On page 164 of his book **Models of Peano Arithmetic**, Kaye states Friedman's Theorem:


Let $M{\vDash}PA$ be nonstandard and countable, let $a\in M$ and let $n\in {\mathbb N}$. Then there is a proper initial segment $I\subseteq\_c M$ ($\subseteq\_c$ means "cofinal in") containing $a$ such that $I\cong M$ and $I<\_{\Sigma\_n} M$.


However, on the next page he writes "Nor can we expect in general to get initial segments $I$ with $M\cong I < M$ and $M\neq I$, i.e., elementary for all formulas. For example if $M=K\_T$ (where $T\neq Th({\mathbb N})$ is a complete extension of PA) then $M$ has no proper elementary substructures, and so certainly has no proper elementary initial segments!"


I am confused. If two models in the same language are isomorphic, are they not elementarily equivalent? Of course the converse need not be true.


An isomorphism of models is a bijective homomorphism of the language's algebraic portion (constants and functions) which preserves **and reflects** all relations of the language (Hodges, **Model Theory**, p5). Therefore by induction on the structure of any ${\mathcal L}\_{\omega,\omega}$ formula, the isomorphism will both preserve and reflect it. So an isomorphism preserves **all** formulas (Hodges, Theorem 2.4.3(c)).


If the proper initial segment is isomorphic as a model to the entire model, how could any first-order sentence possibly be true in one and not in the other?


Thanks,


* a



models of PA which are isomorphic but not elementarily equivalent?
There is a major difference between *elementary equivalence* and *elementary embedding*. Moreover, in this case, the actual embedding is somewhat ambiguous. First, let me recap some often confused terminology.


Two models are *elementary equivalent* if they satisfy the same first-order sentences. Any two isomorphic models are always elementary equivalent. An *elementary embedding* is a map j:A→B such that, for all first-order formulas φ(v1,...,vk) and all a1,...,ak ∈ A, A ⊧ φ(a1,...,ak) iff B ⊧ φ(j(a1),...,j(ak)). An isomorphism is always an elementary embedding.


The notation A ≺ B means that A is an *elementary submodel* of B, i.e. the *inclusion map* A ⊆ B is an elementary embedding from A into B. In your context, the isomorphism (or its inverse) is *not* the proposed elementary embedding, it is the inclusion map which is in question: it is elementary for Σn formulas, but not elementary for all first-order formulas.


