# Satisfaction is not absolute

• J. D. Hamkins and R. Yang, “Satisfaction is not absolute,” , pp. 1-34. (under review)
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author = {Joel David Hamkins and Ruizhi Yang},
title = {Satisfaction is not absolute},
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Abstract. We prove that the satisfaction relation $\mathcal{N}\satisfies\varphi[\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal{N}$ and the formulas $\varphi$ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; 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 $\langle H_{\omega_2},{\in}\rangle$ or the same rank-initial segment $\langle V_\delta,{\in}\rangle$, yet disagree on which assertions are true in these structures.

On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure $\N=\{ 0,1,2,\ldots\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

Many mathematicians and philosophers regard the natural numbers $0,1,2,\ldots\,$, along with their usual arithmetic structure, as having a privileged mathematical existence, a Platonic realm in which assertions have definite, absolute truth values, independently of our ability to prove or discover them. Although there are some arithmetic assertions that we can neither prove nor refute—such as the consistency of the background theory in which we undertake our proofs—the view is that nevertheless there is a fact of the matter about whether any such arithmetic statement is true or false in the intended interpretation. The definite nature of arithmetic truth is often seen as a consequence of the definiteness of the structure of arithmetic $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$ itself, for if the natural numbers exist in a clear and distinct totality in a way that is unambiguous and absolute, then (on this view) the first-order theory of truth residing in that structure—arithmetic truth—is similarly clear and distinct.

Feferman provides an instance of this perspective when he writes (Feferman 2013, Comments for EFI Workshop, p. 6-7) :

In my view, the conception [of the bare structure of the natural numbers] is completely clear, and thence all arithmetical statements are definite.

It is Feferman’s thence’ to which we call attention.  Martin makes a similar point (Martin, 2012, Completeness or incompleteness of basic mathematical concepts):

What I am suggesting is that the real reason for confidence in first-order completeness is our confidence in the full determinateness of the concept of the natural numbers.

Many mathematicians and philosophers seem to share this perspective. The truth of an arithmetic statement, to be sure, does seem to depend entirely on the structure $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$, with all quantifiers restricted to $\N$ and using only those arithmetic operations and relations, and so if that structure has a definite nature, then it would seem that the truth of the statement should be similarly definite.

Nevertheless, in this article we should like to tease apart these two ontological commitments, arguing that the definiteness of truth for a given mathematical structure, such as the natural numbers, the reals or higher-order structures such as $H_{\omega_2}$ or $V_\delta$, does not follow from the definite nature of the underlying structure in which that truth resides. Rather, we argue that the commitment to a theory of truth for a structure is a higher-order ontological commitment, going strictly beyond the commitment to a definite nature for the underlying structure itself.

We make our argument in part by proving that different models of set theory can have a structure identically in common, even the natural numbers, yet disagree on the theory of truth for that structure.

Theorem.

• Two models of set theory can have the same structure of arithmetic $$\langle\N,{+},{\cdot},0,1,{\lt}\rangle^{M_1}=\langle\N,{+},{\cdot},0,1,{\lt}\rangle^{M_2},$$yet disagree on the theory of arithmetic 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 a well-order.
• Two models of set theory that have the same natural numbers and the same reals, yet disagree on projective truth.
• Two models of set theory can have a transitive rank initial segment in common $$\langle V_\delta,{\in}\rangle^{M_1}=\langle V_\delta,{\in}\rangle^{M_2},$$yet disagree about whether it is a model of ZFC.

The proofs use only elementary classical methods, and might be considered to be a part of the folklore of the subject of models of arithmetic. The paper includes many further examples of the phenomenon, and concludes with a philosophical discussion of the issue of definiteness, concerning the question of whether one may deduce definiteness-of-truth from definiteness-of-objects and definiteness-of-structure.

# The role of the foundation axiom in the Kunen inconsistency

• A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek, “The role of the foundation axiom in the Kunen inconsistency.” (under review)
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title = {The role of the foundation axiom in the {Kunen} inconsistency},
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In this article, we examine the role played by the axiom of foundation in the well-known Kunen inconsistency, the theorem asserting that there is no nontrivial elementary embedding of the set-theoretic universe to itself. All the standard proofs of the Kunen inconsistency make use of the axiom of foundation (see Kanamori’s books and also Generalizations of the Kunen inconsistency), and this use is essential, assuming that $\ZFC$ is consistent, because as we shall show there are models of $\ZFCf$ that admit nontrivial elementary self-embeddings and even nontrivial definable automorphisms. Meanwhile, a fragment of the Kunen inconsistency survives without foundation as the claim in $\ZFCf$ that there is no nontrivial elementary self-embedding of the class of well-founded sets. Nevertheless, some of the commonly considered anti-foundational theories, such as the Boffa theory $\BAFA$, prove outright the existence of nontrivial automorphisms of the set-theoretic universe, thereby refuting the Kunen assertion in these theories.  On the other hand, several other common anti-foundational theories, such as Aczel’s anti-foundational theory $\ZFCf+\AFA$ and Scott’s theory $\ZFCf+\text{SAFA}$, reach the opposite conclusion by proving that there are no nontrivial elementary embeddings from the set-theoretic universe to itself. Our summary conclusion, therefore, is that the resolution of the Kunen inconsistency in set theory without foundation depends on the specific nature of one’s anti-foundational stance.

