[FOM] ZC vs. ZFC: an algebraic example
A.R.D.Mathias
ardm at univ-reunion.fr
Sun Jan 31 04:16:11 EST 2010
On Saturday 30 January 2010 05:12:39 Jeremy Bem wrote:
> (I studied logic at Berkeley, where I was required to pass exams on
> analysis and algebra -- but I don't believe that any of that material
> required replacement.)
The following is a slight variant of Example 9.32 in my paper "The strength of
Mac Lane set theory" (Annals of Pure and Applied Logic 110 (2001) pp
107--234):
========
let Q_0 = Q[t] be the space of all polynomials in the variable t with rational
coefficients.
In Z you can prove that for each positive integer k there is a sequence
(Q_n | n <= k) such that for each n < k, Q_{n+1} is the dual of Q_n;
further, for each n, there is a natural injection of Q_n into its bidual
Q_{n+2}), and in ZC you can show that there is a natural projection of
Q_{n+3} onto Q_{n+1}.
======== the following statement cannot be proved in ZC:
the infinite sequence (Q_n | n a natural number) exists
(as the existence of such a sequence implies in MAC the consistency of Z)
the injective (or direct) limit, X, of the sequence (Q_n | n an even natural
number ) exists
the projective (or inverse) limit , Y, of the sequence (Q_n | n an odd natural
number ) exists
Hence you need some replacement to discuss the following algebraic assertion:
Y is isomorphic to the dual of X.
Adrian Mathias
(the FOM server is blocking my posts for some reason which I have yet to
disentangle)
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