logic, it turns out, is less absolute than might have been supposed. Two models of set theory, for example, can agree on their natural numbers and on what they think is the standard model of arithmetic $⟨ ℕ, +, \cdot, 0, 1, < ⟩$ , yet disagree on their theories of arithmetic truth, the first-order truths of this structure. Two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth. Two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree about whether it is well-ordered. Two models of set theory can have a transitive rank initial segment $𝑉 𝛿$ in common, yet disagree about whether it is a model of ZFC. The arguments rely mainly on elementary classical methods.

This is joint work with Ruizhi Yang (Fudan University, Shanghai), and our manuscript will be available soon, in which we prove these and several other very general facts showing that satisfaction is not absolute. On the basis of these mathematical results, we mount a philosophical argument that a commitment to the determinateness of truth in a structure, such as the case of arithmetic truth in the standard model of arithmetic, cannot result solely from the determinateness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

University of Connecticut Logic Seminar | Article

Workshop on paraconsistent set theory, Connecticut, October 2013

Posted on August 17, 2013 by Joel David Hamkins

I’ll be participating in a workshop at the University of Connecticut, Storrs, philosophy department on October 26-27, 2013, on paraconsistent set theory, organized by Graham Priest and JC Beall. I am given to understand that part of the goal is to develop additional or improved model-construction methods, with which one might expand the range of possible behaviors that we know about.

Resurrection axioms and uplifting cardinals

Posted on July 15, 2013 by Joel David Hamkins

[bibtex key=HamkinsJohnstone2014:ResurrectionAxiomsAndUpliftingCardinals]

Abstract. We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of uplifting cardinal.

Many classical forcing axioms can be viewed, at least informally, as the claim that the universe is existentially closed in its forcing extensions, for the axioms generally assert that certain kinds of filters, which could exist in a forcing extension $𝑉 [𝐺]$ , exist already in $𝑉$ . In several instances this informal perspective is realized more formally: Martin’s axiom is equivalent to the assertion that $𝐻 𝔠$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $𝐻 𝔠 ≺ Σ 1 𝑉 [𝐺]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $𝐻 𝜔 2$ is existentially closed in all proper forcing extensions, or $𝐻 𝜔 2 ≺ Σ 1 𝑉 [𝐺]$ ; and there are other similar instances.

In model theory, a submodel $𝑀 \subset 𝑁$ is existentially closed in $𝑁$ if existential assertions true in $𝑁$ about parameters in $𝑀$ are true already in $𝑀$ , that is, if $𝑀$ is a $Σ 1$ -elementary substructure of $𝑁$ , which we write as $𝑀 ≺ Σ 1 𝑁$ . Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.

Elementary Fact. If $M$ is a submodel of $N$ , then the following are equivalent.

The model $M$ is existentially closed in $N$ .
$M \subset N$ has resurrection. That is, there is a further extension $M \subset N \subset M +$ for which $M ≺ M +$ .

We call this resurrection because although certain truths in $M$ may no longer hold in the extension $N$ , these truths are nevertheless revived in light of $M ≺ M +$ in the further extension to $M +$ .

In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $M +$ arising in the fact, and in this context the equivalence of (1) and (2) breaks own, although the converse implication $(2) \to (1)$ always holds, and every instance of resurrection implies the corresponding instance of existential closure. This key observation leads us to the main unifying theme of this article, the idea that

resurrection may allow us to formulate more robust forcing axioms

than existential closure or than combinatorial assertions about filters and dense sets. We therefore introduce in this paper a spectrum of new forcing axioms utilizing the resurrection concept.

Main Definition. Let $Γ$ be a fixed definable class of forcing notions.

