Re: Cantor's definition of set

On Sun, 28 Oct 2007 12:05:45 +0100, G. Frege <nomail@invalid> wrote:


So I hope you don't mind that I still maintain that "these days we are
talking about "sets" instead of "extensions of concepts", but both
concepts (no pun intended) mean (essentially) the same thing."

Note that this "point of view" is rather common... Otherwise you would
have to question claims like the following:

"If we remember that the extension of a concept is something like the
set of objects that fall under the concept, then we could replace
Frege's talk of 'extensions' by talk of 'sets' and use the following
'set notation' to refer to the set of objects that when added to 4 yield
5 and the set of objects that when added to 22 yield 5, respectively:

{x | x + 4 = 5}

{x | x + 22 = 5}

In what follows, we sometimes render Frege's notation in this more
modern notation." [1]

or

"Frege next explained how Basic Law V implies the Naive Comprehension
Axiom for extensions or sets, which Russell's Paradox shows to be
inconsistent."

(Ed Zalta, Frege's Logic, Theorem, and Foundations for Arithmetic
http://plato.stanford.edu/entries/frege-logic/)


F.

____________________

[1] As already mentioned, NF seems to be a first order variant of this
approach - at least if the considered concept can be expressed by a
stratified formula! There ARE sets like {x | x = x} etc. But, of course
the formula expressing the concept /not an element of itself/ is not
stratified; hence there is no set {x : x !e x} in NF/NFU.


Comment: Imho, NF is as close to a "naive set theory" (or rather a set
theory with "naive comprehension" in the spirit of Frege's original
approach) as it's possible for a first-order set theory.

Of course we MUST exclude predicates like "x !e x" from comprehension;
for there simply CAN'T be a Russell _set_. (At least if we want to
maintain /consistency/.)

--

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

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