# [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.