Are numbers universal in Set Theory and nominalist in Category Theory?

The number 3, when considered as a universal, abstracts the property of 3'ness from all groups of 3 objects. One supposes that this universal, by conceptual neccessity of what is understood by the idea of 'universal' is unique. The problem of universals is the ontological status of this universal which can be broadly characterised as being of three kinds - real (following Plato), nominal (following Ockham) and Ideal (following Kant).

In the usual theory of Sets, one notices, that for example the empty set by the axiom of extensionality is unique.

But what about the number 3 in theory of Sets?

If one follows a particular construction of the integers, for example by Von Neumanns nested sets, then the number 3 is determined and constructed uniquely. But one can always choose another construction. And one notes that the detailed construction of the integers do not matter - at least where one is not concerned with their ontological status - but how they function in a theory which makes use of certain of their properties.

One might say, rather than use one particular construction amongst many, use all possible constructions.

In this sense, the number 3 appears to lose its universality and appears in the world of concrete objects. That is the number 3 is thought of not as the instantiation of the abstract property of 3'ness shared by all groups of 3 objects, but as all groups of three objects but understood both indefinately and particularly. Indefinate as we choose not some represenative amongst them, and particularly because we keep hold of their distinctness.

Is this a form of nominalism?

What I've described above is how numbers are understood in Topos Theory - aka generalised Set Theory.

1. Can one say, at least roughly, that Set Theory attempts to ground itself on Platonic Realism, which states that universals exists. And that Category/Topos Theory attempts to ground itself on nominalism?

2. That grounding of Set Theory on Platonic Realism appears to be a failure for the reasons I've given above: In that the only universal it contains is the empty set, and that it doesn't provide us with the universal example of the number 3?

3. That the attempt to ground the the uniqueness that should be inherent in the realist position on universals, at least in set theory, is/was understood to be captured by the axiom of extensionality. One notices in particular that the uniqueness of the empty set is essentially reliant on this axiom.

4. Does Category/Topos Theory capture the nominalist position - or are there failures here? One notes that the theory speaks of 'universal properties' which has a precise technical definition, and one notes that they do not produce a unique object but simply characterises those objects that have those properties - and show, as one should readily note, that any two such objects, so far as these properties are under consideration, are equivalent.

• Imagine a universe with three objects in it. Wouldn't that be an example of a universe in which 3 is a universal? In which case your concern is simply that there are too many things in the universe; allowing for the possibility that there are many sets of cardinality 3. I'm not sure if this remark is sensible. I was just struck by your concern that there's only one empty set but there are lots of sets of cardinality 3. But that's a problem with the universe; not with 3. Feb 8 '14 at 6:19
• Please, see Frege's Die Grundlagen der Arithmetik (1884) [The Foundations of Arithmetic] for a (for me) definitive demolition of the conception of "number x, when considered as a universal, abstracts the property of x'ness from all groups of x objects." Feb 8 '14 at 9:47
• It could be argued that category theory can be used to attempt to repair the damage to done to Platonic realism. That is we define "three" to be any instance of three-ness that we can find, and that any two instances of three-ness are the same (read uniquely isomorphic). If we look at a type system like HoTT, then this becomes that any two threes are equal. I could also see where this argument can be taken apart. Feb 10 '14 at 9:00
• @Baby Dragon: I can see the angle you're coming from. It matters, I think, to what degree uniqueness of universals matter in Platonism. Another way of looking at it is seeing Category Theory as bridging Platonism & nominalism. Feb 11 '14 at 20:05
• Hey, Mozibur! +1 This is an awesome question like usual :). I will think it over but where do I even begin because there is so much that can be said. Thanks for the work. Feb 15 '14 at 20:46