Re: set builder

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 07/03/04 

Date: Sat, 03 Jul 2004 14:52:42 -0500

On Sat, 3 Jul 2004 11:35:52 -0400, "Dan Christensen"
<dchris@allstream.net> wrote:

>
>"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.

I'm not an expert. You're wrong.

(Or you certainly seem to be wrong - we need a definition of
exactly what you mean by "constructing" a set to be certain.)

>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
>

************************

David C. Ullrich

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