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.
Category Archives: Peano Arithmetic
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
Nonstandard Models of Arithmetic 24
MW: Indicators: we don’t need to discuss these, to prove the Paris-Harrington theorem. But I think they offer valuable insight.
Filed under Conversations, Peano Arithmetic
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
