Author Archives: Joel David Hamkins

Algebraicity and implicit definability in set theory

Posted on by
2

[bibtex key=HamkinsLeahy2016:AlgebraicityAndImplicitDefinabilityInSetTheory]

We aim in this article to analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity and its companion concept of implicit definability. In place of the class HOD of hereditarily ordinal definable sets, for example, we consider the class HOA of hereditarily ordinal-algebraic sets. In place of the pointwise definable models of set theory, we examine its (pointwise) algebraic models. And in place of Gödel’s constructible universe L, obtained by iterating the definable power set operation, we introduce the implicitly constructible universe Imp, obtained by iterating the algebraic or implicitly definable power set operation. In each case we investigate how the change from definability to algebraicity affects the nature of the resulting concept. We are especially intrigued by Imp, for it is a new canonical inner model of ZF whose subtler properties are just now coming to light. Open questions about Imp abound.

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

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

The main theorems of the paper are:

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

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

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

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

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

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

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

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

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

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

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

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

Panelist at the Infinity Salon for the World Science Festival, NYC, June 2013

Posted on by
3

The World Science Festival is coming to New York (May 29 — June 2), with numerous interesting events, including the infinity salon, an intimate, free-wheeling discussion about infinity, entitled The future of Infinity:  Solving Math’s most notorious problem, for an audience of experts.

I shall be on the panel along with Steven Strogatz, W. Hugh Woodin and others,  moderated by Keith Devlin.

I’m looking forward to it!

(Note that tickets are required for many events.)

From the salon web site:

Saturday, June 1, 2013

3:30 PM – 5:00 PM

 

In 1873, Georg Cantor proved that there are different sizes of infinity. Cantor tried to answer the question by proposing the Continuum Hypothesis. A solution of sorts was found in 1963, but the answer—proof that there was no proof—raised questions about the foundations of mathematics. Most deemed that counting the infinite was beyond mathematical precision. Recently, progress has been made, and the Continuum Hypothesis might have a definite answer—true or false.

 

The World Science Festival’s annual salon series offers in-depth conversations with leading scientists, extending the discussion of the Festival’s premiere public programs to graduate students, postdocs, faculty and well-informed members of the general public.

Keith Devlin’s blog post about the discussion

More math for six-year-olds: Win at Nim!

Posted on by
1

The latest installment of math for six-year-olds…

Win at Nim!

Win at Nim!
Fold up the bottom flap to prevent parents from learning the super-secret strategy.

This morning once again I went into my daughter’s first-grade classroom, full of inquisitive six-and-seven-year-old girls, and made a mathematical presentation on the game of Nim.   

                   Win at Nim!

The game of Nim, I explained, begins with one player setting up a number of stacks of blocks,while the opponent chooses whether to go first or second.  Taking turns, each player removes one or more blocks from a stack of their choosing. (It is fine to take a whole stack on your turn.) The player who takes the last block wins.

 

DSC00078

We demonstrated the game by playing a number of exhibition rounds, and then the girls divided into pairs to play each other and also me.  They were surprised that I was able to win against them every single time.  In explanation, I told them that this was because in the game of Nim, there is a super-secret mathematical strategy!  Did they want to learn?  Yes!  I took as my goal that they would all learn the Nim strategy, so that they could go home and confound their parents by beating them again and again at the game.

Since this was a first-grade class, we concentrated at first on games with stacks of heights 1, 2 and 3 only, a special case of the game which can still challenge adults, but for which six-year-olds can easily learn the winning strategy.

Two balanced stacks

After gaining some familiarity with the game by playing several rounds amongst each other, we gathered again for the secret strategy session. We began by thinking about the case of a game of Nim with only two stacks. They had noticed that sometimes when I played them, I had made copying moves; and indeed I had purposely said, “I copy you,” each time this had occurred.  The copying idea is surely appealing when there are only two stacks.  After some discussion, the girls realized that with just two stacks, if one played so as to equalize them, then one would always be able to copy the opponent’s move.  In particular, this copying strategy would ensure that one had a move to make whenever the opponent did, and so one would win the game.

