*The theorem remains true without the countability assumption.* The quickest way to see this is to deduce it from the following well-known facts about the regularity scheme (see here for a definition of the regularity scheme).

**1.** *Suppose $\cal{M }$ is an ordered structure (of any cardinality) that has a proper elementary end extension, then $\cal{M}$ satisfies the regularity scheme.* This is a routine exercise.

**2**. *A model $\cal{M}$ of $I\Delta_{0}$ (of any cardinality) satisfies $PA$ iff $M$ satisfies the regularity scheme.* See, e.g., Theorem 7.3 of Kaye's text on models of $PA$ for a proof.

**Remark.** The converse of Lemma 1 is true for countable $\cal{M}$, and is due to Keisler. The proof uses the omitting types theorem. It is known that the converse fails without the countability assumption.