Re: The Difference between a Set and an Element

On 16 Jan 2007 05:13:28 -0800, Paul Holbach said:
Chris Menzel wrote:
Paul Holbach wrote:

Gödel: "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.

As far as arbitrary infinite collections are concerned, we would
indeed have to posit infinite disjunctive concepts. But that's a
matter of theoretical preference, isn't it?

Did I imply anything else?

Gödel's notion of concept is very robust:

"We don' t make concepts, they are there."
"Concepts have an objective existence."

[Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: The MIT Press. (pp. 273+316)]

So why not let those infinite disjunctive concepts simply be there?
Isn't the unrestricted principle <for every set there is a concept
whose extension it is> worth it?

I can see a philosophical role for concepts of some sort, but I can't
imagine what it buys you to have a disjunctive concept correspondingly
uniquely to every set beyond a lot of metaphysical bloat. But then
again, I'm not very imaginative.

.

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