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Joel David Hamkins @joeldavidhamkins.bsky.social · 2d

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

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

mathoverflow.net

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Joel David Hamkins @joeldavidhamkins.bsky.social · 2d

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

Joel David Hamkins @joeldavidhamkins.bsky.social · 2d

And the author reply to critics:

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

(Note: CMS nested quote rule.)

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Joel David Hamkins @joeldavidhamkins.bsky.social · 2d

"On 'On'"

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Joel David Hamkins @joeldavidhamkins.bsky.social · 3d

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

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Joel David Hamkins @joeldavidhamkins.bsky.social · 3d

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

Joel David Hamkins @joeldavidhamkins.bsky.social · Aug 26

For your enjoyment, here is the reading list for my undergraduate core seminar this semester in Philosophy, Science, and Mathematics. We shall focus on topics in the philosophy of mathematics, philosophy of science, and philosophy of computability and AI.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 3d

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

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Joel David Hamkins @joeldavidhamkins.bsky.social · 4d

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

Artist Chalon dans la rue, l’homme orchestre, festival 2021

YouTube video by Svenleboubou

youtu.be

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Joel David Hamkins @joeldavidhamkins.bsky.social · 4d

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

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Joel David Hamkins @joeldavidhamkins.bsky.social · 4d

Detail from student quiz on Lakatos, Proofs and Refutations.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 4d

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

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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.

substack.com

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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.

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Joel David Hamkins @joeldavidhamkins.bsky.social · 9d

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."

Michael Kinyon @profkinyon.bsky.social · 9d

I hope someday people refer to "Kinyon's Third Theorem" and can describe it easily, but the reason no can remember what the First or Second Theorem were is that they never existed.