Re: set builder

From: Dan Christensen (dchris_at_allstream.net)
Date: 07/03/04 

Date: Sat, 3 Jul 2004 11:35:52 -0400

"john Smith" <use.dd@free.de> wrote in message
news:40e690fb$0$31978$636a15ce@news.free.fr...
> Hi,
>
>
> Question :
>
> When we define a set by using the set builder notation, do we suppose
> that the set is a subset of another ?
>
> or can we write
>
> Let S1:={x/ P(x)} instead of S1:={x\in S/ P(x)}
>
> where P is a predicate that a set x can or cannot verify
>
> thanks
>

The experts here can correct me if I am wrong, but as I understand it, the
most widely accepted set theory (ZFC) starts by postulating only the
existence of an empty set phi. Using various axioms, it is allows you to
construct a set that satisfies Peano's Axioms for the natural numbers, for
example.

0=phi, 1={0}, 2={0,1}, 3={0,1,2} and so on

The only sets that can be shown to exist in ZFC are those that can be
constructed, starting with phi.

In my DC Proof system, I don't start by postulating the existence of any
sets, not even the empty set. And you cannot actually prove the existence of
any set. But this turns out not to be a serious limitation. Quite the
contrary. If, for example, you want to do number theory in my system, you
assume the existence of a set n with the properties defined in Peano's
Axioms as an initial premise. All other sets are are then constructed from
the set n. It seems easier to master than ZFC and avoids the known
contradictions of naive set theory (such as David points out here).

Dan

Download DC Proof 1.0 at http://www.dcproof.com

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