Exploring the Frontiers of Incompleteness, Harvard, August 2013

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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

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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.

University of Connecticut Logic Seminar | Article

Workshop on paraconsistent set theory, Connecticut, October 2013

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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

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[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 $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $H_{\frak{c}}\prec_{\Sigma_1}V[G]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $H_{\omega_2}$ is existentially closed in all proper forcing extensions, or $H_{\omega_2}\prec_{\Sigma_1}V[G]$; 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 $\Sigma_1$-elementary substructure of $N$, which we write as $M\prec_{\Sigma_1} N$. Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.

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

  1. The model $\mathcal{M}$ is existentially closed in $\mathcal{N}$.
  2. $\mathcal{M}\subset \mathcal{N}$ has resurrection. That is, there is a further extension $\mathcal{M}\subset\mathcal{N}\subset\mathcal{M}^+$ for which $\mathcal{M}\prec\mathcal{M}^+$.

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

In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $\mathcal{M}^+$ 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 $\Gamma$ be a fixed definable class of forcing notions.

  1. The resurrection axiom $\text{RA}(\Gamma)$ is the assertion that for every forcing notion $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, with $\vdash_{\mathbb{Q}}\mathbb{R}\in\Gamma$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
  2. The weak resurrection axiom $\text{wRA}(\Gamma)$ is the assertion that for every $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.

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 $\kappa$ is uplifting, if there are arbitrarily large inaccessible cardinals $\gamma$ for which $H_\kappa\prec H_\gamma$.  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:

  1. There is an uplifting cardinal.
  2. $\text{RA}(\text{all})$.
  3. $\text{RA}(\text{ccc})$.
  4. $\text{RA}(\text{semiproper})+\neg\text{CH}$.
  5. $\text{RA}(\text{proper})+\neg\text{CH}$.
  6. For some countable ordinal $\alpha$, the axiom $\text{RA}(\alpha\text{-proper})+\neg\text{CH}$.
  7. $\text{RA}(\text{axiom-A})+\neg\text{CH}$.
  8. $\text{wRA}(\text{semiproper})+\neg\text{CH}$.
  9. $\text{wRA}(\text{proper})+\neg\text{CH}$.
  10. For some countable ordinal $\alpha$, the axiom $\text{wRA}(\alpha\text{-proper})+\neg\text{CH}$.
  11. $\text{wRA}(\text{axiom-A})+\neg\text{CH}$.
  12. $\text{wRA}(\text{countably closed})+\neg\text{CH}$.

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

In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $A\subset\frak{c}$ and ask for extensions of the form $\langle H_{\frak{c}},{\in},A\rangle\prec\langle H_{\frak{c}}^{V[g\ast h]},{\in},A^\ast\rangle$, for some $A^\ast\subset\frak{c}^{V[g\ast h]}$ 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

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[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, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ 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 $\kappa$ can be made indestructible by ${\lt}\kappa$-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, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals, when $n\geq 3$. In fact, all these large cardinal properties are superdestructible, in the sense that if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ 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 $\kappa$ will definitely prevent it from being superstrong—as well as preventing it from being uplifting, $\Sigma_3$-correct, $\Sigma_3$-extendible and so on with all the large cardinal properties mentioned above—in the forcing extension.

Main Theorem. 

  1. Superstrong cardinals are never Laver indestructible.
  2. Consequently, almost huge, huge, superhuge and rank-into-rank cardinals are never Laver indestructible.
  3. Similarly, extendible cardinals, $1$-extendible and even $0$-extendible cardinals are never Laver indestructible.
  4. Uplifting cardinals, pseudo-uplifting cardinals, weakly superstrong cardinals, superstrongly unfoldable cardinals and strongly uplifting cardinals are never Laver indestructible.
  5. $\Sigma_n$-reflecting and indeed $\Sigma_n$-correct cardinals, for each finite $n\geq 3$, are never Laver indestructible.
  6. Indeed—the strongest result here, because it is the weakest notion—$\Sigma_3$-extendible cardinals are never Laver indestructible.

In fact, each of these large cardinal properties is superdestructible. Namely, if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the mentioned large cardinal properties with target $\theta$ 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 $\kappa$ 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 $\Sigma_3$-definable in the extension, and this fact can be seen as explaining the main theorem above.

