# Absoluteness and categories

From the wikipedia article on the Skolem paradox:

A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed this is not possible, creating doubts about the use of set theory as a foundation of mathematics.

I would like to know if the term "categorical" for that property here lead to the naming of category theory. Maybe because all the problems of non-absoluteness don't happen there? How is the relation?

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I doubt it is related. "Category" is a common word and it appears in math in unrelated senses like many other words, e.g. Lusternik-Schnirelmann category, Baire category... –  Qiaochu Yuan Jun 28 '12 at 21:06

This is not the same category as in category theory.

This is a model theoretic concept of categoricity. We say that a theory is categorical if it has exactly one model up to isomorphism. We know that a first-order theory is categorical if and only if its only model is finite.

So for first-order theories we have a weaker notion, $\kappa$-categorical, namely all models of cardinality $\kappa$ are isomorphic.

Skolem's paradox showed that ZFC is not categorical, in the model-theoretical sense of the word, although historically I am not sure if the term was coined at the time.

On the other hand, category theory deals with notions of categorizing mathematical objects, "groups" or "compact topological rings".

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Thanks. But is it true what I guessed, that there are no absoluteness analog problems in category theory? If that makes sense. –  NikolajK Jun 28 '12 at 21:47
I have no idea about absoluteness in categories. At least not in the sense of set theoretical absoluteness (e.g. $\varphi$ is true in the large universe if and only if it is true in a smaller one; $\Delta_0$ etc.) –  Asaf Karagila Jun 28 '12 at 21:49
Note that Zermelo proved a quasi-categoricity result for $\mathrm{ZFC}_2$, ZFC formulated in second order logic: the models of $\mathrm{ZFC}_2$ are all and only those $V_\kappa$ where $\kappa$ is an inaccessible cardinal. –  Benedict Eastaugh Jun 28 '12 at 23:13
@Benedict: Interesting, a although reasonable theorem. Either way, second-order set theory is a philosophically weird creature since second-order logic requires a notion of sets to exists, and at least in modern times we define sets as elements of a universe of set theory... –  Asaf Karagila Jun 28 '12 at 23:17
@Asaf you can construe the second-order part as being about classes rather than sets, although of course there are still well-known philosophical difficulties with that. In any case, the proof is in Zermelo's 1930 paper 'On Boundary Numbers and Domains of Sets'. –  Benedict Eastaugh Jun 28 '12 at 23:55