[FOM] Might there be no inaccessible cardinals?

[Vaughan Pratt]

I ran across an interesting paper by Jesus Mosterin titled "How set 
theory impinges on logic" at
http://philsci-archive.pitt.edu/archive/00001620/01/How_Set_Theory_Impinges_on_Logic.pdf
The following sentence caught my eye.

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

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