Re: Is Zermelo-Fraenkel set theory inconsistent?

[Mike Oliver <oliver@math.ucla.edu>]

Paul Taylor wrote:
>         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.

How do you conclude, from the fact that L(infinity) |- "L(n) is consistent", that
L(n) is in fact consistent?

Generally, if T1 |- "T2 is consistent", then to conclude "T2 is consistent",
we use the following argument:  Suppose T2 is inconsistent.  Then
there is some proof by which T2 |- false.  Assuming T1 is strong enough
to formalize the deductive system being used, then it follows
that T1 |- "T2 is inconsistent".   But by hypothesis, T1 |- "T2 is consistent",
therefore T1 is inconsistent.

But this is not a contradiction unless we were already *assuming* the
consistency of T1 !  I.e. it follows from T1+Con(T1) that
if T1 |- Con(T2), then Con(T2), but it does *not* in general
follow from T1 alone.

So the step from
	L(infinity) |- "L(n) is consistent"
to
	L(n) is consistent

cannot be formalized in any obvious way in L(infinity), and therefore
you cannot (again in any obvious way) conclude
L(infinity) |- "L(infinity) is consistent."

-- 
Disclaimer:  I could be wrong -- but I'm not.  (Eagles, "Victim of Love")

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