Another post from the History Book Club.
(Why ‘atomic bomb’ rather than ‘nuclear bomb’? See this post.)
The Decision to Drop the Bomb
Filed under History Book Club, Physics, Reviews
The Making of the Atomic Bomb
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.)
Filed under History Book Club, Physics, Reviews
Nonstandard Models of Arithmetic 23
MW: OK! So, we’re trying to show that M, the downward closure of B in N, is a structure for L(PA). In other words, M is closed under successor, plus, and times. I’m going to say, M is a supercut of N. The term cut means an initial segment closed under successor (although some authors use it just to mean initial segment).
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 22
MW: So we have our setup: B⊆M⊆N, with N a model of PA, B a set of “diagonal indiscernibles” (whatever those are) in N, and M the downward closure of B in N. So B is cofinal in M, and M is an initial segment of N. I think we’re not going to go over the proof line by line; instead, we’ll zero in on interesting aspects. Where do you want to start?
Filed under Conversations, Peano Arithmetic
Topics in Nonstandard Arithmetic 7: Truth (Part 3)
Last time we looked at Tarski’s inductive definition of truth formalized inside ZF set theory. Continue reading
Filed under Peano Arithmetic
Topics in Nonstandard Arithmetic 6: The Axioms
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.
Filed under Peano Arithmetic
Non-standard Models of Arithmetic 21
Bruce Smith joins the conversation, returning to a previous topic: the Paris-Harrington theorem. (Discussion of the Enayat paper will resume soon.)
Filed under Conversations, Peano Arithmetic
Topics in Nonstandard Arithmetic 5: Truth (Part 2)
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).
Filed under Peano Arithmetic
Topics in Nonstandard Arithmetic 4: Truth (Part 1)
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.
Filed under Peano Arithmetic
Non-standard Models of Arithmetic 20
Trudy Campbell
MW: OK, let’s recap the setup: we have a three-decker ωU⊂U⊂V. 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.
Filed under Conversations, Peano Arithmetic
Diagonal Argument 