[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

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