Related MO question | CUNY talk

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

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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 $\mathcal{M}$ is universal for a class of structures, if every structure in that class arises as (isomorphic to) a substructure of $\mathcal{M}$.  For example, Cantor proved that the rational line $\mathbb{Q}$ 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

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

  • The extension $W\subset V$ satisfies the $\delta$-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 $\delta$ in $W$, then $A\in W$.
  • The extension $W\subset V$ satisfies the $\delta$-cover property if whenever $A\subset W$ is a set of size less than $\delta$ in $V$, then there is a covering set $B\in W$ with $A\subset B$ and $|B|^W\lt\delta$.

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

Proof. First, notice that the $\delta$-approximation property trivially implies the $\gamma$-approximation property for any larger cardinal $\gamma$. So we need only verify the $\gamma$-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 $\gamma=\delta^+$, meaning $(\delta^+)^V$, but we must allow that $\delta$ may be singular here.

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

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

Thus, in either case we’ve established the $\gamma$-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 $\delta$; 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 $\delta$-approximation property, it is not true that the $\delta$-cover property transfers upward. For example, every extension has the $\aleph_0$-cover property.

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

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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

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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.)

Embeddability amongst the countable models of set theory, plenary talk for ASL / Joint Math Meetings in Baltimore, January 2014

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A one-hour plenary talk for the ASL at the Joint Math Meetings, January 15-18, 2014 in Baltimore, MD.

Saturday January 18, 2014, 2:00 p.m.-2:50 p.m, Room 319 BCC

Abstract. A surprisingly vigorous embeddability phenomenon has recently been uncovered amongst the countable models of set theory.  In particular, embeddability is linear:  for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  In particular, every countable model of set theory, including every well-founded model, is isomorphic to a submodel of its own constructible universe, so that there is an embedding $j:M\to L^M$ for which $x\in y\iff j(x)\in j(y)$. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraïssé limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers.

Slides | Article | JMM Schedule

Largest-number contest: what is the largest number that you can describe on an index card?

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Playful Paradox lectureMy recent talk, Playful Paradox with large numbers, infinity and logic, was a romp through various paradoxical topics in mathematics and logic for an audience of about 150 mathematics and philosophy undergraduate students at Fudan University in Shanghai, and since Fudan is reportedly a top-three university in China, you can expect that the audience was sharp.

For a bit of fun, and in keeping with the theme, we decided to hold a Largest-Number Contest:  what is the largest number that one can describe on an ordinary index card?

Playful paradoxMy host Ruizhi Yang made and distributed announcement flyers for the contest a week before the talk, with a poster of the talk on one side, and a description of the contest rules on the other, and including a small card to be used for submissions. The rules were as follows:

  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.

The submissions were collected at the beginning of my talk, which I began with a nod to Douglas Hofstadter, who once held a similar contest in his column at Scientific American magazine.

To win, I began, it seemed clear that the submitted number must be very large, and since the prize was to be one million dollars divided by the winning number, then this would mean that the prize should become small.  So the goal is just to win, and not necessarily to win any money.  The real prize is Pride.

So, what kind of numbers should one submit?  Of course, anyone could think to write $99999$, or one million ($10^6$), one billion ($10^9$) or one trillion ($10^{12}$).  Perhaps someone would want to write a googol, which is a common name for $10^{100}$.  Written in decimal, this is $$10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,$$ which is one character too long for the rules, so one should use the more compact formulations.  But let’s just write it as googol.

But we can easily describe much larger numbers.  A googol plex is the number $10^{\text{googol}}$, written in decimal with a $1$ followed by a googol many $0$s. A googol plex is also describable as $10^{10^{100}}$.  An iterated exponential $a^{b^c}$ could be seen as ambiguous, referring either to $(a^b)^c$ or to $a^{(b^c)}$. These are not always the same, and so exponentiation is not an associative operation.  But which is bigger?  If the aim is to make big numbers, then generally one wants to have as big an exponent as possible, and so $a^{(b^c)}$ will be much larger than $(a^b)^c$, which is just $a^{bc}$, and so we will always associate our exponentials upward. (Nevertheless, the upward associated form is not always larger, which you can see in the case of $2^{1^{100}}$.) In general, we can apply the plex operation to any number, so $x$ plex just means $10^x$.

A googol bang means googol !, or googol factorial, the number obtained by multiplying googol $\cdot$ (googol -1) $\cdots 2\cdot 1$, and in general, $x$ bang means $x!$.  Which is bigger, a googol plex or a googol bang? If you think about it, a googol plex is 10 times 10 times 10, and so on, with a googol number of terms. Meanwhile, the factorial googol bang also has a googol terms in it, but most of them (except for 10) are far larger than 10, so it is clear that a googol bang is much larger than a googol plex.