The resurrection axiom $R A (Γ)$ is the assertion that for every forcing notion $ℚ \in Γ$ there is further forcing $ℝ$ , with $⊢ ℚ ℝ \in Γ$ , such that if $𝑔 * ℎ \subset ℚ * ℝ$ is $𝑉$ -generic, then $𝐻 𝔠 ≺ 𝐻 𝑉 [𝑔 * ℎ] 𝔠$ .
The weak resurrection axiom $w R A (Γ)$ is the assertion that for every $ℚ \in Γ$ there is further forcing $ℝ$ , such that if $𝑔 * ℎ \subset ℚ * ℝ$ is $𝑉$ -generic, then $𝐻 𝔠 ≺ 𝐻 𝑉 [𝑔 * ℎ] 𝔠$ .

The main result is to prove that various formulations of the resurrection axioms are equiconsistent with the existence of an uplifting cardinal, where an inaccessible cardinal $𝜅$ is uplifting, if there are arbitrarily large inaccessible cardinals $𝛾$ for which $𝐻 𝜅 ≺ 𝐻 𝛾$ . This is a rather weak large cardinal notion, having consistency strength strictly less than the existence of a Mahlo cardinal, which is traditionally considered to be very low in the large cardinal hierarchy. One highlight of the article is our development of “the world’s smallest Laver function,” the Laver function concept for uplifting cardinals, and we perform an analogue of the Laver preparation in order to achieve the resurrection axiom for c.c.c. forcing.

Main Theorem. The following theories are equiconsistent over ZFC:

There is an uplifting cardinal.
$R A (a l l)$ .
$R A (c c c)$ .
$R A (s e m i p r o p e r) + \neg C H$ .
$R A (p r o p e r) + \neg C H$ .
For some countable ordinal $𝛼$ , the axiom $R A (𝛼 - p r o p e r) + \neg C H$ .
$R A (a x i o m - A) + \neg C H$ .
$w R A (s e m i p r o p e r) + \neg C H$ .
$w R A (p r o p e r) + \neg C H$ .
For some countable ordinal $𝛼$ , the axiom $w R A (𝛼 - p r o p e r) + \neg C H$ .
$w R A (a x i o m - A) + \neg C H$ .
$w R A (c o u n t a b l y c l o s e d) + \neg C H$ .

The proof outline proceeds in two directions: on the one hand, the resurrection axioms generally imply that the continuum $𝔠$ is uplifting in $𝐿$ ; and conversely, given any uplifting cardinal $𝜅$ , we may perform a suitable lottery iteration of $Γ$ forcing to obtain the resurrection axiom for $Γ$ in a forcing extension with $𝜅 = 𝔠$ .

In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $𝐴 \subset 𝔠$ and ask for extensions of the form $⟨ 𝐻 𝔠, \in, 𝐴 ⟩ ≺ ⟨ 𝐻 𝑉 [𝑔 * ℎ] 𝔠, \in, 𝐴 * ⟩$ , for some $𝐴 * \subset 𝔠 𝑉 [𝑔 * ℎ]$ in the extension. In that article, we prove the equiconsistency of various formulations of boldface resurrection with the existence of a strongly uplifting cardinal, which we prove is the same as a superstrongly unfoldable cardinal.

Superstrong and other large cardinals are never Laver indestructible

Posted on July 14, 2013 by Joel David Hamkins

[bibtex key=BagariaHamkinsTsaprounisUsuba2016:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible]

Abstract. Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, $1$ -extendible cardinals, $0$ -extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, $Σ 𝑛$ -reflecting cardinals, $Σ 𝑛$ -correct cardinals and $Σ 𝑛$ -extendible cardinals (all for $𝑛 \geq 3$ ) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if $𝜅$ exhibits any of them, with corresponding target $𝜃$ , then in any forcing extension arising from nontrivial strategically $< 𝜅$ -closed forcing $ℚ \in 𝑉 𝜃$ , the cardinal $𝜅$ will exhibit none of the large cardinal properties with target $𝜃$ or larger.

