[cxm7@po.CWRU.Edu: Re: FOM: bad set membership]
Colin Mclarty
cxm7 at po.cwru.edu
Sat Jan 24 16:28:48 EST 1998
Reply to message from friedman at math.ohio-state.edu of Fri, 23 Jan
>
>This is a response to McLarty 11:15AM 1/23/98. By the way, we still don't
>have a concession from you that you need much more than a topos with
>natural number object to naturally do undergraduate real analysis. How
>about it?
You must have missed a lot of my posts, but here is an idea:
Read my book. It has been around for 6 years saying just what I said
here, firmly fixed on paper. Below I append a summary of the relevant
passages with page numbers.
I just got my author's copies from the third printing and I'd
be happy to send you one Harvey. (I can't do this for everyone.) Give me
your surface address if you like. It might make the discussion more
constructive.
>Also, we don't have a concession from you that there is no
>coherent conception of the mathematical universe that underlies categorical
>foundations in your sense. You should make it more clear that you are very
>proud of the fact that there is no such coherent conception of the
>mathematical universe on which you base your approach so that the rest of
>us can stop thinking that you are even trying to do foundations of
>mathematics.
You have my repeated statements that categorical foundations
spring from many disparate examples--you quote one a few lines further
on in your post (I do not requote it here). You are right that I am
proud of this, though it was hardly my doing. I'll give you this,
call it a concession if you like:
I find many things relevant to foundations that you find
irrelevant.
>You would have greater credibility if you would say, e.g., that you are
>doing o.o.m. instead of f.o.m. O.o.m. is organization of mathematics. Even
>here, you have a lot of convincing to do to make it stick that it is all
>that illuminating. Simply wrapping yourself up in Grothendieck is
>insufficient, according to my mainstream mathematical sources, who are
>perfectly satisfied with the usual set theoretic foundations of
>mathematics.
Grothendieck uses perfectly usual set theoretic foundations
himself. (I assume you consider inaccessible ordinals usual.) Is that
supposed to prove his categorical ideas are not widely used to
organize math?
You have specific objections to category theory, and specific
complaints about the motives of categorists, wildly disproportionate
to your knowledge of either the theory or the theorists. You have a
right to ignore whatever you want, but then what is the point of
arguing about it?
>How cute. Kids at 3 grasp three cards on a table as a unit, and later the
>idea that any of those cards could even be a unit. You can generate just
>about everything in set theory from that. Your kids at 3 grasped - well I
>don't know what.
Indeed, you do not know. I have said several times that I
believe they grasp the very same thing. At this level Zermelo style
set theory and categorical do not differ. I even referred to an article
I published on this very point. ("Numbers can be just what they have to"
NOUS 27:4, 1993, pp.487-498.) You may consider the article a concession
if you like. You do not know what I think about this, because you
do not read my posts.
Summary of the statements on real analysis
in toposes, from my book ELEMENTARY CATEGORIES,
ELEMENTARY TOPOSES (Oxford, 1992).
pages 128-132 describe differences between the logic valid in any
topos, and the classical logic which is valid in certain ones.
pages 177-78 discuss arithmetized real analysis in any topos with a
natural number object. The usual definitions are used, but in the weaker
topos logic they do not give all the classical results. For a start,
the Cauchy and Dedekind reals do not agree.
pages 211-218 describe the topos of sets (that is, categorical set
theory). Here, because the logic is classical, arithmetized analysis
gives all the results in classical analysis that do not use the
axiom of choice. If you add the axiom of choice to the topos axioms,
you get all of classical analysis.
page 235 briefly mentions recursive real analysis, using the
standard definitions for arithmetization but in a topos of recursive
sets.
There are references for more information, and I specify that little
is published on categorical set theory per se because it "is not
very different from other set theories, being just a little bit
closer to naive practice" (page 255).
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