[FOM] Why not NF?

Thomas Forster T.Forster at dpmms.cam.ac.uk
Mon Feb 20 05:19:54 EST 2006

On Sun, 19 Feb 2006, Martin Davis wrote:

> NF suffers from at least two grave faults:
> 1. It's inconsistent with AC

  Why is this a fault?

> 2. Cantor's theorem 2^x > x fails.

  Well, this is of course a consequence of there being a universal set!
Cantors theorem fails in NF beco's NF aspires to discuss collections that
ZF doesn't aspire to discuss.  There is no evidence that it fails for 
small (eg wellfounded) sets. (Actually the same goes for (1): the point 
about AC)

If there is a real problem with NF it could be argued that it is there is
no intuitive picture of a set-theoretic universe for which NF is the
formal version.  Granted, there are non-trivial mathematical ideas
underlying NF but they require more sophistication in the audience than
the cumulative hierarchy does, and i can quite understand why the
Mathematician On The Clapham Omnibus prefers to direct his/her attention

  URL: www.dpmms.cam.ac.uk/~tf   Tel: +44-1223-337981
  (U Cambridge); +44-20-7882-3659 (QMW) +44-7887-701-562


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