The large cardinal indestructibility phenomenon, occurring when certain preparatory forcing makes a given large cardinal become necessarily preserved by any subsequent forcing from a large class of forcing notions, is pervasive in the large cardinal hierarchy. The phenomenon arose in Laver’s seminal result that any supercompact cardinal $𝜅$ can be made indestructible by $< 𝜅$ -directed closed forcing. It continued with the Gitik-Shelah treatment of strong cardinals; the universal indestructibility of Apter and myself, which produced simultaneous indestructibility for all weakly compact, measurable, strongly compact, supercompact cardinals and others; the lottery preparation, which applies generally to diverse large cardinals; work of Apter, Gitik and Sargsyan on indestructibility and the large-cardinal identity crises; the indestructibility of strongly unfoldable cardinals; the indestructibility of Vopenka’s principle; and diverse other treatments of large cardinal indestructibility. Based on these results, one might be tempted to the general conclusion that all the usual large cardinals can be made indestructible.

In this article, my co-authors and I temper that temptation by proving that certain kinds of large cardinals cannot be made nontrivially indestructible. Superstrong cardinals, we prove, are never Laver indestructible. Consequently, neither are almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals and $1$ -extendible cardinals, to name a few. Even the $0$ -extendible cardinals are never indestructible, and neither are weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, strongly uplifting cardinals, superstrongly unfoldable cardinals, $Σ 𝑛$ -reflecting cardinals, $Σ 𝑛$ -correct cardinals and $Σ 𝑛$ -extendible cardinals, when $𝑛 \geq 3$ . In fact, all these large cardinal properties are superdestructible, in the sense that if $𝜅$ exhibits any of them, with corresponding target $𝜃$ , then in any forcing extension arising from nontrivial strategically $< 𝜅$ -closed forcing $ℚ \in 𝑉 𝜃$ , the cardinal $𝜅$ will exhibit none of the large cardinal properties with target $𝜃$ or larger. Many quite ordinary forcing notions, which one might otherwise have expected to fall under the scope of an indestructibility result, will definitely ruin all these large cardinal properties. For example, adding a Cohen subset to any cardinal $𝜅$ will definitely prevent it from being superstrong—as well as preventing it from being uplifting, $Σ 3$ -correct, $Σ 3$ -extendible and so on with all the large cardinal properties mentioned above—in the forcing extension.

Main Theorem.

Superstrong cardinals are never Laver indestructible.
Consequently, almost huge, huge, superhuge and rank-into-rank cardinals are never Laver indestructible.
Similarly, extendible cardinals, $1$ -extendible and even $0$ -extendible cardinals are never Laver indestructible.
Uplifting cardinals, pseudo-uplifting cardinals, weakly superstrong cardinals, superstrongly unfoldable cardinals and strongly uplifting cardinals are never Laver indestructible.
$Σ 𝑛$ -reflecting and indeed $Σ 𝑛$ -correct cardinals, for each finite $𝑛 \geq 3$ , are never Laver indestructible.
Indeed—the strongest result here, because it is the weakest notion— $Σ 3$ -extendible cardinals are never Laver indestructible.

In fact, each of these large cardinal properties is superdestructible. Namely, if $𝜅$ exhibits any of them, with corresponding target $𝜃$ , then in any forcing extension arising from nontrivial strategically $< 𝜅$ -closed forcing $ℚ \in 𝑉 𝜃$ , the cardinal $𝜅$ will exhibit none of the mentioned large cardinal properties with target $𝜃$ or larger.

The proof makes use of a detailed analysis of the complexity of the definition of the ground model in the forcing extension. These results are, to my knowledge, the first applications of the ideas of set-theoretic geology not making direct references to set-theoretically geological concerns.

Theorem 10 in the article answers (the main case of) a question I had posed on MathOverflow, namely, Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals? I had been specifically interested there to know whether a cardinal $𝜅$ necessarily becomes definable after adding a Cohen subset to it, and theorem 10 shows indeed that it does: after adding a Cohen subset to a cardinal, it becomes $Σ 3$ -definable in the extension, and this fact can be seen as explaining the main theorem above.

Universal structures: the countable random graph, the surreal numbers and the hypnagogic digraph, Swarthmore College, October 2013

Posted on July 10, 2013 by Joel David Hamkins

I’ll be speaking for the Swarthmore College Mathematics and Statistics Colloquium on October 8th, 2013.

