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.

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 (plural!) 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 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 T -standard model of PA PA ”, where T T is some set of axioms extending ZF ZF . We conclude with a brief digression into Hermetic philosophy: “as above, so below”.

Part 8: We discuss the tight relation between PA PA and ZFC 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.

Part 11: We finally get serious about working through Ali Enayat’s paper Standard models of arithmetic. Michael introduces PA T PA^T , the set of all closed formulas in the language of Peano arithmetic that hold in all T T -standard models where T T is a consistent recursively enumerable extension of ZF. He explains how to recursively axiomatize PA T PA^T using “Craig’s trick”, and as a bonus explains “Rosser’s trick”.

Part 12: Some examples. PA ZF PA^ZF is strictly stronger than PA PA . PA ZFC + GCH = PA ZF PA^{ZFC+GCH} = PA^ZF where GCH GCH is the generalized continuum hypothesis. PA ZFL = PA ZF PA^{ZFL} = PA^{ZF} where ZFL ZFL is ZF ZF plus an axiom saying all sets are constructible in Gödel’s sense. But PA ZFI PA^{ZFI} is strictly stronger than PA ZF PA^{ZF} , where ZFI ZFI is ZF ZF plus the existence of an inaccessible cardinal!