Continuing down this road, how about a googol plex bang, or a googol bang plex? Which of these is larger? Please turn away and think about it for a bit……..Have you got it?  I think like this:  A googol plex bang has the form $(10^g)!$, which is smaller than $(10^g)^{10^g}$, which is equal to $10^{g10^g}$, and this is smaller than $10^{g!}$ when $g$ is large. So a googol plex bang is smaller than a googol bang plex.

There is an entire hierarchy of such numbers, such as googol bang plex bang bang bang and googol bang bang plex plex plex.  Which of these is bigger?  Do you have a general algorithm for determining which of two such expressions is larger?

A googol stack is the number

$$10^{10^{10^{\cdot^{\cdot^{10}}}}}{\large\rbrace} \text{ a stack of height googol},$$

where the height of the exponential stack is googol.  Similarly, we may speak of $x$ stack for any $x$, and form such expressions as:

googol stack bang plex

googol bang plex stack

googol plex stack bang

Which of these is larger?  If you think about it, you will see that the stack operator is far more powerful than bang or plex, and so we want to apply it to the biggest possible argument.  So the second number is largest, and then the third and then the first, simply because the stack operator will overrule the other effects.  Let us call this hierarchy, of all expressions formed by starting with googol and then appending any number of plex, bang or stack operators, as the googol plex bang stack hierarchy.

Question. Is there a feasible computational procedure to determine, of any two expressions in the googol stack bang plex hierarchy, which is larger?

This is an open question in the general case.  I had asked this question at math.stackexchange, Help me put these enormous numbers in order, and the best currently known (due to user Deedlit) is that we have a simple algorithm that works for all expressions of length at most googol-2, and also works for all expressions having the same number of operators.  But this algorithm breaks down for large expressions of much different lengths, since it seems that one must come to know the exact threshold where an enormous iteration of bangs and plexes can ultimately capture a stack.  Of course, in principle one can compute these numbers exactly, given sufficient time and space resources, but the point of the question is to have a feasible algorithm, such as one that takes only polynomial time in the length of the input expressions.  And a feasible such general algorithm is not known.

The fact that we seem to have difficulty comparing the relative sizes of expressions even in the simple googol plex bang stack hierarchy suggests that we may find it difficult to select a winning entry in the largest-number contest.  And indeed, I’ll explain a bit later why there is no computable procedure to determine the winner of arbitrary size largest-number contests.

The numbers we’ve seen so far are actually rather small, so let’s begin to describe some much larger numbers.  The most basic operation on natural numbers is the successor operation $a\mapsto a+1$, and one can view addition as repeated succession:

$$a+b=a+\overbrace{1+1+\cdots+1}^{b\text{ times}}.$$

Similarly, multiplication is repeated addition

$$a\cdot b=\overbrace{a+a+\cdots+a}^{b\text{ times}}$$

and exponentiation is repeated multiplication

$$a^b=\overbrace{a\cdot a\cdots a}^{b\text{ times}}.$$

Similarly, stacks are repeated exponentiation.  To simplify the expressions, Knuth introduced his uparrow notation, writing $a\uparrow b$ for $a^b$, and then repeating it as

$$a\uparrow\uparrow b=\overbrace{a\uparrow a\uparrow\cdots\uparrow a}^{b\text{ times}}= a^{a^{a^{.^{.^a}}}}{\large\rbrace}{ b\text{ times}}.$$

So googol stack is the same as $10\uparrow\uparrow$ googol. But why stop here, we can just as easily define the triple repeated uparrow

$$a\uparrow\uparrow\uparrow b=\overbrace{a\uparrow\uparrow a\uparrow\uparrow\ \cdots\ \uparrow\uparrow a}^{b\text{ times}}.$$

In this notation, $10\uparrow\uparrow\uparrow$ googol means

$$\overbrace{10^{10^{.^{.^{10}}}}{\large\rbrace}10^{10^{.^{.^{10}}}}{\large\rbrace}10^{10^{.^{.^{10}}}}{\large\rbrace}\cdots\cdots{\large\rbrace} 10^{10^{.^{.^{10}}}}{\large\rbrace}10}^{\text{iterated googol times}},$$

where the number of times the stacking process is iterated is googol. This number is a vast number, whose description $10\uparrow\uparrow\uparrow$ googol fits easily on an index card.

