# Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't have to be {0,1}.

But I wonder, aren't all the multivalued logics also part of/can be modeled in set theory? Is there some new logic coming in with topoi which weren't there before? Did it just help discovering new ideas? Fuzzy stuff etc. are all existent in "conventional set theory mathematics" already, right?

-
Goldblatt's book Topoi: The Categorial Analysis of Logic is very accessible and is available in an inexpensive paperback edition from Dover. –  MJD Oct 26 '12 at 11:31
@MJD: Haha, I'm reading the book and this is where the question popped up. ;) –  NikolajK Oct 26 '12 at 11:31

Category theory itself, and thus topos theory, can be formalized in set theory(*). So, in a weak sense, everything in topos theory is already "in" set theory. Of course, set theory can be formalized in topos theory using the category Set, and so set theory is also "in" topos theory. The real question that matters is which formalization is useful for a particular purpose. For some purposes, topos theory provides a useful framework to the people who use it, and they prefer this framework over the equivalent framework where everything is rephrased in terms of set theory.

The key point about formalization in set theory is that category theory, and topos theory, are formal axiomatic systems, and any axiomatic system can be studied using set theory as a metatheory.

(*): There is a minor issue that some things in topos theory may use axioms that appear to be large cardinal axioms from the point of view of set theory. But this is not an impediment to formalizing things in set theory if we simply assume the necessary large cardinal axioms.

-

Topos theory provides several kinds of morphisms of toposes, which allow the comparison of models. Most toposes we study are indeed classes of functions that exist in some model of higher order constructive logic. These models can be described using set theory. But from that perspective, it is harder to see the relations between the models.

I have not read Goldblatt's book, but the main criticism I have heard about it, is that it doesn't cover morphisms of toposes. You should try MacLane and Moerdijk's "Sheaves in Geometry and Logic" instead.

-
How so? You certainly can model fuzzy sets using normal sets, indeed, afaik, they're defined as functions from a normal set into $[0,1]$, and as such they are defined by their representations in set-theory, and I see no reason for other logics (including intuitionistic logic) to be impossible to model in set theory. –  tomasz Oct 27 '12 at 11:04