Re: The Difference between a Set and an Element

On 15 Jan 2007 11:15:58 -0800, Paul Holbach
<paulholbachDELETETHENAME@xxxxxxxxxx> said:
Chris Menzel schrieb:

Granted, there might be those who would
say that {5,17,1383} is the concept of being identical to 5, 17 or 1383
(though I think that would be an odd thing to say). So consider
instead, for example, an arbitrary infinite set S of natural numbers for
which -- unlike, say, {5,17,1383} or the set of prime numbers -- there
is no description that characterizes exactly the members of S, and no
procedure that lists them. What concept is S? Sure doesn't seem like
anything I'd call a concept.

I agree with you insofar as, in the Fregean sense, sets are objects
and not concepts. As we know, there isn't a set for every concept.
But isn't it the case that there is a (defining) concept for every
set, as Gödel conjectured:

"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.

.

Relevant Pages

  • Re: The Difference between a Set and an Element
    ... "A plausible conjecture is: Every set is the extension of a concept." ... 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. ...
    (sci.logic)
  • Re: The Difference between a Set and an Element
    ... (though I think that would be an odd thing to say). ... "A plausible conjecture is: Every set is the extension of a concept." ...
    (sci.logic)