But why stop there? We could of course continue with $10\uparrow\uparrow\uparrow\uparrow$ googol and so on, but I’d rather define

$$10\Uparrow x=10\ \overbrace{\uparrow\uparrow\cdots\uparrow}^{x\text{ times}}\ 10.$$

So that $10\Uparrow$ googol is vaster than anything we’ve mentioned above. But similarly, we will have $10\Uparrow\Uparrow$ googol and so on, leading eventually to a triple uparrow
$$10 \uparrow\hskip-.55em\Uparrow\text{ googol } = 10 \overbrace{\Uparrow\Uparrow\cdots\Uparrow}^{\text{googol iterations}} 10,$$
and so on. These ideas lead ultimately to the main line of the Ackerman function.

So we’ve described some truly vast numbers.  But what is the winning entry?  After all, there are only finitely many possible things to write on an index card in accordance with the limitation on the number of symbols.  Many of these possible entries do not actually describe a number–perhaps someone might submit a piece of poetry or confess their secret love–but nevertheless among the entries that do describe numbers, there are only finitely many. Thus, it would seem that there must be a largest one. Thus, since the rules expressly allow descriptions of numbers in English, it would seem that we have found our winning entry:

The largest number describable on an index card according to the rules of this contest

which uses exactly 86 symbols. We seem to have argued that this expression describes a particular number, and accords with the rules, so we conclude that this is indeed provably the winning entry.

But wait a minute! If the expression above describes a number, then it would seem that the following expression also describes a number:

The largest number describable on an index card according to the rules of this contest, plus one

which uses 96 symbols.  And this number would seem to be larger, by exactly one, of the number which we had previously argued must be the largest number that is possible to describe on an index card. Indeed, how are we able to write this description at all? We seem to have described a number, while following certain rules, that is strictly larger than any number which it is possible to describe while following those rules. Contradiction!  This conundrum is known as Berry’s paradox, the paradox of “the smallest positive integer not definable in under eleven words.”

Thus, we can see that this formulation of the largest-number contest is really about the Berry paradox, which I shall leave you to ponder further on your own.

I concluded this part of my talk by discussing the fact that even when one doesn’t allow natural language descriptions, but stays close to a formal language, such as the formal language of arithmetic, then the contest ultimately wades into some difficult waters of mathematical logic.  It is a mathematical theorem that there is no computable procedure that will correctly determine which of two descriptions, expressed in that formal language, describes the larger natural number.  For example, even if we allow descriptions of the form, “the size of the smallest integer solution to this integer polynomial equation $p(x_1,\ldots,x_k)=0$,” where $p$ is provided as a specific integer polynomial in several variables. The reason is that the halting problem of computability theory is undecidable, and it is reducible to the problem of determining whether such a polynomial equation has a solution, and when it does, the size of the smallest solution is related to the halting time of the given instance of the halting problem.  Worse, the independence phenomenon shows that not only is the comparison problem undecidable, but actually there can be specific instances of descriptions of numbers in these formal languages, such that both definitely describe a specific number, but the question of which of the two numbers described is larger is simply independent of our axioms, whatever they may be. That is, the question of whether one description describes a larger number than another or not may simply be neither provable nor refutable in whatever axiomatic system we have chosen to work in, no matter how strong. In such a case, I would find it to be at least a debatable philosophical question whether there is indeed a fact of the matter about whether one of the numbers is larger than the other or not. (See my explanation here). It is my further considered belief that every largest-number contest, taken seriously, unless severely limited in the scope of the descriptions that entries may employ, will founder on precisely this kind of issue.

For further reading on this topic, there are a number of online resources:

Googol plex bang stack hierarchy |  Largest number contest on MathOverflow

Scott Aaronson on Big Numbers (related:  succinctly naming big numbers)

Meanwhile, lets return to our actual contest, for which we collected a number of interesting submissions, on which I’ll comment below.  Several students had appreciated the connection between the contest and Berry’s paradox.

 

05Your number is quite large, since we can surely describe an enormous number of digits of $\pi$.  Researchers have announced, for example, that they have now found 10 trillion digits of $\pi$.  If by “describe physically” you meant that we should say the digits aloud, however, then supposing that we speak very quickly and say five digits per second (which is quite a lot if sustained for any length of time), then one person would be able to pronounce a little over 4 million digits this way in 10 days. If all humanity were to speak digits in parallel, then we could rise up to the order of $10^{16}$ digits in ten days.  This is a big number indeed, although not nearly as big as a googol or many of the other numbers we have discussed.

