Talk:Impredicativity

citations

There are citations missing from this article. — Preceding unsigned comment added by 64.229.238.112 (talk) 01:49, 7 March 2013 (UTC)

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).

The definition of "greatest lower bound" here is also suspect. Usually glb = {⟨X,y⟩ ∈ P(A) × A | for all x ∈ X, y ≤ x}. That is, glb is a relation defined on subsets of some (quasi)ordered set A, not on A itself (it is a partial function if A is a partial order and a total function if A is a meet-semilattice).

I recently attended a workshop including a lecture on constructive set theory (CZF) in which it was explained that ordinary topology can't be formalised in CZF, a predicative theory, because it invokes the power set operator, which is impredicative (it would be a function P : Set → Set defined by P(A) = {X ∈ Set | forall x ∈ X, x ∈ A}, where Set is the class of all sets, including the power set being defined). Hairy Dude (talk) 15:03, 17 July 2008 (UTC)

Another possible source: Section 2.2 "Predicativity" in the article "The crisis in the foundations of mathematics" by José Ferreirós, in collection "The Princeton companion to mathematics" (T. Gowers, editor), Princeton University Press 2008. Some quotes:

"Informally, a definition is impredicative when it introduces an element by reference to a totality that already contains that element." (Page 146).

(About Poincaré) "In this kind of approach to the foundations of mathematics, all mathematical objects (beeyond the natural numbers) must be introduced by explicit definitions. If a definition refers to a presumed totality of which the object being defined is itself a member, we are involved in a circle: the object itself is then a constituent of its own definition. (...) Important authors such as Russel and Weyl accepted this point of view and developed it." (Pages 146–147.)

"But Zermelo insisted that these definitions are innocuous, because the object being defined is not "created" by the definition; it is merely singled out (...)." (Page 147.)

"Poincaré's suggestions also became a key principle for the interesting foundational approach proposed by Weyl in his book Das Kontinuum (1918). The main idea was to accept the theory of the natural numbers as they were conventionally developed using classical logic, but to work predicatively from there on." (Page 147.)

"Predicative systems lie between those that countenance all of the modern methodology and the more stringent constructivistic systems. This is one of several foundational approaches that do not fit into the conventional but by now outdated triad of logicism, formalism, and intuitionism." (Page 147.)

Boris Tsirelson (talk) 10:09, 6 February 2009 (UTC)

To Hairy dude: Indeed ordinary topology can't be formalized with predictives only. Look at how closure of A is defined, the intersection of ALL closed sets containing A, but then the closure is itself a closed set containing A so we are necessarily quantifying over a collection already containing the thing we want to define. Breath of the Dying (talk) 19:38, 23 October 2009 (UTC)