FOM: Re: Explaining Simpson and McLarty to each other
cxm7 at po.cwru.edu
Sat Jan 17 14:19:46 EST 1998
Reply to message from JSHIPMAN at bloomberg.net of Fri, 16 Jan
> Colin, I think Steve really wants to know the following:
> If Set Theory did not exist, would it be necessary to invent it? That is,
>can you come up with an alternative Analysis 101 curriculum, in which you prove
>useful theorems like Fubini's theorem, Stokes's theorem (bridge to topology),
>some useful formulation of Fourier Analysis (bridge to engineering), basic
>properties of the Riemann zeta function (bridge to Number Theory), Chebyshev's
>inequalities (bridge to probability and statistics), and so on, starting from
>categories or toposes rather than set theory? If all you need are natural
>number objects and right inverses those are not hard to motivate, the question
>is can this really be done in a technically manageable fashion?
That's a nice openning. Of course we would need to invent set
theory in some sense. We would not need the cumulative heirarchy. Think
of some of Peter Aczel's points or notice Cantor did not use iterated
membership (let alone a cumulative hierarchy) in his own work.
To get the results you name you would most naturally use natural
numbers and right inverses, I think. (Steve Awodey has pointed out that
something significantly weaker than right inverses will do, called the
"internal axiom of choice", and he is the expert on this. But I find
right inverses more obvious.)
I will mention that the parts of this math not requiring choice
could as well be done in a topos assuming natural numbers and
Booleanness, i.e. assuming the law of excluded middle. That is a very
much weaker assumption than right inverses. And I suppose Steve would
consider it well motivated.
As to how it works: Whether you start with ZF or with a Boolean
topos with natural numbers, you need a lot of notational conventions
(even abuses of notation) before your math will look anything like
a textbook on real analysis. This is the "mess" Vaughan Pratt referred
to. I find that mess a bit simpler in a topos than in ZF--but we are
not talking about orders of magnitude difference, and this kind
of "simplicity" is somewhat vague and even subjective/psychological.
Anyway, once that mess is behind you, the current textbooks
on differential equations, Fourier analysis, number theory and so
on can be used unchanged. The foundational interpretation of the
notations will be different but the notations, statements, and proofs
can stand exactly as stated today.
> (Harvey and
>Steve would maintain, correctly, that you'd still have to invent set theory for
>certain natural theorems whose logical strength corresponds to big ordinals).
More often I hear about large cardinals, which are defined by
combinatoric or topological properties which are just as easily stated
in a topos as in ZF (indeed stated in virtually the same way). When it
comes to ordinals there is some difference of method, though no
difference in expressive power. In ZF every set has a partial order
on its elements, the membership relation. And some sets are well-ordered
this way, "the ordinals". In the topos approach no set, or object,
has any intrinsic ordering. But you can still define an ordered set as
a set with a specified order relation, and you can ask which well
orderings provably exist.
The difference is that when you talk about ordinals in
ZF you use a lot of the apparatus you needed from the start to deal
with sets. In a topos when you talk about well orderings you are
raising an issue you did not need for a lot of other purposes.
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