Re: Another Inconvenient Truth

On Aug 20, 6:43 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Jesse F. Hughes wrote:
Try to recall: You claimed that in your theory there are no infinite
sets *and* that the proof of this fact is more or less Russell's
paradox.

Yes.

Were you drunk?

This discussion reminds me of something from metamath.org:

http://us.metamath.org/mpegif/omon.html

"The class of natural numbers (omega) is either an ordinal
number (if we accept the Axiom of Infinity) or the proper
class of all ordinals (if we deny the Axiom of Infinity)."

So in ZF-Infinity+~Infinity (or more precisely we want
NBG-Infinity+~Infinity since we need proper classes) omega
is not the Russell class of all sets that do not contain
themselves, but rather the Burali-Forti class of all
possible ordinals.

Of course, as Jesse Hughes points out, this doesn't by
itself prove ~Infinity, any more than the fact that an
inaccessible cardinal can be a model for all sets in ZFC
doesn't preclude the existence of an inaccessible.

.

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