Re: Elementary equivalence and elementary substructures

On 24 helmi, 08:56, "mordov" <-knowled...@xxxxxxxxxx> wrote:
There's a really common fact inmodeltheory:

(1) Let A and B be L-structures with L a first-order language. That A
is a substructure of B and A and B are elementarily equivalent does
not imply that A is an elementary substructure of B.

According to Hodges "Ashortermodeltheory", that must mean that,
supposing A is a substructure of B and they're elementarily
equivalent, that the substructure embedding must not be elementary
(i.e. preserve all first-order formulas of L). Well, I can't think of
an example (or, the examples I can think of I"m not sure of). Can
anyone give a concrete example?

Hinman (2005) gives this example: M_1 = ("naturals"/{0}, =<) and M_2 =
("naturals", =<). Now there is isomorphism so they are elementaryli
equivalent, but FORALLy(x=<y)[1] is not valid in both structures.


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