This is joint work with Ali Sadegh Daghighi, Mohammad Golshani, myself and Emil Jeřábek, which grew out of our interaction on Ali’s question on MathOverflow, Is there any large cardinal beyond the Kunen inconsistency?

# Just do it?

November 3, 2013
BARBARA GAIL MONTERO interviewed by Joe Gelonesi along with Richard Menary on The Philosopher’s Zone.

Famed choreographer George Balanchine was reputed to have said, “don’t think, dear: just do”. The idea that champion performers switch off their brains to achieve their best has taken hold in popular imagination. Just do it promises an existential zone where real players hit the heights whilst the rest shuffle to the back of the pack. We explore Expert action, a philosophical football punted between those for automatic responses and those who hear the whirring cogs.

→ go listen to Just Do It

Barbara was previously interviewed on Leading Minds, with David Brendel.

# Rubik’s cube competition, CSI, November 14, 2013

Come and compete in the CSI Rubik’s cube competition!

November 14, 2013, College of Staten Island of CUNY, 1S-107, 2:30 pm.

Sponsored by MTH 339, and the CSI Math Club.

As a part of the undergraduate course in abstract algebra (MTH 339), which I am teaching this semester at the College of Staten Island, we shall hold a Rubik’s cube competition on November 14th.  In class, I have used the Rubik’s cube as a source of examples to explain various group-theoretic concepts, and I have encouraged the students to learn to solve the cube.  Several have now already mastered it, and there seems lately to be a lot of Rubik’s cube activity in the math department.  (I am giving extra credit for any student who can solve a scrambled cube in my office.)

Several students have learned how to solve the cube from the following video, which explains one of the layer-based solution methods:

The Competition.  On November 14, 2013, we will have the Rubik’s cube competition, with several rounds of competition, to see who can solve the cube the fastest.  Prizes will be awarded, and best of all, there will be free pizza!

Results Of the Competition

The event has now taken place. We had 15 competitors, from all around the College and beyond.  We organized two qualifying heats of 7 and 8 competitors, respectively, taking the top four from each qualtifying heat to form the quarterfinalist competitors. The top four of these formed the semifinalist competitors. And the top two of these headed off in the championship round.  The champion, Sam Obisanya, won all the rounds in which he competed, and his cube was a blaze of lightning color as he solved it.  Honorable mention goes especially to Oveen Joseph, who faced Sam in the championship round and who came out to the college from middle school I72, where he is in the 7th grade, and also to Justin Mills, who had extremely fast times.

## Quarterfinals:

Itiel Cohen (CSI math major)

William George (CSI math major)

Oveen Joseph (middle school I72, 7th grade)

Wing Yang Law (CSI math major)

Justin Mills (CSI psychology major)

Mike Siozios (CSI math major)

Sam Obisanya (CSI nursing major)

James Yap (CSI math major)

Oveen Joseph

Justin Mills

Sam Obisanya

James Yap

Oveen Joseph

Sam Obisanya

## Final Champion:

Sam Obisanya

Congratulations to our champion and to all the competitors.

# Win the game of Nim! CSI Math Club, October, 2013

This will be a talk for the CSI Math Club on October 31, 2013 at 2:30 pm in room 1S-107.

Abstract  Come and learn how to play and win the game of Nim!  The game has two players, faced with several small piles of blocks.  Each player, on their turn, can remove one or more blocks from one pile, but only one pile. (Removing a whole pile is fine.)  The player who removes the last block wins.  This simple-to-describe game is maddening for those who don’t know the secret mathematical winning strategy.  Come and learn the mathematical secret that will allow you to win every time against someone who doesn’t know it.

