Every specialty has its tricks of the trade. They become second nature to practitioners, so they often don’t make it into the textbooks. Quantifiers rule in logic; here are some of the games we can play with them. I’ll start with tricks that apply in logic generally, then turn to those specific to Peano arithmetic.
Author Archives: Michael Weiss
The Second French Revolution
Delacroix’s Liberty Leading the People
Another post from the History Book Club. Continue reading
Filed under History Book Club, Reviews
Topics in Nonstandard Arithmetic 8: Extensions and Substructures
Substructures and extensions loom large in math: subgroups, subrings, extension fields, submanifolds, subspaces of topological spaces… So too in the model theory of PA.
Filed under Peano Arithmetic
The Decision to Drop the Bomb
Another post from the History Book Club.
(Why ‘atomic bomb’ rather than ‘nuclear bomb’? See this post.)
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
Topics in Nonstandard Arithmetic 7: Truth (Part 3)
Previous “Truth” post Next “Truth” post
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
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
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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
