Mathematics and philosophy of the infinite. Professor of Logic @NotreDame @UniofOxford #ProofandtheArt #PhilMaths #InfinitelyMore #BookOfInfinity
University of Notre Dame
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I am very pleased to announce that
The Book of Infinity
is available for pre-order!
All my favorite paradoxes and conundrums.
#BookOfInfinity #InfinitelyMore
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It seems that I shall be giving the Skolem Lecture next year in Oslo. Looking forward! More news later.
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The Largest Tweetable Number: Why 280 Characters Beat 280 Digits
Professor @JDHamkins introduces the paradox of the largest tweetable number, showing how 280 characters can describe numbers far larger than 280 digits of nines.
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My daughter Hypatia is now a double major in mathematics and philosophy (at Caltech), befitting her eponym.
en.wikipedia.org/wiki/Hypati…
I always encouraged her that she could do absolutely whatever she wanted in intellectual matters, but ultimately she found her home as I have.
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Here is a rough outline and syllabus of my games course this semester. It's going great! jdh.hamkins.org/philosophy-a…
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Excited for the chess guest lecture today in my Philosophy and Logic of Games class by FIDE Master and Oxford philosopher Daniel Gallagher!
@BishopPair
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Galileo's paradox of Infinity—Lectures on Infinity (lecture 6)
infinitelymore.xyz/p/galileo… #InfinitelyMore
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Suppose that A is a computably enumerable set that is undecidable. Of course every instance of n∈A is provable in PA, since we verify that the algorithm proceeded as it did. Nevertheless, for any consistent theory extending (or interpreting) PA, such as ZFC or ZFC plus large cardinals, there will be infinitely many instances of n∉A that are not provable in the theory, for otherwise we could decide A by searching for proofs. The point being that proofs of n∉A in a consistent theory extending PA are reliable, since upon finding the proof we couldn't afterward find that n∈A, since this would reveal inconsistency in the theory. If theory is furthermore sound for existential assertions, this means that n∈A will be often independent of the theory.
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In short, every c.e. undecidable set A is saturated also with logical independence, in that for any of our aspirational foundational theories, if both consistent and sound for Sigma_1 assertions, will have many independent instances n∈A. Thus every instance of computable undecidability is saturated also with logical undecidability, even with respect to our best aspirational foundational theories.
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Nevertheless, finally, I want to point out that "consistent" is not enough in this argument, and that one does need existential soundness. Let A be the set of halting Turing machine programs (on empty input), unless a proof of a contradiction in PA is discovered, in which case every number is enumerated into A. So actually, A is c.e. and undecidable. But there are no instances of n∈A that are independent of the theory T=PA+¬Con(PA), since this theory proves that A is everything.
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Today in Philosophy and Logic of Games, we shall discuss Thi Nguyen's book Games: Agency as Art.
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Suppose you are standing on the shore of Lake Michigan at Milwaukee. How tall would a lighthouse on the opposite shore have to be to directly visible? maps.app.goo.gl/J23rpf6tEftF…
15%
50 meters
15%
100 meters
21%
250 meters
49%
More than 1 km tall
100 votes • Final results
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There is a specific finite group presentation, which we can write down, for which it is consistent with PA that it presents the trivial group if and only if ZFC+∃inaccessible is consistent.
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Next month in New York:
On the fundamental theorem of finite games.
jdh.hamkins.org/fundamental-…
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Latest result: ZFC is equiconsistent (over PA) with the assertion that a certain computably enumerable set is nonempty. The same is true of ZFC plus large cardinals, or indeed any computably axiomatizable theory extending PA. Will explain more soon...
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