On Wed, 2010-02-03 at 13:36 -0500, Patrick Cassidy wrote:
> Rob,
> [RF] > > Examples of incompatible theories: the axiomatic set theories of
> > > mathematics.
> >
> The math theories you mention may or may not be all describable in terms
> of the same primitives - but I can't visualize the incompatibilities you are
> referring to (and decide whether they are not expressible by common
> primitives) unless I can see the logical axioms that demonstrate a logical
> contradiction. My first attempt to find them came up null. (01)
It is very easy to find examples: ZF set theory is incompatible with
AFA, the theory that results from removing the axiom of foundation from
ZF and replacing it with an anti-foundation axiom that asserts the
existence of, e.g., self-membered sets. Again, ZF + the axiom of choice
is inconsistent with ZF + the axiom of determinacy. (02)
There is really only one axiom common to all set theories, viz.,
extensionality: sets with the same members are identical and perhaps
that is the only primitive principle you need. That said, virtually all
set theories in broad use seem to be based in a common set of intuitions
and share a pretty solid axiomatic core -- Kripke-Platek set theory
would probably be a good shot at that core: KP is basically ZF without
the axioms of infinity and power set and with weaker versions of the
axiom schemas of separation and replacement. (See Barwise, _Admissible
Sets and Structures_ to see KP in action (more exactly KPU, KP + "there
are things other than sets")). (03)
There are much sharper incompatibilities between ZF-style theories and
theories based on Quine's NF (notably, in these theories, there is a set
of all sets) but NF-style theories are not in wide use and are mostly of
theoretical interest (see, e.g., Thomas Forster's excellent book _Set
Theory with a Univerals Set). (04)
Chris Menzel (05)
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