Re: Zermelo-Fraenkel Theory Of Sets


Jesse F. Hughes wrote:
> "smn" <smnewberger@xxxxxxxxxxx> writes:
>
> > There are 2 views of equality for set theory.The less well known but in
> > some sense more self contained in set theory is the one given by Abian
> > and also in the book -Introduction to Axiomatic set theoy by Takeuti
> > and Zaring (who have a sequal called Axiomatic set theory which is very
> > formal Mathematical Logic) .
> > In this view variables like x,y always sets and x=y is defined as
> > Abian says.Then Abian calls an axiom, a special case of the
> > characreristic property of equal sets ,namely that they have the same
> > properties so that one can interpret x=y as meaning that "x" and "y'
> > denote the same object. For fixed z the statement : xez states a
> > property of x and the axiom says that if x=y then y must have the same
> > property which is expressed by saying that if you replace x by y in
> > the statement "xez" to get "yez" then the truth of the first statement
> > implies the truth of the second.
> > Using this and recursive proceedures for building statements from the
> > logical connectives like and ,or if__then__ ,not,if and only if and
> > the quantifiers; one can prove the fact (A) that if x=y then
> > P(x)-->P(y) where P(x) abreviates a statement and is read x has the
> > property P . (see post 9 by Hughes)
>
> Great -- I thought that might be what's going on here, but it is good
> to see confirmation.
>
> It is, as you say, surely a minority approach. Again, as you say, (A)
> is almost uniformly considered a logical axiom (or as good as...).
> --
> Jesse F. Hughes
> "Conviction of fraud can mean jail time. It can mean social censor. It
> can mean big headlines where mathematicians take "perp walks" before a
> jeering public." -- JSH on the "censor" that awaits mathematicians.
Thanks for your remarks and telling me about the identification number
..Although a post by de Bruijn said it doesn't help with the netscape
browser I was able to find the number.On google if I press options at
the post and then click original message the id number is right there.I
am happy you found the post .I thought since I posted it a minute later
it would be right above the post with the additional
remark.Regards,Stuart

.

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