MW: Last time we learned about the “back-and-forth” condition for two countable structures M and N for a (countable) language L:
Category Archives: Math
Nonstandard Models of Arithmetic 30
MW: Time to finish off Enayat’s Theorem 7:
Theorem 7: Every countable recursively saturated model N of PA+ΦT is a T-standard model of PA.
Filed under Conversations, Peano Arithmetic
Nonstandard Models of Arithmetic 29
MW: We’re still going through Enayat’s proof of his Theorem 7:
Theorem 7: Every countable recursively saturated model N of PA+ΦT is a T-standard model of PA.
Filed under Conversations, Peano Arithmetic
Nonstandard Models of Arithmetic 28
MW: I ended the last post with a puzzle. Here it is again, in more detail.
Filed under Conversations, Peano Arithmetic
Nonstandard Models of Arithmetic 27
MW: Enayat’s second major result is:
Theorem 7: Every countable recursively saturated model of PA+ΦT is a T-standard model of PA.
Filed under Conversations, Peano Arithmetic
Very Unique
My unique (but not very unique) microwave
Everyone has their pet peeves, and peeves about language abound. My pet peeve is with people who object that “very unique” is illogical. For example, this pithy statement:
Uniqueness is a binary condition. Something is unique or it is not. There are no degrees of uniqueness. Something cannot be partly unique, mostly unique, very unique, etc.
Nonstandard Models of Arithmetic 26
MW: Continuing the recap… Continue reading
Filed under Peano Arithmetic
Nonstandard Models of Arithmetic 25
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.
Filed under Peano Arithmetic
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
Diagonal Argument 