[FOM] About Paradox Theory
Aatu Koskensilta
Aatu.Koskensilta at uta.fi
Tue Sep 20 04:18:31 EDT 2011
Quoting Daniel Mehkeri <dmehkeri at gmail.com>:
> I don't think it is what you want anyway. The classical foundation
> axiom is not constructively valid. Yes it is true we can make it
> valid by contraposition: no inhabited set intersects all of its
> members. However (and this is the sort of thing Nik Weaver keeps
> reminding us about) this is weaker than set induction over all
> first-order formulas.
Arguably even classically foundation isn't the right formal
expression of the idea that sets are obtained from the empty set (or
urelements) by repeatedly applying the "set-of"-operation:
second-order Zermelo set theory with foundation, for instance, has
ill-founded models. (This is because without replacement we can't
prove every set has a transitive closure.)
--
Aatu Koskensilta (aatu.koskensilta at uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
More information about the FOM
mailing list