DSC00076

A balanced position

In short order, the girls also realized that if one had any number of pairs of such balanced stacks—so that every stack had a partner—then the whole position was also winning (for one to give to the other player), since one could copy a move on any stack by making the corresponding move on the partner stack.  Thus, we deduced that if we could match up stacks of equal height in pairs, then we had a winning strategy, the strategy to copy any move on a partner stack.

In particular, this balancing idea provides a complete winning strategy in the case of Nim games for which all stacks have height one or two.  One should play so as to give a balanced position to one’s opponent, namely, a position with an even number of stacks of height one and an even number of stacks of height two.  Any unbalanced position can always be balanced in this way, and any move on a balanced position will unbalance it.

DSC00074

1+2+3 counts as balanced

To handle positions with stacks of height three, the super-secret trick is that one can balance a stack of height three either with another stack of height three, of course, but also with two stacks:  one of height one and one of height two.   Thus, one should regard a stack of height three as consisting of two sub-stacks, one of height one and one of height two, for the purposes of balancing. Thus, the Nim position 1+2+3 counts as balanced, since the 3 counts as 2+1, which balances the other stacks.  The 1+2+3 position has two stacks of height two and two of height one, when one regards the stack of height three as having a substack of height two and a substack of height one.

In this way, one arrives at a complete winning strategy for Nim positions involving stacks of height at most three, and furthermore, this is a strategy that can be mastered by first-graders. The strategy is to strive to balance the position.  If you always give your opponent a balanced position, then  you will win!  Faced with an unbalanced position, you can always find a balancing move, and any move on an balanced position will unbalance it.  If the game is just starting, and you are deciding whether to go first or second, you should determine whether it is balanced yet or not.  If it unbalanced, then you should go first and make the balancing move; if it is already balanced, then you should go second and adopt the copying strategy, in which you re-balance the position with each move.

More advanced players will want to consider Nim positions with taller stacks than three, and we talked about this a little in the classroom.  Some of the girls realized that the copying strategy and the idea of balanced positions still worked with taller stacks.  One can balanced stacks of height four against other stacks of height four, and so one, but the trick for these taller stacks is that one may balance 5 with 4+1; balance 6 with 4+2; and 7 with 4+2+1. Mathematicians will recognize here the powers of two.

To teach the strategy to children, it is a great opportunity to talk about the powers of two. Any child knows how to count 1, 2, 3, 4 and so on, and most can count by twos 2, 4, 6, 8, 10, …; by fives 5, 10, 15, 20, …; by tens, by threes; by sevens; and so on.  , The powers of two are the numbers 1, 2, 4, 8, 16, 32, 64, 128, and so on, doubling each time.  Climbing this exponential growth, children are often amazed at how quickly one reaches very large numbers:

One plus one is two;

two plus two is four;

four plus four is eight;

eight plus eight is sixteen;

sixteen plus sixteen is thirty-two;

thirty-two plus thirty-two is sixty-four;

sixty-four plus sixty-four is one hundred twenty-eight.

For Nim, we don’t in practice need such big powers of two, since one doesn’t usually encounter stacks of height eight or larger, and usually just 1s, 2s and 4s suffice. The relevant fact for us here is that every natural number is uniquely expressible as a sum of distinct powers of two, which of course is just another way of talking about binary representation of a number in base two.  We regard a Nim stack as consisting of its power-of-two substacks.  Thus, a stack of height 3 counts as 2+1; a stack of height 5 counts as 4+1; a stack of height 6 counts as 4+2; and a stack of height 7 counts as 4+2+1.

Ultimately, the winning general strategy for Nim is always to play so as to balance the position, where one regards every stack as being composed of its power-of-two sub-stacks, and a position counts as balanced when these stacks and sub-stacks can be matched up in pairs. This is a winning strategy, since every unbalanced position can be balanced, and any move on a balanced position will unbalance it.  To balance an unbalanced stack, play on any stack containing the largest size unbalanced power of two substack, and reduce it so as to balance the parity of all the stacks.  If one thinks about it, at bottom what we are doing is ensuring that if we represent the stack heights in their binary representation, then we should play so as to ensure that the position has a even number of one digits in each place.

