In Models of Peano Arithmetic by Kaye, p153, it states : Tennenbaum's theorem may be regarded as the model-theoretic analogue of the Godel-Rosser incompleteness theorem.
In Cohen Set Theory and the Continuum Hypothesis, p49 it states "By a similar argument using Tarski's theorem on the un-definability of truth, (and without using recursively inseparable sets) one "easily" shows "There is no non-standard model M for the integers in which + and * are given by functions definable in Z1 (i.e. PA) and such that all the true statements of Z1 (PA) hold in M (a non standard model of Z1 (PA)).
My first question is - is there a proof of Cohens "easy" proof somewhere I could read?
My second question is on Kayes p153 statement : Is it saying that models of PA exist (viewed purely from the Metatheory) which satisfy additional statements to those that are derivable from Z1 (PA) and that these models include non-isomorphic models, called "Non-standard" models, and these additional statements are "not recursive" in that they aren't made up from a finite set of expressions based upon Z1 (PA) axioms. ? Are there examples of what these "additional statements" look like, or where they sit in the arithmetical higher-archy ? I suspect its saying more than this.