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From: Stefan Kahrs <smk@dcs.ed.ac.uk>
Subject: Re: Non-standard proofs (was Penrose and human mathematical capabilities)
In-Reply-To: ikastan@alumni.caltech.edu's message of 31 Jul 1995 21:14:28 GMT
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Date: Tue, 1 Aug 1995 11:37:39 GMT
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Ilias Kastanas writes:
[...]
>>> sort of machine). It would also tell us something about what has
>>> to go on when a child learns about numbers as forming an infinite
>>> set, and gets the *right* set, not one of the non-standard ones.


ikastan> 	There is no danger.  The most precocious child would
ikastan> still _not_ manage to work out Compactness, or ultrafilters,
ikastan> _before_ going through the business of repeated +1's.  No
ikastan> matter how generously interpreted, +1's will not yield the
ikastan> non-standard part of a model of PA ...

I wouldn't be so sure.  We see the numbers, but do we see their
properties?  We surely see the primitive-recursive ones, probably
Sigma-01 and Pi-01, and possibly one level beyond that.
But then?  Take a Sigma-03 sentence Ex.P(x) that is classically but not
intuitionistically provable.  (Up to Pi-02 provability of classical
and intuitionistic logic coincide, at least so I've been told.)
Do we "see" its truth, do we see the number x that witnesses its
truth? [There isn't even a classical proof of P(x) since any P(x) is
in Pi-02.]
In a certain sense, one could argue that if you see such an x then
you actually see a non-standard model.


But apart from that discussion, I always wondered what people think
about the "standard model of the" real numbers.
Do you see that?  Do you believe other people see the same?
The reals and their definition often made me feel uneasy.
On the one hand, they are given non-constructively, on the other
infinitesimals are regarded as non-standard elements.  [That is, they
_are_ non-standard elements, but I wonder whether this is just an
accident of a particular axiomatisation in a particular foundation of
mathematics.]  The naive intuition I had as a teenager about the reals
actually included infinitesimals - and, no, I didn't know a bit about
ultrafilters at the time.
[BTW there are other notions of infinitesimals that do not rely on
ultrafilters, e.g. if you do analysis in topos theory rather
than set theory. But then, I didn't know a thing about topos theory
at the time either.]

So I'm arguing here that you can see non-standard elements without
having a grasp of constructing these in set theory.  The point about
"seeing" a model is that it is a primitive concept --- you don't have
to encode it in set theory to see it.

-- 
Stefan Kahrs
