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Let $\mathcal{L}$ be a finite first-order language. When we say structure we mean $\mathcal{L}$-structure.

Question. Can someone lists different ways which we may use to show two given structures are elementarily equivalent?

For example:

1) doing induction on complexity of formulas,

2) using Ehrenfeucht-Fraisse games,

3) ....

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  1. Finding a theory satisfied by both structures and then proving that this theory is complete, for example by Vaught's test.

  2. Building a forcing extension of the set-theoretic universe in which the structures become isomorphic. (This actually proves $L_{\infty,\omega}$-equivalence, and it tends to be similar to an argument about infinitely long Ehrenfeucht-Fraïssé games.)

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    $\begingroup$ This is a subset of 3. (and 1.), but can be useful: showing that the theory of each has elimination of quantifiers and that they satisfy the same quantifier-free sentences (possibly after expanding the language a bit). Another method is finding an elementary embedding or a common elementary extension (with which q.e. also helps). $\endgroup$ – tomasz May 16 at 14:34

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