### Nonstandard Models of Arithmetic

#### Posted by John Baez

A nice quote:

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 from Matthew Katz and Jan Reimann’s nice little 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.

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 $T$-standard model of $PA$”, where $T$ is some set of axioms extending $ZF$. 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.

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

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

Part 13: Enayat’s “natural” axiomatization of $PA^T$, and his proof that this works. A digression into Tarski’s theorem on the undefinability of truth, and how to work around it. For example, while truth is not definable, we can define truth for statements with at most a fixed number of quantifiers.

## Re: Nonstandard Models of Arithmetic

A joke:

If we add up all of a nonstandard set of natural numbers, do we still get $-\frac{1}{12}$?