There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time.
This is a quote from Matthew Katz and Jan Reimann’s book An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics. I’ve been been talking to my old friend Michael Weiss about nonstandard models of Peano arithmetic on his blog. We just got into a bit of Ramsey theory. But you might like the whole series of conversations, which are precisely about this murky abyss.
Here it is so far:
• Part 1: I say I’m trying to understand ‘recursively saturated’ models of Peano arithmetic, and Michael dumps a lot of information on me. The posts get easier to read after this one!
• Part 2: I explain my dream: to show that the concept of ‘standard model’ of Peano arithmetic is more nebulous than many seem to think. We agree to go through Ali Enayat’s paper Standard models of arithmetic.
• Part 3: We talk about the concept of ‘standard model’, and the ideas of some ultrafinitists: Alexander Yessenin-Volpin and Edward Nelson.
• Part 4: Michael mentions “the theory of true arithmetic”, and I ask what that means. We decide that a short dive into the philosophy of mathematics may be required.
• Part 5: Michael explains his philosophies of mathematics, and how they affect his attitude toward the natural numbers and the universe of sets.
• Part 6: After explaining my distaste for the Punch-and-Judy approach to the philosophy of mathematics (of which Michael is thankfully not guilty), I point out a strange fact: our views on the infinite cast shadows on our study of the natural numbers. For example: large cardinal axioms help us name larger finite numbers.
• Part 7: We discuss Enayat’s concept of “a T-standard model of PA”, where T is some axiom system for set theory. I describe my crazy thought: maybe your standard natural numbers are nonstandard for me. We conclude with a brief digression into Hermetic philosophy: “as above, so below”.
• Part 8: We discuss the tight relation between PA and ZFC with the axiom of infinity replaced by its negation. We then chat about Ramsey theory as a warmup for the Paris–Harrington Theorem.
• Part 9: Michael sketches the proof of the Paris–Harrington Theorem, which says that a certain rather simple theorem about combinatorics can be stated in PA, and proved in ZFC, but not proved in PA. The proof he sketches builds a nonstandard model in which this theorem does not hold!
• Part 10: Michael and I talk about “ordinal analysis”: a way of assigning ordinals to theories of arithmetic, that measures how strong they are.