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3 ideas
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
Full Idea: In the superior realist and simple theory of types, the place of the axiom of reducibility is not taken by the axiom of classes, Zermelo's Aussonderungsaxiom. | |
From: report of Kurt Gödel (Russell's Mathematical Logic [1944], p.140-1) by Bernard Linsky - Russell's Metaphysical Logic 6.1 n3 | |
A reaction: This is Zermelo's Axiom of Separation, but that too is not an axiom of standard ZFC. |