Universality, saturation and the surreal number line, Shanghai, June 2013

Posted on by
Reply

Fudan bridgeThis will be a short lecture series given at the conclusion of the graduate logic class in the Mathematical Logic group at Fudan University in Shanghai, June 13, 18 (or 20), 2013.

I will present an elementary introduction to the theory of universal orders and relations and saturated structures.  We’ll start with the classical fact, proved by Cantor, that the rational line is the universal countable linear order.  But what about universal partial orders, universal graphs and other mathematical structures?  Is there a computable universal partial order?  What is the countable random graph? Which orders embed into  the power set of the natural numbers under the subset relation $\langle P(\mathbb{N}),\subset\rangle$? Proceeding to larger and larger universal orders, we’ll eventually arrive at the surreal numbers and the hypnagogic digraph.

Fudan University seal

 

Playing go with Jiachen

Playful paradox with large numbers, infinity and logic, Shanghai, June 2013

Posted on by
1

Playful paradox

This will be a talk at Fudan University in Shanghai, China, June 12, 2013, sponsored by the group in Mathematical Logic at Fudan, for a large audience of students.

Abstract: For success in mathematics and science, I recommend an attitude of playful curiosity about one’s subject. We shall accordingly explore a number of puzzling conundrums at the foundations of mathematics concerning issues with large numbers, infinity and logic. These are serious issues—and we’ll have serious things to say—while still having fun. Can one complete a task involving infinitely many steps? Are there some real numbers that in principle cannot be described? Is every true statement provable? Does every mathematical problem ultimately reduce to computational procedure? What is the largest natural number that can be written or described in ordinary type Fudan University seal on an index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every sentence either true or false or neither true nor false? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics and philosophy.   The Largest-Number Contest.  In preparation for the talk, and with a nod to Douglas Hofstadter, we shall be holding a contest:  Who can describe the largest number on an ordinary index card?   See the contest announcement poster.

  1. A submission entry consists of the description of a positive integer written on an ordinary index card, using common mathematical operations and notation or English words and phrases.
  2. Write legibly, and use at most 100 symbols from the ordinary ASCII character set.  Bring your submission to the talk.
  3. Descriptions that fail to describe a number are disqualified.
  4. The submission with the largest number wins.
  5. The prize will be $1 million USD divided by the winning number itself, rounded to the nearest cent, plus another small token prize.

Example submissions: 

99999.

10*(10*99)+5

The population of Shanghai at this moment.

Read a more detailed account of the contest and its results.

Quoted in Simons Foundation Science News Feature

Posted on by
Reply

I was quoted in The Global Math Commons, Simons Foundation, Science News Features article by Erica Klarreich, May 18, 2013, concerning MathOverflow.

In memory, Clark John Hamkins (1930- 2013)

Posted on by
2
Clark John Hamkins, Hoyerswort 2005

Clark John Hamkins at            Hoyerswort, 2005

My father, Clark John Hamkins (1930 – 2013) passed away May 11, 2013 at his home in Brunswick, Maine. He was a good man, witty, kind, intelligent, honest, hard-working, caring, curious, patient, philosophical, mathematical, practical, fair.  The world has just lost a great human being. He was a loving family man, married to my mother, Monica, for 55 years (their wedding anniversary was yesterday), with six children and thirteen grandchildren.

US Patent 3756058

US Patent 3756058

He was an engineer’s engineer, one who could take apart any machine and put it back together and tell you all about the ways in which the design was flawed or how it was clever.  I think of his work as a kind of meta-engineering, for he designed the machines for manufacturing a product, bending pipes and twisting wire, rather than the product itself.   He held a number of patents for his inventions, including some for his design of a manufacturing apparatus for winding semi-toroidal transformers. When I was a kid, he designed and built in his wood shop a hand-crank centrifugal honey-extractor, which we used to harvest the crop from our bee hives.  He taught all his kids their way around a wood shop, how to cut, saw, nail, drill, screw, sand, bore, file, buff, solder, join, plane, dowel, glue, measure, how to use all manner of hand tools and the jig saw, the band saw, the table saw, the mitre saw, the drill press, the belt sander. He was beyond serious, with a lathe in his shop.  “Use the right tool for the job,” I can still hear him say, and “measure twice, cut once; measure once, cut twice,” a warning to those who would rush their work.  He made all manner of toys for his children and then for his grandchildren.  He made cabinets, tools, furniture, items of all kinds, large and small, fine and plain.  Dad's dulcimerHe kept and used a wood shop his entire adult life, insisting even in his final days that he go down into it.  One of his final projects was to design and build a beautiful, melodious whale-motif dulcimer.

