[FOM] Question for Nik Weaver
nweaver at math.wustl.edu
Thu Feb 23 05:23:07 EST 2006
Bill Taylor asked (quoting me):
> Nik, you made this comment in an article yesterday...
> > it is possible to treat things like the real line and
> > the power set of N essentially as proper classes
> Presumably this apposition implies that you do NOT think
> of P(N) and R as being essentially isomorphic?
I do think there's a straightforward bijection between them.
Sorry for misleading you.
> Incidentally, I think this identification, accepted by many
> (most?) mathematicians, and especially by almost all "pure
> mathematicians", is a clear-cut example of what Friedman
> called "coding" just the other day. He commented as that
> mathematicians loathe coding whenever they see it, and try
> to avoid it at all costs. This seems to be a counterexample.
Steve Simpson made the point somewhere that there is plenty of
coding in classical mathematics. (I don't have the reference at
hand.) I think what he meant by this was that, for example, any
classical "construction" of the real line (as Cauchy sequences of
rationals, Dedekind cuts, decimal expansions) is really a "coding".
Probably most mathematicians, to the extent they are interested
in foundations at all, would prefer a foundational scheme that
involves as little coding as possible. I don't see that as a
really fundamental issue though.
Friedman's particular beef was that in my J_2 model, some coding
is involved in forming the dual of a separable Banach space. That
could be a legitimate esthetic criticism. I don't insist that
J_2 is the "best" predicative setting in which to develop core
mathematics, though I do think it is rather nice.
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