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I wonder what the minimal axiomatization of a set of structures mean? I came across this term from Wikipedia:

For a theory $T\in A,$ let $F(T)$ be the set of all structures that satisfy the axioms $T$; for a set of mathematical structures $S$, let $G(S)$ be the minimal axiomatization of $S$. We can then say that $F(T) $ is a subset of $S$ if and only if T logically implies $G(S)$: the "semantics functor" $F$ and the "syntax functor" $G$ form a monotone Galois connection, with semantics being the lower adjoint.

My guess is that given a set of structures, its minimal axiomatization is the set of axioms such that the set of all structures satisfying the set of axioms is the given set of structures, isn't it?

Thanks and regards!

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I'm not convinced minimal axiomatisations exist. The intersection of two axiomatisations need not be an axiomatisation of the same class of models. –  Zhen Lin Feb 16 '12 at 21:44
    
@ZhenLin: What does an axiomatization of a class of models mean? –  Tim Feb 16 '12 at 21:49
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Let $T$ be as in my comment above. Define a set $S$ of axioms for $T$ to be minimal if (i) $S$ is a subset of $T$ from which every sentence of $T$ can be derived and (ii) there is no proper subset $R$ of $S$ such that all the sentences of $S$ (and hence of $T$) can be derived from $R$. Except for pretty trivial $T$, there will be infinitely many such minimal $S$. –  André Nicolas Feb 16 '12 at 22:20
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@André: The lecture notes by Peter Smith referenced in footnote 5 of the Wikipedia article also clearly use the maximal axiomatization. –  Brian M. Scott Feb 16 '12 at 23:27
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It’s implicit in Definition 3.2.1 and the little discussion that immediately follows it: $f^*$ ‘takes a bunch of $L$-structures and looks for the biggest bunch of $L$-sentences that are true of all of those structures’ [my emphasis]. That’s the maximal axiomatization. –  Brian M. Scott Feb 17 '12 at 5:09

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I think the Wikipedia is just wrong. It is trying to explain Lawvere's notion that syntax and semantics can be thought of as a pair of adjoint functors. The description of the functor F from syntax to semantics is correct (except it should probably say "class of structures"). The functor G from semantics to syntax should send a class S of structures to the set of sentences that hold in every structure in S. G(S) is not an axiomatization of S in general. You will find a fairly detailed account of all this at http://www.logicmatters.net/resources/pdfs/Galois.pdf.

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