Re: The Difference between a Set and an Element

On Tue, 16 Jan 2007 00:55:38 +0000 (UTC), Chris Menzel
<cmenzel@xxxxxxxxxxxxxxxxxxxx> wrote:


"A plausible conjecture is: Every set is the extension of a concept."

It would take a *very* robust and fine-grained notion of "concept" for
that to be so, e.g., one on which there are infinite disjunctive
concepts whose disjuncts are of the form "being identical with A" for
arbitary objects A. I don't see any other way of justifying the claim
that, e.g., every arbitrary subset of N is the extension of a concept.

You may be right.

Btw, imho it should be possible to prove that claim in some system of
2OL (comprising a theory of sets).

Theorem:

Ax(set x -> EF(x = {y : Fy})).
"Every set is the extension of a concept."

Just a sketch of a proof:

Let /a/ be a set: set a.

Then we have (easy to show):

a = {y : y e a}. (*)

Now in a 2OL we have the principle of abstraction:

EFAx(Fx <-> ...x...),

where "...x..." is a formula in x.

Hence we have especially:

EFAx(Fx <-> x e a).

Let G be such an F, then we have:

Ax(Gx <-> x e a).

Hence [...]:

{y : Gy} = {y : y e a}.

Hence from (*) we have:

a = {y : Gy}.

Existential introduction gives:

EF(a = {y : Fy}).

Conditional introduction gives:

set a -> EF(a = {y : Fy}).

Finally universal introduction gives:

Ax(set x -> EF(x = {y : Fy})).

qed.


F.

--

E-mail: info<at>simple-line<dot>de
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Relevant Pages

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