[FOM] Might there be no inaccessible cardinals?

Vaughan Pratt pratt at cs.stanford.edu
Fri Nov 2 02:03:05 EDT 2007

I ran across an interesting paper by Jesus Mosterin titled "How set 
theory impinges on logic" at
The following sentence caught my eye.

"In ZFC we can neither prove nor disprove the existence of inaccessible 

Can anyone (Mosterin perhaps?--I don't have his email address) enlighten 
me as to the meaning of "cannot" ("can neither") here?

In the case of "prove" there is no question: "cannot" means it is 
impossible, since there are models of ZFC too small to include an 
inaccessible cardinal.

At first I assumed that he meant the same for "disprove."  But I 
couldn't immediately come up with an equally convincing argument, nor 
was Google of much help.

Has this been shown?  Or did Mosterin merely mean that we can't 
*currently* disprove the existence of inaccessible cardinals?

Vaughan Pratt

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