220px-Swarthmore_College_Logo_Current

Abstract. I’ll be giving an introduction to universal structures in mathematics, where a structure $M$ is universal for a class of structures, if every structure in that class arises as (isomorphic to) a substructure of $M$ . For example, Cantor proved that the rational line $ℚ$ is universal for all countable linear orders. Is a corresponding fact true of the real line for linear orders of that size? Are there countably universal partial orders? Is there a countably universal graph? directed graph? acyclic digraph? Is there a countably universal group? We’ll answer all these questions and more, with an account of the countable random graph, generalizations to the random graded digraphs, Fraïssé limits, the role of saturation, the surreal numbers and the hypnagogic digraph. The talk will conclude with some very recent work on universality amongst the models of set theory.

Poster

Approximation and cover properties propagate upward

Posted on July 1, 2013 by Joel David Hamkins

I should like to record here the proof of the following fact, which Jonas Reitz and I first observed years ago, when he was my graduate student, and I recall him making the critical observation.

It concerns the upward propagation of the approximation and cover properties, some technical concepts that lie at the center of my paper, Extensions with he approximation and cover properties have no new large cardinals, and which are also used in my proof of Laver’s theorem on the definability of the ground model, and which figure in Jonas’s work on the ground axiom.

The fact has a curious and rather embarrassing history, in that Jonas and I have seen an unfortunate cycle, in which we first proved the theorem, and then subsequently lost and forgot our own proof, and then lost confidence in the fact, until we rediscovered the proof again. This cycle has now repeated several times, in absurd mathematical comedy, and each time the proof was lost, various people with whom we discussed the issue sincerely doubted that it could be true. But we are on the upswing now, for in response to some recently expressed doubts about the fact, although I too was beginning to doubt it again, I spent some time thinking about it and rediscovered our old proof! Hurrah! In order to break this absurd cycle, however, I am now recording the proof here in order that we may have a place to point in the future, to give the theorem a home.

Although the fact has not yet been used in any application to my knowledge, it strikes me as inevitable that this fundamental fact about the approximation and cover properties will eventually find an important use.

Definition. Assume $𝛿$ is a cardinal in $𝑉$ and $𝑊 \subset 𝑉$ is a transitive inner model of set theory.

The extension $𝑊 \subset 𝑉$ satisfies the $𝛿$ -approximation property if whenever $𝐴 \subset 𝑊$ is a set in $𝑉$ and $𝐴 \cap 𝑎 \in 𝑊$ for any $𝑎 \in 𝑊$ of size less than $𝛿$ in $𝑊$ , then $𝐴 \in 𝑊$ .
The extension $𝑊 \subset 𝑉$ satisfies the $𝛿$ -cover property if whenever $𝐴 \subset 𝑊$ is a set of size less than $𝛿$ in $𝑉$ , then there is a covering set $𝐵 \in 𝑊$ with $𝐴 \subset 𝐵$ and $| 𝐵 | 𝑊 < 𝛿$ .

Theorem. If $𝑊 \subset 𝑉$ has the $𝛿$ -approximation and $𝛿$ -cover properties and $𝛿 < 𝛾$ are both infinite cardinals in $𝑉$ , then it also has the $𝛾$ -approximation and $𝛾$ -cover properties.

Proof. First, notice that the $𝛿$ -approximation property trivially implies the $𝛾$ -approximation property for any larger cardinal $𝛾$ . So we need only verify the $𝛾$ -cover property, and this we do by induction. Note that the limit case is trivial, since if the cover property holds at every cardinal below a limit cardinal, then it trivially holds at that limit cardinal, since there are no additional instances of covering to be treated. Thus, we reduce to the case $𝛾 = 𝛿 +$ , meaning $(𝛿 +) 𝑉$ , but we must allow that $𝛿$ may be singular here.