 

 

06Surely $\infty$ is a large number, but it is not ordinarily considered to be a positive integer or a natural number. And so I think that your submission does not accord with the rules.

 

 

03This is charming, and I’m sure that your girlfriend will be very happy to see this. Depending on the units we use to measure your affection, I find it very likely that this number is extremely large indeed, and perhaps you win the contest. (But in order to stay within the rules, we should pick the units so that your affection comes out exactly as a positive integer; in particular, if your affection is actually infinite, as you seem to indicate, then I am afraid that you would be disqualified.)  One concern that I have, however, about your delightful entry—please correct me if I am wrong—is this:  did you write at first about your affection for your “girlfriends” (plural), and then cross out the extra “s” in order to refer to only one girlfriend? And when you refer to your “girlfriend(s), which values at least $\omega$,” do you mean to say that your affection for each girlfriend separately has value $\omega$, or are you instead saying that you have affection for $\omega$ many girlfriends? I’m not exactly sure. But either interpretation would surely lead to a large number, and it does seem quite reasonable that one might indeed have more affection, if one had many girlfriends rather than only one! I suppose we could accept this entry as a separate submission about your affection, one value for each girlfriend separately, but perhaps it may be best to combine all your affection together, and consider this as one entry. In any case, I am sure that you have plenty of affection.

 

 

07You’ve submitted Graham’s number, an extremely large number that is also popularly known as the largest definite number ever to be used in a published mathematical proof. This number is certainly in the upper vicinity of the numbers we discussed above, obtained by iterating the Knuth uparrow. It is much larger than anything we can easily write in the googol plex bang stack hierarchy.  This is essentially in line with iterating the $\Uparrow$ operator $64$ times, using base $3$. This number is extremely large, and I find it to be the largest number submitted by means of a definite description whose numerical value does not depend on which other numbers have been submitted to the contest. Meanwhile, we discussed some even larger numbers above, such as the triple uparrow $10 \uparrow\hskip-.55em\Uparrow$ googol.

 

 

02Your clever submission shows that you realize that it isn’t necessary to actually mention the largest possible number to describe, as we discussed in Berry’s paradox above, but it suffices rather to exceed only the largest actually occurring numbers that are described by others in the contest.  What are we to do, however, when two people have this same clever idea?  If either of them succeeds in describing a number, then it would seem that they both must succeed, but in this case, they must both have described numbers larger than the other. This is the same contradiction at the heart of Berry’s paradox.  Another risky feature of this entry is that it presumes that some numbers were in fact described by others.  How are we to interpret it if all other submissions are disqualified?

 

 

04Your description has the very nice feature that it avoids the self-reference needed in the previous description to refer to all the “other” entries (that is, all the entries that are not “this” one).  It avoids the self-reference by speaking of “second-largest” entry, which of course will have to be the largest of the other entries, since this entry is to be one larger.  Although it avoids self-reference, nevertheless this description is what would be called impredicative, because it quantifies over a class of objects (the collected entries) of which it is itself a member.  Once one interprets things as you seem to intend, however, this entry appears to amount to essentially the same description as the previous entry.  Thus, the problematic and paradoxical situation has now actually occurred, making it problematic to conclude that either of these entries has succeeded in describing a definite number. So how are we to resolve it?

 

 

01Your description is very clever, trying to have it both ways.  Of course, every participant in the contest wants to win, but you also to win with a very low winning number, so that you can maximize the prize payout. With this submission, of course you hope that all other submissions might be disqualified, in which case you will win with the number 1, achieving the million dollar payout!  In the case of our actual contest, however, there are other cards that are qualified, such as the Graham’s number submission, and so your submission seems to be in case 2, making it similar in effect to the previous submission.  Like the previous two entries, this card is problematic when two or more players submit the same description.  And indeed, even with the other descriptions submitted above, we are already placed into the paradoxical situation that if this card describes a number in its case 2, then it will be larger than some numbers that are larger than it.

