# [FOM] Formalization Thesis

Larry Stout lstout at iwu.edu
Wed Jan 9 11:50:21 EST 2008

Here's a candidate:

Every category has a skeleton; that is, given any category C there is
a category Skel(C) in which the only isomorphisms are the identities
and a functor F:C\to Skel(C) which preserves and reflects isomorphism.

If you start with the category of finite sets this would give the
natural numbers with a category structure somewhat richer than the
order.  The obvious construction of Skel(C) in ZFC would be to make
objects equivalence classes of objects in C under isomorphism.  Doing
that in the case of C=FiniteSets would make each of the objects a
Frege number.  But Roy Cook showed at the PHILMATH conference at
Notre Dame last October that the existence of Frege numbers (in
particular 1 and 2) as sets is inconsistent with ZFC.

Larry Stout

On Jan 3, 2008, at 6:42 PM, joeshipman at aol.com wrote:

> I repeat my earlier challenge: can anyone who disputes Chow's
> Formalization Thesis respond with a SPECIFIC MATHEMATICAL STATEMENT
> which they are willing to claim is not, despite its expressiblity in
> English text on the FOM discussion forum, "faithfully
> representable" or
> "adequately expressible" as a sentence in the formal system ZFC?
>
> I don't want an argument that such a statement exists, and I don't
> want
> a METAmathematical statement, I want an actual English sentence within
> quotation marks that is claimed to be mathematical but not
> formalizable
> in Chow's sense.
>
> -- JS
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