# Wacky Non-Standard Models of PA

Peano arithmetic is unable to pin down the natural numbers, despite its countable infinity of axioms. In fact, assuming its consistency Peano arithmetic has models of every cardinality, meaning that as far as PA is aware, there might be uncountably many real numbers. (If PA is not consistent then it has no models.) I want to take a look at these non-standard models of PA, especially the countable ones. A natural question is, how many countable non-standard models are there alongside the standard models?

Assuming the consistency of PA (which I will leave out from now on, as it’s assumed in all that follows), there are continuum many non-isomorphic countable models. That’s a lot! That means that there’s a distinct countable non-standard model of arithmetic for every real number. This is our first result: the number of nonstandard models of PA of cardinality ℵ0 is 20. Interestingly, this result generalizes! For every infinite cardinal κ, there are 2κ non-isomorphic nonstandard models of cardinality κ. That’s a lot of nonstandard models! In fact, since any model of cardinality κ involves a specification of some number of constants, binary relations over κ and functions from κ to κ, we know that the maximum number of models of cardinality κ is 2κ. So in this sense, Peano arithmetic has as many nonstandard models of each cardinality as it can have!

Let’s take a closer look at the countable non-standard models. It turns out that we can say a lot about their order type. Namely, all countable non standard models of PA have order type ω + (ω*+ω)·η. What does this notation mean? Let me break down each of the order types involved in that formula:

• ω: order type of the naturals
• ω*: order type of the negative integers
• η: order type of the rationals

So ω*+ω is the order type of the integer line, and (ω*+ω)·η is the order type of a structure that resembles an integer line for each rational number. Thus, every countable non-standard model of PA looks like a copy of natural numbers followed by as many copies of integers as there are rational numbers. The order on this structure is lexicographic: two nonstandards on the same integer line are judged according to their position on the integer line, and two nonstandards on different integer lines are judged according to the position of these integer lines on the rational line. It’s not the easiest thing to visualize, but here’s my attempt: