[FOM] Naturality of Con(ZFC)

[MartDowd at aol.com]

This posting is in response to Vaughn Pratt's comment:
 
> Meanwhile I will continue to ponder the possibility you raise that  
> consistency of ZFC is unnatural.
 
Although I haven't read Harvey's manuscripts, he says in #513
 
:It was shown in #75 that the Divine system corresponds to something  a
little weaker than a measurable cardinal
 
Con(ZFC) is much weaker.  Indeed it is implied by "there exists a  strongly 
inaccessible cardinal".  There are now strong arguments that these  exist, 
due to the fact that they comprise the first step in a "natural", indeed  
"inevitable", process of extending the cumulative hierarchy.  I believe it  is 
an important question of modern set theory, whether weakly compact 
cardinals  are natural in this sense.  Again, these are much weaker than measurable  
cardinals.
 
- Martin Dowd
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