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Regarding structures such as the natural numbers, complex numbers, groups, etcetera.

Would it make sense to say that a collection of properties is a characterization of sets? I know that there are axiomatizations (ZFC, NBG, New foundations, etcetera) but I have never heard them referred to as characterizations, is that convention or due to a fundamental difference?.

It's much better for me if you can also mention a reference that discusses the topic of axiomatization vs characterization.

Regards and thanks.

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I think of a characterization generally as describing a structure uniquely up to isomorphism. Thus for example the system of real numbers is characterized as the (unique up to isomorphism) complete ordered field. Universal properties also serve to characterize a structure. Or, a characterization (perhaps more often called a classification) may specify a particular class of structures; for example one may classify finite simple groups in terms of well-known families.

An axiomatization is just part of a theory: the signature of a theory describes the data for a certain class of structures one may be interested in, and axioms describe properties we would like the structures to satisfy because we consider them particularly relevant to a given subject. Thus the signature for the theory of monoids consists of a constant (for the identity element) and a binary function symbol; the axioms are associativity and unit axioms, which we feel are particularly relevant for capturing what we expect of endomorphisms (e.g., functions from a set to itself). The same would apply to more complicated theories such as ZFC. There are many non-isomorphic models which satisfy these axioms, so we don't have anything like a characterization in the sense of the first paragraph.

I can't think of any references which discuss this distinction, but it's the one that I'm familiar with.

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A characterization of a structure is to give some properties of the structure, but the structure must then first be defined.

The axioms define a structure and of course give it it's form, or if you want: it's character.

So axiomatization is about forming a structure while characterization is about describing an already formed structure.

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