🚀 go-pugleaf

In article <1992Nov11.175842.123427@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:
>In article <1992Nov10.001234.18488@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:
>>In article <1992Nov09.172532.43648@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:
>>>All right; but what would you put in it's place?  I'm not willing to call
>>>something "set theory" unless it has *some* recognizable form of the axiom
>>>of comprehension.  Can you formalize your ideas?
>>
>>ii.  NFU:
>
>Yes, Randall, I know that's what *you* would put in its place.  If you can
>show that NF[U]'s comprehension rule is free from the kind of objection David
>is making, I will grant that you have given a good reason for taking NFU
>seriously.  It isn't enough to show that a few examples work, however.

I agree with your remarks absolutely.  The case for NFU does not rest
on a few examples.  There is an equiconsistency proof with set
theories of the usual type.  Rosser developed the foundations of
mathematics in _Logic for Mathematicians_ in a way which can be
adapted to NFU (he used NF itself).  Finally, there is a philosophical
argument of sorts (due to myself (unfortunately for my attempt to
cloak it with Authority!)) indicating why unstratified formulae should
not be understood to define legitimate properties of sets.

NFU is impredicative, which may cause Gudeman to object to it.  If
impredicativity is rejected, mathematics is weakened considerably.
The predicative fragment NFI of NF is consistent (with full
extensionality!) but weaker than Peano arithmetic.  (When I say NFU, I
mean NFU + at least the Axiom of Infinity; NFU by itself is also
weaker than arithmetic, as is the theory of types without infinity).
But it allows a much wider range of definitions of sets than the usual
set theory (for example, most natural categories are sets in NFU).


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu