Date: Jan 23, 2013 12:44 AM
Subject: Re: ZFC and God

On Jan 22, 1:39 pm, Virgil <> 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.


Ross Finlayson