If $𝛿$ is singular, then we claim that the $𝛿$ -cover property alone implies the $𝛿 +$ -cover property: if $𝐴 \subset 𝑊$ has size $𝛿$ in $𝑉$ , then by the singularity of $𝛿$ we may write it as $𝐴 = ⋃ 𝛼 \in 𝐼 𝐴 𝛼$ , where each $𝐴 𝛼$ and $𝐼$ have size less than $𝛿$ . By the $𝛿$ -cover property, there are covers $𝐴 𝛼 \subset 𝐵 𝛼 \in 𝑊$ with $𝐵 𝛼$ of size less than $𝛿$ in $𝑊$ . Furthermore, the set ${𝐵 𝛼 ∣ 𝛼 \in 𝐼}$ itself is covered by some set $B \in 𝑊$ of size less than $𝛿$ in $𝑊$ . That is, we cover the small set of small covers. We may assume that every set in $B$ has size less than $𝛿$ , by discarding those that aren’t, and so $𝐵 = ⋃ B$ is a set in $𝑊$ that covers $𝐴$ and has size at most $𝛿$ there, since it is small union of small sets, thereby verifying this instance of the $𝛾$ -cover property.

If $𝛿$ is regular, consider a set $𝐴 \subset 𝑊$ with $𝐴 \in 𝑉$ of size $𝛿$ in $𝑉$ , so that $𝐴 = {𝑎 𝜉 ∣ 𝜉 < 𝛿}$ . For each $𝛼 < 𝛿$ , the initial segment ${𝑎 𝜉 ∣ 𝜉 < 𝛼}$ has size less than $𝛿$ and is therefore covered by some $𝐵 𝛼 \in 𝑊$ of size less than $𝛿$ in $𝑊$ . By adding each $𝐵 𝛼$ to what we are covering at later stages, we may assume that they form an increasing tower: $𝛼 < 𝛽 \to 𝐵 𝛼 \subset 𝐵 𝛽$ . The choices $𝛼 \mapsto 𝐵 𝛼$ are made in $𝑉$ . Let $𝐵 = ⋃ 𝛼 𝐵 𝛼$ , which certainly covers $𝐴$ . Observe that for any set $𝑎 \in 𝑊$ of size less than $𝛿$ , it follows by the regularity of $𝛿$ that $𝐵 \cap 𝑎 = 𝐵 𝛼 \cap 𝑎$ for all sufficiently large $𝛼$ . Thus, all $𝛿$ -approximations to $𝐵$ are in $𝑊$ and so $𝐵$ itself is in $𝑊$ by the $𝛿$ -approximation property, as desired. Note that $𝐵$ has size less than $𝛾$ in $𝑊$ , because it has size $𝛿$ in $𝑉$ , and so we have verified this instance of the $𝛾$ -cover property for $𝑊 \subset 𝑉$ .

Thus, in either case we’ve established the $𝛾$ -cover property for $𝑊 \subset 𝑉$ , and the proof is complete. QED

(Thanks to Thomas Johnstone for some comments and for pointing out a simplification in the proof: previously, I had reduced without loss of generality to the case where $𝐴$ is a set of ordinals of order type $𝛿$ ; but Tom pointed out that the general case is not actually any harder. And indeed, Jonas dug up some old notes to find the 2008 version of the argument, which is essentially the same as what now appears here.)

Note that without the $𝛿$ -approximation property, it is not true that the $𝛿$ -cover property transfers upward. For example, every extension has the $ℵ 0$ -cover property.

Research on the weak embedding phenomenon in set theory, PSC-CUNY grant award, 2013 – 2014

Posted on June 30, 2013 by Joel David Hamkins

J. D. Hamkins, Research on the weak embedding phenomenon in set theory, PSC-CUNY grant award 44, traditional B, 2013 – 2014.

Quoted in New Scientist magazine, June 2013

Posted on June 26, 2013 by Joel David Hamkins

I was quoted briefly in Mathematicians think like machines for perfect proofs, New Scientist, by Jacob Aron, June 26, 2013. (Actually, my quote there is a little out of context, as my remark there was referring only to research in set theory, where anyone would view the switch to another foundation as a distraction.)

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share:

Share: