Some notes on uncountable models of arithmetic
Abstract
Historically, logicians have found it very difficult to understand the structure of uncountable models of arithmetic. I argue that if a model $M$ of arithmetic has an element $a$ with $|[0,a]^M| = \aleph_0$, then it resembles a countable model. As evidence, I present two theorems. The first states that if $M$ is such a model with its standard system contained in some Scott set $\mathcal {S}$ and $X \in \mathcal{S}$, then $M$ admits an elementary extension $N$ with $X \in \mathrm{SSy}(N) \subseteq \mathcal{S}$. The second states that if $M$ is such a model and $L$ is a finite lattice such that every countable $M_0$ has an elementary extension $N_0$ with $\mathrm {Lt}(N_0/M_0) \cong L$, then $M$ can also be extended this way. An alternative proof of Ehrenfeucht’s lemma is also included.
