Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Topic: The quantifier "Most"
Replies: 32   Last Post: Aug 11, 2012 5:57 PM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,029 Registered: 5/20/10
Re: The quantifier "Most"
Posted: Aug 3, 2012 4:17 PM

On Aug 4, 4:56 am, Zuhair <zaljo...@gmail.com> wrote:
> The quantifier "most" lets denote it as M. how is it defined?
>
> It is easy to do that per a set, for example: "most elements of X
> satisfy phi" can be formulated as:
>
> Define: (Mx. x e X -> phi(x)) <->
> (Exist K,R,S: K subset of X & R subset of X & K disjoint R & ((Exist
> r. r e R & K equinumerous to R\{r}) or R equinumerous to K) & X=K U R
> & S={x| x e X & ~phi(x)} & S strictly subnumerous to R)
>
> However how can we say matters relative to the universe of discourse?
> I think this can be done for set theories with a universal class like
> VGB\MK and NF; but how can it be done in those that lack such sets
> like ZF for example? i.e. how can we express the idea "Most x. phi(x)"
> in ZF? is it by stipulating that S exists, i.e. Mx.phi(x) <-> Exist S.
> for all s. s e S <-> ~phi(s). What if ~phi range over a proper class
> that is strictly subnumerous to the universe of discourse of ZF? how
> can we express Mx.phi(x) in that situation?
>
> Zuhair

You can define many sets impervious to contradiction

e.g. a Russell's Set of everything that doesn't belong to itself (bar
itself)

ALL-BUT-1(x) x e RUSSELL <-> x ~e x

e.g. a consistent theory bar Godel's Statement

EXIST(THEORY) ALL-BUT-1(t): THEORY|- TRUE(t)

Herc