Re: Set existence

On 12 feb, 20:00, tc...@xxxxxxxxxxxxx wrote:
In article <fose39$hq...@xxxxxxxxxxxxxxxx>,  <malc...@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.

What does it mean that ZFC has an uncountable model? Doesn't it mean
that from ZFC axioms we can formally prove the existence of a set (or
class) M and a relation (or set-theoretic relation) R such that every
axiom of ZFC is true under the interpretation?

.

Relevant Pages

  • Re: Set existence
    ... proved from ZFC axioms alone? ... If you're asking whether ZFC can prove the existence of an uncountable model ...
    (sci.math.research)
  • Re: axioms of mathematical logic
    ... ZFC is in fact inconsistent as to what constitutes a set! ... we cannot in ZFC claim the existence of a set ... has any such statement as "You get sets only from the ZFC axioms". ...
    (sci.logic)
  • Re: Extrapolating linear ratios
    ... I very much like the philosophical implications of this fact. ... So even in our world sets (as "described" by ZFC) might very ... the meaning of "existence" here is debatable. ... we can "emulate" non-Eucleadean geometry within ...
    (sci.logic)
  • Re: what makes it true?
    ... we can also assert the existence of such an entity in ZFC ... > negation is consistent with the rest of PA. ...
    (sci.math)
  • Re: what makes it true?
    ... >> entity in ZFC without introducing any new inconsistency. ... >> consistent statement asserting the existence of a natural number ... opaquely invoke the Axiom of Infinity to assert the existence of N, ... PA allow 'natural numbers' that are *not* modelled by finite ordinals. ...
    (sci.math)