Re: The Difference between a Set and an Element

On Wed, 17 Jan 2007 22:28:28 +0100, G Frege <nomail@invalid> said:
...
I don't think so. We are talking about 2OL here. 2OL is not set
theory.

That's a matter of some dispute, but in any case you need to specify a
semantics for a second-order language before the discussion can even go
anywhere. I was just talking relative to a standard, extensional
second-order semantics.

But suppose we assume a countable language and that every
formula with at least one free variable signifies a concept.

??? Comprehension just guaranties that for every such formula A[x]
there _is_ an concept F such that for all x: Fx <-> A[x].

If that's your semantics. You have to tell me what a concept is on that
semantics, of course.

On those assumptions, there are at most countably many concepts

I don't think so. Comprehension just guaranties that there are _at
least_ countable many concepts.

Yes, of course. The point there was simply that the most that the
assumptions in question could guarantee was a much smaller universe of
concepts than the one envisioned in which every set is the extension of
a concept.

.

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