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Suppose the study of some kind of mathematical object has evolved in an "organic" fashion, until it reaches the level of maturity that one wants to take an axiomatic approach to the study of those objects.

How does one go about finding a "minimal" set of axioms so that the axiomatically-constructed object has all the desired properties of the "organic" object, and no more?

To give some context to the question: in my math class, we spent the first half of the semester studying the (real and sometimes the complex) projective plane. Now we're looking at the axiomatic approach (begun by Hilbert?) to projective geometry, starting by defining the axioms for an "incidence geometry".

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closed as not a real question by lhf, LVK, tomasz, Quixotic, Austin Mohr Sep 4 '12 at 16:15

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I don't think there is an answer to your question... You might just as well ask "How does one do mathematical work?". –  Mariano Suárez-Alvarez Apr 5 '12 at 22:43
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One discards redundant axioms by finding proofs of them from the others. One shows that an axiom $\phi$ is not redundant by finding a model $M$ of the remaining axioms such that $\phi$ is false in $M$, and such that $M$ is not something we want to allow. –  André Nicolas Apr 5 '12 at 22:44
    
@AndréNicolas, Thank you, that's a great way to put it. –  Will Apr 5 '12 at 22:50
    
@Will: There are complications. The Hilbert axiomatization of geometry was during the prehistory of logic, and was second-order. Nowadays, there is a preference for first-order formulations, which cannot be categorical, though they can be complete. Thus one gives up the notion of capturing all the intuition one has about a particular structure. But the general idea (prove, or find a model that exhibits the independence) works in principle for second-order. However, funny things can happen. For instance, the Pasch Axiom is a theorem if one denies Axiom of Choice sufficient strongly. –  André Nicolas Apr 5 '12 at 23:31
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One discards redundant axioms by finding proofs of them from the others.

One shows that an axiom $\varphi$ is not redundant by finding a model $M$ of the remaining axioms such that $\varphi$ is false in $M$, and such that $M$ is not something we want to allow.

Here I was dealing only with the term minimal in a set-theoretic sense. There are fuzzier notions of smallness that involve digging into the structure of individual axioms. Witness for example, in Group Theory, the (only apparent) weakenings of the requirement of the existence of an inverse.

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