Author Archives: Michael Weiss

The Making of the Atomic Bomb

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For a few years, I belonged to a history book club. Unlike many book clubs, we didn’t all read the same book. Instead, we’d pick a topic for the next meeting, at which the participants would each give short presentations on books of their choosing.

Recently I ran across my write-ups. As the internet has yet to run out of space, I thought I’d post them. I begin with two on the atomic bomb.

(Why ‘atomic bomb’, rather than ‘nuclear bomb’? See this post.)

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Topics in Nonstandard Arithmetic 7: Truth (Part 3)

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Previous “Truth” post

Last time we looked at Tarski’s inductive definition of truth formalized inside ZF set theory. Continue reading

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Topics in Nonstandard Arithmetic 6: The Axioms

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This is a “reference” post. With all the posts already filed under Peano Arithmetic, I realize I never explicitly stated the axioms. Of course you can find them on Wikipedia and at a large (but finite) number of other places, but I thought I should put them down somewhere on this site.

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Topics in Nonstandard Arithmetic 5: Truth (Part 2)

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Next “Truth” post

Last time we looked at Tarski’s inductive definition of truth, expressed informally. We saw how for models of PA, it can be formalized as an infinite sequence of formulas True0, True1, …, formulas belonging to L(PA) itself. But not as a single formula in L(PA).

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Topics in Nonstandard Arithmetic 4: Truth (Part 1)

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In post 15 of the Conversation, I observed:

  • Gödel’s two most famous results are the completeness theorem and the incompleteness theorem.
  • Tarski’s two most famous results are the undefinability of truth and the definition of truth.

The second bullet has occupied its share of pixels in the Conversation. Time for a summing up.

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Non-standard Models of Arithmetic 20

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Trudy Campbell

MW: OK, let’s recap the setup: we have a three-decker ωUUV. So far as U is concerned, ωU is the “real, true omega”. V knows it isn’t. Enayat’s question: what properties must an omega have, for it to be the omega of a model of T? Here T is a recursively axiomatizable extension of ZF, and U is a model of it.

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Topics in Nonstandard Arithmetic 3: The Arithmetic Hierarchy (Part 2)

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Last time I defined ∃n and ∀n prefixes and formulas; Σn, Πn , and Δn relations (and functions) on ℕ; Σn(PA), Πn(PA), and Δn(PA) formulas in L(PA); and Σn(N), Πn(N), and Δn(N) relations (and functions) on a model N of PA. I won’t repeat all that, but a few bullet points may help load it into working memory:

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Topics in Nonstandard Arithmetic 2: The Arithmetic Hierarchy (Part 1)

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Nonstandard Arithmetic: A Long Comment Thread

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Posts 7 and 8 developed an extensive comment thread, mainly between Bruce Smith and John Baez. It was hard to follow in that format, so I converted it to a separate webpage.

Topics: (a) Why do standard models of ZF have standard ω’s? (b) Interactions between the Infinity Axiom and the Foundation Axiom (aka Regularity). (c) The compactness theorem. (d) The correspondence between PA and “ZF with infinity negated”: nonstandard numbers vs. ill-founded sets, and the Kaye-Wong paper (cited in post 8).

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Topics in Nonstandard Arithmetic 1: Table Setting

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John Baez and I have been having a conversation about nonstandard models of Peano arithmetic (PA). It started with “John’s dream”, as I’ll call it, which begat a goal: understand a paper by Enayat. For more on the dream and the goal, check out the second post.

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