On Jul 31, 2007, at 5:09 PM, Kathryn Blackmond Laskey wrote:
> [Pat wrote:]
>> Set theory is ..., usually in the form of
>> Zermelo-Fraenkel (Z-F) set theory
> or the equally powerful, somewhat less intuitive, but finitely
> axiomatizable von Neuman / Godel / Bernays (NGB) set theory
>> ... still the widely
>> accepted mathematical foundational language that
>> is as near to consistent as anything can be.
> It either is or is not consistent. You can't be "near to consistent"
> any more than you can be "almost pregnant." (01)
Pat of course knows that (as I am sure you are yourself aware).
Surely all he meant was that, in over 100 years of extensive
theoretical examination and extremely heavy use, no set theoretic
paradoxes have arisen in ZF (likewise, it follows, for the equi-
consistent NGB). While that of course is not a formal proof, this
gives logicians great confidence that the theory is consistent.
(There is also a very intuitive informal model of ZF -- the so-called
cumulative hierarchy -- whose role is analogous to the natural number
structure for Peano Arithmetic. We can't prove PA consistent (in
PA), but it is quite intuitively clear that it *is* consistent given
our belief in the existence of the natural number structure.
Similarly for the cumulative hierarchy and the consistency of ZF.) (02)
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