He was an artist, and as a young man produced oil paintings, which move me to this day.  Long ago, before having kids, he made architectural drawings for the family home he had planned, and it is clear in retrospect that he hadn’t planned at that time on having such a large family.  (The joke in our family is that Dad wanted four kids and Mom wanted two, and they both got their wish!)  Later, he would inevitably win our family games of pictionary, where one is given the task to draw the meaning of a word selected randomly from the dictionary. His drawings always started with a clear simple line, capturing the essence of the concept, which was then fleshed out in fuller artistic flair until his partner said the word.  I found an old math book of his, from his own school days, with a chapter on Polar Coordinates in which he had drawn a wonderful Eskimo, carrying a harpoon and fish, and an igloo.

He was a teacher at heart.  He explained the slide rule to me when I was young, and gave a lesson on logarithms and log tables and the accompanying explanation of linear interpolation.  He explained to me as a child what $x^2$ and $x^3$ meant, and then, with a twinkle in his eye, as I listened wide-eyed and dumbstruck, about what $x^{2.3}$ meant.  Some of my fondest young memories with him include viewings of the moon and planets through a telescope, as I shivered in my pajamas in the cool night air, pondering the craters and drinking hot cocoa. He would discuss the various historical astronomical theories, comparing Copernicus with Ptolemy and the role of Tycho Brahe.  He loved a good scientific controversy, and liked even more to figure things out himself.  He liked to put a scientific explanation in a historical context.

He was a programmer, programming computers in the earliest days. I brought his cast-off computer cards, punched paper tapes and so on into my third-grade classroom for show-and-tell.  I remember him poring over expansive flowcharts spread across the kitchen table in the evenings. He made sure that his kids were learning how to program.

He was a voracious reader, consuming books in science, philosophy, literature, history, archeology, biology, physics, mathematics. You name it; he read it.

He was a skeptic.

He was a rebel physicist, refusing to accept the conclusion of the Michelson-Morley experiment on relativistic length foreshortening and time contraction.  And he backed up his beliefs by writing an account of this part of physics, re-developing the theory from an alternative perspective of his own invention, involving a notion of time translation, which avoided the need for those paradoxical conclusions, but ended up deriving the same fundamental equations.

I will miss him.

My younger brother Jon’s eulogy

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

Posted on by
3

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

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

This is joint work with Cole Leahy (MIT).

NYlogic abstract | MathOverflow post

The theory of infinite games, with examples, including infinite chess

Posted on by
2

This will be a talk on April 30, 2013 for a joint meeting of the Yeshiva University Mathematics Club and the  Yeshiva University Philosophy Club.  The event will take place in 5:45 pm in Furst Hall, on the corner of Amsterdam Ave. and 185th St.

Abstract. I will give a general introduction to the theory of infinite games, suitable for mathematicians and philosophers.  What does it mean to play an infinitely long game? What does it mean to have a winning strategy for such a game?  Is there any reason to think that every game should have a winning strategy for one player or another?  Could there be a game, such that neither player has a way to force a win?  Must every computable game have a computable winning strategy?  I will present several game paradoxes and example infinitary games, including an infinitary version of the game of Nim, and several examples from infinite chess.

NYlogic entry | Yeshiva University | Infinite chess | Video

 

Norman Lewis Perlmutter

Posted on by
Reply

Norman Lewis Perlmutter successfully defended his dissertation under my supervision and will earn his Ph.D. at the CUNY Graduate Center in May, 2013.  His dissertation consists of two parts.  The first chapter arose from the observation that while direct limits of large cardinal embeddings and other embeddings between models of set theory are pervasive in the subject, there is comparatively little study of inverse limits of systems of such embeddings.  After such an inverse system had arisen in Norman’s joint work on Generalizations of the Kunen inconsistency, he mounted a thorough investigation of the fundamental theory of these inverse limits. In chapter two, he investigated the large cardinal hierarchy in the vicinity of the high-jump cardinals.  During this investigation, he ended up refuting the existence of what are now called the excessively hypercompact cardinals, which had appeared in several published articles.  Previous applications of that notion can be made with a weaker notion, what is now called a hypercompact cardinal.