# Doubled, squared, cubed: a math game for kids or anyone

Doubled, squared, cubed is a great math game to play with kids or anyone interested in math.  It is a talking game, requiring no pieces or physical objects, played by a group of two or more people at almost any level of mathematical difficulty, while sitting, walking, boating or whatever.  We play it in our family (two kids, ages 7 and 11) when we are sitting around a table or when walking somewhere or when traveling by train.  I fondly recall playing the game with my brothers and sisters in my own childhood.

The game proceeds by first agreeing on an allowed number range.  For youngsters, perhaps one wants to allow the integers from 0 to 100, inclusive, but one will want to have negative numbers soon enough, and of course much more sophisticated play is possible. Eventually, one lessens or even abandons the restriction altogether. The first player offers a number, and each subsequent player in turn offers a mathematical operation, which is to be applied to the current number, which must not be mentioned explicitly.  The resulting number must be in the allowed number range.

The goal of the game is successfully to keep track of the number as it changes, and to offer an operation that makes sense with that number, while staying within the range of allowed numbers.  The point is to have some style, to offer an operation that proves that you know what the number is, without stating the number explicitly.  Perhaps your operation makes the new number a nice round number, or perhaps your operation can seldom be legally applied, and so applying it indicates that you know it is allowed to do so.  You must offer only operations that you yourself can compute, and which do not rely on hidden information (for example, “times the number of grapes I ate at breakfast” is not really permissible).

A losing move is one that doesn’t make sense or that results in a number outside the allowed range. In this case, the game can continue without that person, and the last person left wins.  It is not allowed to offer an operation that can always be applied, such as “times zero” or “minus itself“, or which can always be applied immediately after the previous operation, such as saying “times two” right after someone said, “cut in half”.  But in truth, the main point is to have some fun, rather than to win. Part of the game is surely simply to talk about new mathematical operations, and we usually take time out to discuss or explain any mathematical issue that may come up.  So this is an enjoyable way for the kids to encounter new mathematical ideas.

Let me simply illustrate a typical progression of the game, as it might be played in my family:

Hypatia: one

Barbara: doubled

Horatio: squared

Joel: cubed

Hypatia: plus 36

Barbara: square root

Horatio: divided by 5

Joel: times 50

Hypatia: minus 100

Barbara: times 6 billion

Horatio: plus 99

Joel: divided by 11

Hypatia: plus 1

Barbara: to the power of two

Horatio: minus 99

Joel: times itself 6 billion times

Hypatia: minus one

Barbara: divided by ten thousand

Horatio: plus 50

Joel: plus half of itself

Hypatia: plus 25

Barbara: minus 99

Horatio: cube root

Joel: next prime number above

Hypatia: ten’s complement

Barbara: second square number above

Horatio: reverse the digits

Joel: plus 3 more than six squared

Hypatia: minus 100

and so on!

As the kids get older, we gradually incorporate more sophisticated elements into the game, and take a little time out to explain to young Hypatia, for example, what it means to cube a number, to take a number to the power two, or what a prime number is.  I remember playing the game with my math-savvy siblings when I was a kid, and the running number was sometimes something like $\sqrt{29}$ or $2+3i$, and a correspondingly full range of numbers and operations. It is fine to let the youngest drop out after a while, and continue with the older kids with more sophisticated operations; the youngsters will rejoin in the next round.  In my childhood, we had a “challenge” rule, used when someone suspects that someone else doesn’t know the number: when challenged, the person should say the number; if incorrect, they are out, and otherwise the challenger is out.

Last weekend, I played the game with Horatio and Hypatia as we walked through Central Park to the Natural History Museum, and they conspired in whispering tones to mess me up, until finally I lost track of the number and they won…

# Address at the Dean’s List Ceremony

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

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.

The CUNY Graduate Center 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 currently on the doctoral faculty in three Graduate Center programs:

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

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 $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle^{M_1}=\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle^{M_2}$, yet disagree on arithmetic truth: they have a sentence $\sigma$ in the language of arithmetic that $M_1$ thinks is true in the natural numbers, yet $M_2$ thinks $\neg\sigma$ there. Two models of set theory can agree on the natural numbers $\mathbb{N}$ and on the reals $\mathbb{R}$, 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 $V_\delta$ in common, yet disagree about whether this $V_\delta$ 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…

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

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 $\omega_1$-like models of set theory, Collaborative Incentive Research Grant award, 2013-2014

V. Gitman, J. D. Hamkins and T. Johnstone, “Weak embedding phenomena in $\omega_1$-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 $\Delta_0$-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 $\omega_1$-like models of set theory. The study of $\omega_1$-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

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

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 $\mathcal{N}\models\varphi[\vec a]$ 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 $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, 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 $V_\delta$ 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.

# Workshop on paraconsistent set theory, Connecticut, October 2013

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.