[FOM] Adequacy of FOL+ZFC as foundation

[T.Forster@dpmms.cam.ac.uk]

On May 14 2010, joeshipman at aol.com wrote:

>On an earlier thread, I mentioned two ways that first-order logic with 
>the ZFC axioms is problematic as a foundation:
>

>2) NF is not consistent with ZFC so far as we know.

>No one responded to my request for examples of mathematics done in NF 
>as a base theory that were not obviously reinterpretable as arguments 
>from ZFC.


My apologies: we NF-istes have been asleep on the watch. (Marking!) There 
are small sets that can be defined in NF using the suspect large sets of NF 
that do not have any obvious equivalent definitions in ZF(C). Although it 
is certainly true that the refutations of AC in NF involve only the Mrs 
Rochester sets (mad things squirrelled away in the attic `attic' in Andrey 
Bovykin's elegant expression for those sets in NF which are not sets in the 
ZF dispensation) there are sets of reals that can be defined using the Mrs 
Rochester sets as parameters. I can supply details on request. It's 
recondite, agreed, but it may yet turn out to matter. It certainly merits 
investigation, and it isn't getting enough.