```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 / forany / for every / for all" may well be used to correctly formulatestatements where the transfer principle applies, that for example aset 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 onlyevince themselves as properties when the objects are consideredtogether or apart.For example, for each finite integer it is of finitely many for allthey are not finitely many.  Indeed, a careful appropriation of thenatural language phrases describing universal quantification may wellsimplify notation for a wide variety of statements.This is where, in systems, there's a general consideration that theuniversal quantifier is as to all of them:  sometimes necessarily atleast together.Then, the general form might have "for any" with the usualexpectation, that transfer is undecided by the statement, then workingup for each / for every / for all in as to then simply andmechanically carrying the symbological import, for notational brevity,and clarity:  of "the" universal quantifier.Regards,Ross Finlayson
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