<logic> Roughly, when models are identical in form and differ (if at all) only in content. Or when their domains map onto one another in the sense that their elements can be put into one-to-one correspondence and they stand in the same relations. To define isomorphism more precisely, let us say that D and D' are the domains of the two models under comparison, that for every member d of D there is a counterpart d' of D' and vice versa, that every function f defined for D has a counterpart function f' defined for D' and vice versa, and that every predicate P defined for D has a counterpart predicate P' defined for D' and vice versa. Now the two models are isomorphic iff these three conditions are met: (1) D and D' can be put into one-to-one correspondence, (2) for all functions f and f', f(d1...dn) = dn+1 iff f'(d1'...dn') = dn+1' and (3) for all predicates P and P', Pd1...dn iff P'd1'...dn'.
See categoricity of systems, L"wenheim-Skolem theorem
[Glossary of First-Order Logic]
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