[FOM] Fregean set theory and replacement

John McCarthy jmc at cs.Stanford.EDU
Tue Aug 14 20:07:20 EDT 2007

Consider set theory with unrestricted comprehension.  I'll call it F, 
because I don't know a standard name.  Is there one?  Regrettably, F
is inconsistent because of Russell's paradox.

A replacement schema (as well as most axioms) is unneeded, because the
set asserted to exist by replacement is just a comprehension term,
namely {y|x in A implies R(x,y)}.

F is a very natural system.  However, any proof using unrestricted
comprehension is suspect.  However again, a proof in F may be
repairable, e.g. transformed into a proof in ZFC.  This seems likely
if the proof doesn't go near a paradox.

Is there any literature discussing the repair of proofs in F?

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