Date: Wed, 13 Mar 1996 19:30:06 GMT From: Ralph Loader <loader@maths.ox.ac.uk>
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This is an interesting example. You state a question in mathematical English, and then criticise ZF for being able to express this question, while category theory cannot. One wonders what other questions stated in mathematical English---some of them perhaps perfectly sensible---can be stated in the language of ZF, but not in the language of category theory. I'd much rather that my formalisation of mathematics could state nonsensical things (consistency strength isn't an issue here), than to be living in the fear that my formalisation may be inadequate to express some sensible arguments.
[One] example that I think [is] relevant to this debate.
Category theorists are keen on statements to the effect that structures are defined by their universal properties. A typical book on topos theory may define an elementary topos as a category with finite limits and power objects. It then goes on to show that any topos has internal-homs. How? By defining the function space as a certain set of sets of ordered pairs...
This might be a little disingenuous - Of course, in any topos there will be a monic X=>Y --> P(XxY) , so via the "internal language" X=>Y may be seen as a certain set of sets of ordered pairs. But the point is that via the "internal language" you can (and indeed you may) argue about the structure of a topos in a very "set-like" manner. But of course you are not obliged to do so. So in a very real sense, your (first) example illustrates that your fear is unfounded. (Indeed, one can imagine a topos theorist pondering the matter of simple groups being zeros of the Riemann zeta function in suitable toposes; the point of Peter Freyd's observation was (I think) that a topos theorist without any taste is less likely to stumble upon this question than a set theorist without taste, who might well do so... due to the emphasis upon different fundamental notions in the two approaches.) = rags =