Re: Set existence

In article <fose39$hqn$1@xxxxxxxxxxxxxxxx>, <malcobe@xxxxxxxxx> wrote:
Wrong question. It would lead to a contradiction with the Löwenhem-
Skolem theorem. Ok. But can the existence of an uncountable model be
proved from ZFC axioms alone?

If you're asking whether ZFC can prove the existence of an uncountable model
of ZFC, then the first observation is that ZFC can't prove the existence of
any models at all (unless ZFC is inconsistent), by Goedel's 2nd theorem.

However, ZFC (in fact a much weaker system) can prove that if ZFC is
consistent, then it has an uncountable model. This is usually referred
to as the "upward Loewenheim-Skolem theorem," although I think it's
actually due to Tarski.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

.

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