Talk:Impredicativity
From Wikipedia, the free encyclopedia
I feel that the existing (stub) article is inadequate, even as a starting-point. In particular:
(i) The opening sentence is misleading.
(ii) The second sentence, apparently equating "impredicative" with "self-referential", is wrong. The two terms have distinct meanings, the latter being more general. There are self-referential definitions (including simple kinds of recursion) that are not at all problematical in the way impredicative definitions are.
(iii) However, the author is correct in citing Russel's Paradox as an example of impredicativity.
(iv) Ramsey of course will have had a point, but it's impossible to tell here what it is ("absolutely necessary" to what?).
(v) Though in keeping with the already misleading opening sentence, and indeed for that very reason, I cannot accept that "tallest person" sheds any light on what the term "impredicative" means (nor that it falls within any useful definition thereof, nor even of "self-reference" as usually understood).
(vi) Indeed, it appears that Ramsey cited "tallest person" as an example of a definition which, though perhaps superficially reminiscent of it, does NOT involve the kind of problematical circularity characteristic of impredicativity:
"... just as we may refer to a man as the tallest in the group, thus identifying him by means of a totality of which he is himself a member without there being any vicious circle..." [Ramsey, Foundations, page 41]
The Ramsey quote appears in this:
Ramsey and the notion of arbitrary function by Gabriel Sandu http://www.helsinki.fi/filosofia/gts/ramsay.pdf
For useful reading:
Predicativity, by Solomon Feferman in The Oxford Handbook of Philosophy of Mathematics and Logic Oxford University Press, Oxford (2005) 590-624. http://math.stanford.edu/~feferman/papers/predicativity.pdf
Peter010101 19:51, 21 August 2006 (UTC) Peter Burton (wiki@waking.fslife.co.uk).
