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What general set-theoretical definitions of the notion of "structure" are there?

By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that

a set $x$ is a structure iff $\Phi(x)$

The definitions of structures in standard model theory would qualify, but they are restricted to first-order structures (or second-order, at most).

Suppes' set-theoretical predicates à la

a set $x$ is an algebraic structure iff $\phi(x)$

would qualify, but they are obviously too specific. (A rigorous meta-definition of "set-theoretical predicate" seems to be missing.)

Bourbaki's definition of species of structures seems to be the most general one and most strongly related to categories (see here and here). Nevertheless it was never widely accepted and used.

Are there other approaches I'm just not aware of? Or is such a general definition considered useless?

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Are you looking for an internal definition in $ZFC$ (say), or an external definition as metalogic (i.e. we "know" that the function $f$ is acting the way it should, etc. etc.)? –  Asaf Karagila Jun 20 '11 at 10:57
    
I assume I'm looking for an internal definition, but it would be interesting to see an external one, too. –  Hans Stricker Jun 20 '11 at 11:25
    
The usual definition of a structure in model theory extends in a trivial way to third-order logic and upwards to logic in all finite types (in the sense of type theory). To go beyond that, we have to switch to infinitary formulas, at which point the definition of a structure in model theory could also handle infinitary types. –  Carl Mummert Jun 20 '11 at 14:04
    
@Carl: And Bourbaki's species of structures are subsumed under this definition? (If it's so easy, why did Bourbaki make it so hard?) –  Hans Stricker Jun 20 '11 at 14:46
    
@HansStricker, good question - I have yet to encounter a satisfactory answer to this general class of question. –  goblin Apr 19 '13 at 6:59

1 Answer 1

Stewart Shapiro, in his book Philosophy of Mathematics: Structure and Ontology, discusses a general "Theory of Structures". Given its proximity to the standard model theoretic account, I'm not sure why he bothers.

Others in the philosophy of mathematics have tried to characterise structure in terms of category theory. (I just noticed you mentioned this in your question.)

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