[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 


let Q_0 = Q[t] be the space of all polynomials in the variable t with rational 

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 

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