Date: Jan 23, 2013 12:44 AM
Author: ross.finlayson@gmail.com
Subject: Re: ZFC and God
On Jan 22, 1:39 pm, Virgil <vir...@ligriv.com> wrote:

> In article

> Perhaps it is merely a quirk of German-Engish differences but in English

> mathematics one cannot have "for every" without having "for all".

> --

A delineation of the universal quantifier's statements "for each / for

any / for every / for all" may well be used to correctly formulate

statements where the transfer principle applies, that for example a

set of sets is a set.

for each x s.t. P(x): P(x)

for all x s.t. P(X): P(x) and P( {x s.t. P(x) } )

Correspondingly, anti-transfer:

for any x s.t. P(x): P(x) and not P( x: P(x) )

Then there's a consideration as to properties of the objects that only

evince themselves as properties when the objects are considered

together or apart.

For example, for each finite integer it is of finitely many for all

they are not finitely many. Indeed, a careful appropriation of the

natural language phrases describing universal quantification may well

simplify notation for a wide variety of statements.

This is where, in systems, there's a general consideration that the

universal quantifier is as to all of them: sometimes necessarily at

least together.

Then, the general form might have "for any" with the usual

expectation, that transfer is undecided by the statement, then working

up for each / for every / for all in as to then simply and

mechanically carrying the symbological import, for notational brevity,

and clarity: of "the" universal quantifier.

Regards,

Ross Finlayson