Norman Lewis Perlmutter

web page | math genealogy | MathSciNet | ar$\chi$iv | related posts

Norman Lewis Perlmutter, “Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2013.

Abstract.  This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved in both directions by forcing under fairly general assumptions. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing.  A high-jump cardinal is the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\{\ j(f)(\kappa) \mid f: \kappa \to \kappa\ \}$.  Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.

Joining the Doctoral Faculty in Philosophy

Posted on by
4

I am recently informed that I shall be joining the Doctoral Faculty of the Philosophy Program at the CUNY Graduate Center, in addition to my current appointments in Mathematics and in Computer Science.  This means I shall now be able to teach graduate courses in philosophy at the Graduate Center and also to supervise Ph.D. dissertations in philosophy there.  I am pleased to become a part of the GC Philosophy Program, which is so highly ranked in the area of mathematical logic, and I look forward to making a positive contribution to the program.

 

Interviewed by Richard Marshall at 3:AM Magazine

Posted on by
2

I was recently interviewed by Richard Marshall at 3:AM Magazine, which was a lot of fun. You can see that his piece starts out, however, rather over-the-top…

3:AM MAGAZINE

playing infinite chess

Joel David Hamkins interviewed by Richard Marshall.

Joel David Hamkins is a maths/logic hipster, melting the logic/maths hive mind with ideas that stalk the same wild territory as Frege, Tarski, Godel, Turing and Cantor. He thinks we all can go there and that we all should. He gives tips about the Moebius strip to six year olds and plays around with his sons homework. He has discovered all sorts of wonders involving supertasks, infinite-time Turing machines, black-hole computations, the mathematics of the uncountable, the lost melody phenomenon of infintary computability (which really should be the name of a band), set theory and multiverses, infinite utilitarianism, and infinite chess. He’s also thinking about whether we really have an absolute notion of the finite and doubts if any of this is brain melting, which is just a testimony to his modesty. He also thinks that although maths is open to all he thinks mathematicians could use more metaphors and silly terminology to get their ideas across better than they do. All in all, this is the grooviest of the hard core maths/logic groovsters. Bodacious!

→ continue to the rest of the interview

The interview is now available at Marshall’s site 3:16, along with the full collection of his interviews.

Pluralism in mathematics: the multiverse view in set theory and the question of whether every mathematical statement has a definite truth value, Rutgers, March 2013

Posted on by
Reply

This is a talk for the Rutgers Logic Seminar on March 25th, 2013.  Simon Thomas specifically requested that I give a talk aimed at philosophers.

Abstract.  I shall describe the debate on pluralism in the philosophy of set theory, specifically on the question of whether every mathematical and set-theoretic assertion has a definite truth value. A traditional Platonist view in set theory, which I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. I shall try to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Some of this material arises in my recent articles:

The omega one of chess, CUNY, March, 2013

Posted on by
Reply

This is a talk for the New York Set Theory Seminar on March 1, 2013.

This talk will be based on my recent paper with C. D. A. Evans, Transfinite game values in infinite chess.

Infinite chess is chess played on an infinite chessboard.  Since checkmate, when it occurs, does so after finitely many moves, this is technically what is known as an open game, and is therefore subject to the theory of open games, including the theory of ordinal game values.  In this talk, I will give a general introduction to the theory of ordinal game values for ordinal games, before diving into several examples illustrating high transfinite game values in infinite chess.  The supremum of these values is the omega one of chess, denoted by $\omega_1^{\mathfrak{Ch}}$ in the context of finite positions and by $\omega_1^{\mathfrak{Ch}_{\hskip-1.5ex {\ \atop\sim}}}$ in the context of all positions, including those with infinitely many pieces. For lower bounds, we have specific positions with transfinite game values of $\omega$, $\omega^2$, $\omega^2\cdot k$ and $\omega^3$. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true $\omega_1$.