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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?

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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
up vote 4 down vote accepted

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.

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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.

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No, you can't model all multivalued logics in set theory. Set theory models classical propositional logic, but it does not model a logic where say the principle of contradiction fails and its negation fail also. All formal theorems of any multivalued logic exist within classical logic in the sense that if A comes as a formula in multivalued logic, it will also happen in classical logic and thus can get modeled by set theory (the converse does seem to hold for some multivalued logics, but hardly all that many of them). But, the domain of truth values differs for a formula in a multivalued logic than in classical logic.

Fuzzy stuff does NOT exist withing conventional set theory mathematics. The axiom of extensionality does not hold for fuzzy sets.

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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
No, fuzzy subsets are NOT defined as functions. A type-1 fuzzy subset of a reference set R consists of a collection of pairs where each pair has an element x and a membership function exists which assigns a degree of membership of x. You have to have both for a fuzzy set. Fuzzy set theory also permits for ideas like that of a fuzzy theorem. – Doug Spoonwood Oct 27 '12 at 11:55
Isn't the collection of pairs you described a function? Even if it's not, isn't it a set? Unless you allow it to be of proper class size, but I don't think that's really much of an obstacle. – tomasz Oct 27 '12 at 12:04
@tomasz That it fits the definition of a function and that of a set in some contexts comes as irrelevant. In classical set theory the notion of membership gets taken for granted, and every element either belongs or does not belong to a set. If we have {1, 3, 5} and we have the fuzzy subset [(1, .3), (3, 3) (5, .3)]==a set of numbers close to 3, then we have elements 1 and 5 which neither belong to set of numbers close to 3 nor do not belong to the numbers close to 3. – Doug Spoonwood Oct 27 '12 at 14:46
I'm not saying that fuzzy sets are the same thing as regular sets. But you said that you can't model fuzzy sets using regular sets, which is exactly what you just did in your comment. How is it irrelevant? – tomasz Oct 27 '12 at 15:13

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