Re: Largest Set in ZFC?

On Mar 9, 1:55 am, "Peter Webb" <webbfam...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:

Your [reaste's] argument suggests omega is a model of ZF.

No, that's wrong. In ZF we prove that there exists a unique least
inductive set, which we name 'w'. That doesn't entail that in ZF we
prove w is the universe of a model of ZF. Rather, in ZF, we prove that
IF ZF has a model then w is the universe of a model of ZF.

Yes but No. Omega is not X, because it doesn't include w+1, and hence
doesn't satify the requirement that "and whenever y is in X, so is S(y)".

You're TERRIBLY confused about this. To be successor inductive it is
not required that the set have w as a member so it is not required
that the set have w+ as a member. What is required is that IF w is a
member then w+ is a member; but it is not required that w is a member.

As
I indicated, nobody has ever come up with an actual set X that meets the
requirements of the definition.

What are you talking about?! We DO prove the existence of a unique
defined set that is successor inductive.

If they did, they would prove the
consistency of ZF by providing a "model" of ZF(C), viz the set X itself.

No, that is wrong, as I explained earlier in this post.

MoeBlee
.

Relevant Pages

  • Re: Simplifying M theories.
    ... x is countable iff (x equinumerous to Omega or x subnumerous to ... member, and there exist a universe which contain y as a member. ... from connectivity axiom either the universe which contain x is a ...
    (sci.logic)
  • RCU tops 200,000 members
    ... RC Universe has just passed the 200,000 member ... RC Universe, which was created in December of 2001, quickly grew as it ... through and passed the 200,000 member registration milestone. ... Everyone here at RC Universe would like to thank the members and moderators ...
    (rec.models.rc.air)
  • Re: Infinite Crisis 1 SPOILERS SPOILERS SPOILERS
    ... mere re-alignment of the DC universe. ... With the prospect of the ... member of the multi-verse spawned from the events of IC. ...
    (rec.arts.comics.dc.universe)
  • Re: Small Set theory:Revised.
    ... If one wants to use Ax.3 to show that ~xex, ... But *if* you add that axiom, ... This doesn't stop it from having a universe, ... There cannot exist x such that V is a member of it. ...
    (sci.math)
  • Re: Largest Set in ZFC?
    ... inductive set, which we name 'w'. ... IF ZF has a model then w is the universe of a model of ZF. ... that the set have w+ as a member. ... This is not impossible, but after a century of looking, nobody has found one, and it looks pretty unlikely such a set X will ever be found. ...
    (sci.logic)