Re: Is Zermelo-Fraenkel set theory inconsistent?

[Dusko Pavlovic <dusko@kestrel.edu>]

>      Let L(0) be Zermelo set theory (or the axioms for an elementary topos).
>
>         For each n,  let L(n+1)  be   L(n)  plus
>         as much of the axiom-scheme of replacement as is needed
>         to justify the gluing construction that shows that
>
>                 L(n+1) |-  ``L(n) is consistent.''
>
>         Now let   L(infinity)   be   the union of  L(n) over n:N.
>
>         If    L(infinity) |- false   then   L(n) |- false   for some n.
>
>         But   L(infinity) |-  ``L(n) is consistent,''
>
>         so    L(infinity)   proves its OWN consistency,
>         contradicting Godel's theorem.
>
>         However,  L(infinity)   has a standard non-trivial interpretation
>         in Zermelo--Fraenkel set theory, which is therefore inconsistent.

i think there is a gap is in the step

        L(infinity) |- "L(n) is consistent"
        so L(infinity) proves its OWN consistency

formalized in a suitable category of theories and interpretations, paul's
construction, if i understand it correctly, refers to the colimit of the tower

        L(0) --> FL(0)  --> FFL(0)  -->...

where FX = X + replacement_X, so that FX |- "X is consistent".

IF the colimit of this tower, paul's L(infinity), were a fixpoint of F, THEN it
would indeed prove its own consistency. but it doesn't seem to be a fixpoint:
the restrictions of the replacement to L(0), FL(0) etc. do not imply the
replacement for L(infinity).

btw, i bought paul's book and warmly recommend it.

-- dusko