MW: It’s been ages since John Baez and I discussed Enayat’s paper—not since October 2020! John has since moved on to fresh woods and pastures new. I’ve been reading novels. But I feel I owe it to our millions of readers to finish the tale, so here goes.
Category Archives: Math
The −1 Dimensional Reduced Homology Group of the Empty Set
Hatcher’s Algebraic Topology unfortunately hadn’t been written when I studied the subject in grad school, but a few years ago I participated in a meetup that went through about half of it. I wrote up notes on things I puzzled over, or just observations I found helpful. Here they are. Besides the homology group mentioned in the title, some other tidbits: a figure to elucidate the calculation of the fundamental group of the complement of the Alexander Horned Sphere, and more details for the intuitive proof Hatcher sketches of Poincaré duality.
Filed under Topology
Topics in Nonstandard Arithmetic 9: Tricks with Quantifiers
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.
Filed under 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)
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
Non-standard Models of Arithmetic 21
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Previous Paris-Harrington post
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
Diagonal Argument 