 

 

08Your submission is very ambitious, attempting both to win and to win the most money.   (I take the description as the lower part of this card, and the upper note is a remark about it.)  You seem to realize that there will be ties in the game, with multiple people submitting a winning entry.  And indeed, if everyone had submitted this entry, then the winning number would indeed seem to be 1, which is the smallest number that would make them all be one of the members who win the game, so the million dollars would be split amongst that group. Some clumsy person, however, might submit a googol plex bang, however, which would bump up the value of this card to that same value, which although still winning, would win much less money. Your line of reasoning about rationality is very similar to the concept of superrationality defended by Douglas Hofstadter in the context of the prisoner’s dilemma.  Perhaps one might also be tempted to say something like:

the smallest number that makes me win, with the largest possible prize payout to me.

with the idea that you’d get a bigger prize solely for yourself if you avoided a tie situation by adding one more, so as to get the whole prize for yourself. But if this reasoning is correct, then one would seem to land again in the clutches of the Berry paradox.

 

 

To my readers:  please make your submissions or comments below!

 

 

 

 

 

The Myth of Just Do It

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Barbara’s piece this week in the Philosopher’s Stone in New York Times:

OPINION   | June 09, 2013
The Myth of ‘Just Do It’
By BARBARA GAIL MONTERO

The idea that thinking interferes with doing is often taken for granted. But the realities at the highest levels of athletic and artistic performance are more complex.

→ go to The Myth of ‘Just Do It’

How well do you think this applies to expert action in mathematics? Go and post comments at the NYT…

 

Barbara was interviewed by Christie Nicholson at CBS News, Smart Planet, in the Pure Genius series:

CBS News, Smart Planet | June 28, 2013
Innovation / Pure Genius
Q&A: Barbara Montero, philosopher,
on the myth of ‘Just Do It’
Christie Nicholson interviews Barbara Gail Montero

There is a widely held view that thinking about one’s performance while performing ruins our ability to perform well. Many athletes say that once you’ve mastered the skill, one ought to let go of thinking and well, to quote Nike’s tag line, “Just do it.” Professional golfer Dave Hill said, “Golf is like sex. You can’t be thinking about the mechanics of the act while you are performing.”

I first heard that quote from Barbara Montero, associate professor of Philosophy at College of Staten Island and Graduate Center, City University of New York and author of a forthcoming book, Mind, Body, Movement: The Relevance of Consciousness to Expert Performance. (This is a working title, to be published by Oxford University Press.) Montero holds that thinking is not detrimental to successful expert performance. She describes the kind of thinking that might interfere but also the type of thinking that is actually necessary for an expert to improve upon his or her top performance.

SmartPlanet spoke with Montero, to hear more about the ‘just do it’ philosophy and why she feels it is misguided.

→ go to the interview

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact

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[bibtex key=CodyGitikHamkinsSchanker2015:LeastWeaklyCompact]

Abstract.   We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, for nearly any desired function $\kappa\mapsto\theta_\kappa$. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

In this article, we prove that the least weakly compact cardinal can exhibit any of several much stronger large cardinal properties. Namely, the least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact for any desired $\theta$.

Main Theorem.  Assuming a suitable large cardinal hypothesis, the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$.

Meanwhile, the least weakly compact cardinal can never exhibit these extra large cardinal properties in $L$, and indeed, the existence of a weakly measurable cardinal in the constructible universe is impossible. Furthermore, in each case the extra properties are strictly stronger than weak compactness in consistency strength.

We show in addition a more global result, that the entire class of weakly compact cardinals can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals, and with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, with enormous flexibility in the map $\kappa\mapsto\theta_\kappa$.

Our results therefore extend the `identity-crises’ phenomenon—first identified (and named) by Magidor—which occurs when a given large cardinal property can be made in various models to coincide either with much stronger or with much weaker large cardinal notions. Magidor had proved that the least strongly compact cardinal can be the least supercompact cardinal in one model of set theory and the least measurable cardinal in another. Here, we extend the phenomenon to weak measurability, partial near supercompactness and unfoldability. Specifically, the least weakly measurable cardinal coincides with the least measurable cardinal under the GCH, but it is the least weakly compact cardinal in our main theorem. Similarly, the least cardinal $\kappa$ that is nearly $\kappa^{+}$-supercompact is measurable with nontrivial Mitchell order under the GCH, but it is the least weakly compact cardinal here (and similar remarks apply to near $\kappa^{++}$-supercompactness and so on). The least unfoldable cardinal is strongly unfoldable in $L$, and therefore a $\Sigma_2$-reflecting limit of weakly compact cardinals there, but it is the least weakly compact cardinal in our main theorem. The global results of section 6 show just how malleable these notions are.

Algebraicity and implicit definability in set theory

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[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

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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