Joel David Hamkins

Chaos train and monkey madness—fun with quantifiers Learn the language of first-order logic by immersion in some fun logic puzzles

www.infinitelymore.xyz/p/monkey-mad...

Relations between the usefulness of a true statement, its closeness to falsity, and the difficulty of its proof Let us start with a trivial observation: for any statements $A$ and $B$ (in a standard language), the implication $A\implies B$ is true if $A$ is false. So, if one were able to prove a false statem...

I answered a question on MathOverflow about the fruitfulness of considering the closeness-to-falsity of a given statement in mathematics. mathoverflow.net/a/501465/1946

New post-doctoral position at Notre Dame
Core analytic/logic/phil math. philjobs.org/job/show/29622
2 year position, $75K annual salary plus $15K expense account. Please apply! (I am chairing this search.)

I think the best topics will be mathematically interesting and surprising. Should be accessible at first (since undergraduates) but grow in sophistication (since Caltech). I have several ideas, but would be interested to hear people's requests and suggestions.

I am deeply honored to be invited to give the De Prima lecture at Caltech this fall, a distinguished annual math lecture aimed primarily at undergraduates. Help me decide on a topic! What would you request, drawing from topics on which you've seen me write or speak?

Boolean ultrapower - set-theoretic vs algebraic/model-theoretic I've been looking through the Hamkins/Seabold paper "Well-founded Boolean ultrapowers as large cardinal embeddings". The Boolean ultrapowers are defined there in two different ways: in

I answered a question on MathOverflow about the Boolean ultrapower, regarding the algebraic approach to the ultrapower versus the names-based approach. mathoverflow.net/a/501257/1946

I deleted an earlier version of this post, since I had screwed up the quotation marks! I hope it is right now.

And the author reply to critics:

"On 'On "On"' and 'On "On 'on'"'".

(Note: CMS nested quote rule.)

"On 'On'"

What is the view you are pointing out? We are to think of Lotks-Volterra in a Kuhnian light?

Full reading list here. bsky.app/profile/joel...

This week's reading for my undergrad PhilSciMath core seminar. Looking forward to the discussion.

YouTube video by Svenleboubou Artist Chalon dans la rue, l’homme orchestre, festival 2021

If I were a musician...
youtu.be/fjesu41f1mY?...

Attending team summit STOP Boss and colleagues have now boarded the hot-air balloon STOP Ready to cut them loose STOP

Detail from student quiz on Lakatos, Proofs and Refutations.

Overheard: the wife says "You're always wrong" and the husband replies "You're right".

Joel David Hamkins (@joeldavidhamkins) Substack @Substack, when do we get a Mathematics category for bestsellers? We have many great math stacks with huge engagement and audiences.

I have made a post on Substack Notes substack.com/@joeldavidha...
to the effect that we should have a Mathematics category on Substack. If you support math on Substack, please like and restack there.

This is a purely semantical proof of the compactness theorem, not relying on the details of any particular proof system, thereby bypassing proof theory entirely, a purely model-theoretical proof of a central model-theoretic result.

Namely, given a finitely satisfiable theory T, one adds the Henkin assertions ∃xφ(x)→φ(c) and then completes the theory. Each step preserves finite satisfiability, and the final theory is satisfiable by the Henkin model. Thus, compactness: every finitely satisfiable theory is satisfiable.

For example, I view the Henkin argument mainly as providing a proof of the compactness theorem, rather than the completeness theorem. One proves compactness directly via Henkin by proving that every finitely-satisfiable theory is contained in a complete finitely-satisfiable Henkin theory.

Meanwhile, his proof of the completeness theorem is top rate!! I could tell you all about it, and discuss various issues at length.

And although the focus of his talk was on those other two (forgotten) chapters, I must admit, truth be told, that I don't quite recall what those other chapters were about. (I'm sorry Leon!) I had heard his talk, but I am embarrassed to say that I couldn't tell you now the first thing about them.

The main substance was to explain the other two chapters of his dissertation. He talked at length. I recall that all the prominent Berkeley logicians were there, amongst them Vaught, Harrington, Solovay, Woodin, Addison, and many others. The lecture hall had probably 50 people or more.

In the colloquium talk, he gave his famous proof of the completeness theorem, now known as the Henkin proof, with the Henkin constants and what not. He explained how he had come to that proof in a dream, using the constants themselves instead of Skolem functions, as Gödel had done.

Henkin was evidently annoyed that everyone seems always to remember only the third chapter of his dissertation, the one containing his famous proof of the completeness theorem, but people never talked much about the other two chapters.

When I was a graduate student in Berkeley, I remember a talk for the Logic Colloquium by Leon Henkin, long a member of the math department amongst the logic group started by Tarski. The title of his talk was "The other two chapters."

Find the whole series of essays on the infinite subway paradox here.
www.infinitelymore.xyz/t/infinite-s...
#InfinitelyMore #FrivolitiesOfTheGods

The uncountable transfinite subway We explore a more sophisticated version of the infinite subway paradox, with stations all the way to the uncountable ordinals and beyond

The infinite subway paradox, extending to uncountable ordinals.
www.infinitelymore.xyz/p/the-uncoun... #InfinitelyMore #FrivolitiesOfTheGods

Where is the first repetition in the cumulative hierarchy up to elementary equivalence? This is a sequel to my MSE question about elementary equivalences between the $V_α$. Given that there are only $ℶ_1$ first-order theories in the language of set theory, by pigeonhole principle, there

I answered a question about the first repetition of theories in the cumulative hierarchy. mathoverflow.net/a/500938/1946

Next week's reading for the core PhilSciMath seminar, two sessions.