[FOM] About Paradox Theory
Daniel Mehkeri
dmehkeri at gmail.com
Sun Sep 18 13:14:32 EDT 2011
Harry Deutsch offers:
> AyEzAx(F(xz) <--> x=y). Therefore,
>
> -EwAx(F(xw) <--> Au([F(xu) --> Ey(F(yu) & -Ez{F(zu) & F(zy)])]).
about which Vaughan Pratt asks:
> Anyone care to comment on whether it's intuitionistically valid as it
> stands? (I presume one can hedge by inserting a few "surely"'s,
> formalized as not-not, to make it so, via Goedel's translation.)
> Nik Weaver?
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.
I'd say set induction is the intended meaning of well-foundedness.
Thomas Forster seems to agree: "The only context in which this
definition [well-foundedness] makes sense at all is [set] induction, and
the only way to understand the definition or to reconstruct it is to
remember that it is cooked up precisely to justify induction." (_Logic,
Induction and Sets_)
CZF just postulates set induction as a schema. I don't know if there is
a single first-order formula that is equivalent to it over some basic
assumptions. I don't know how the "set of all well-founded sets" paradox
would work.
Daniel Mehkeri
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