display all the ideas for this combination of texts
4 ideas
| 8309 | A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe] |
| Full Idea: A set is 'a number of things', not a 'collection'. Nothing literally 'collects' the members of a set, such as the set of planets of the sun, unless it be a Fregean 'concept' under which they fall. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.6) | |
| A reaction: I'm tempted to say that the sun has collected a set of planets (they're the ones that rotate around it). Why can't we have natural sets, which have been collected by nature? A question of the intension, as well as the extension.... |
| 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) |
| 8322 | I don't believe in the empty set, because (lacking members) it lacks identity-conditions [Lowe] |
| Full Idea: It is not clear to me that the empty set has well-defined identity-conditions. A set has these only to the extent that its members do - but the empty set has none. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 12.3 n8) | |
| A reaction: The empty set is widely used by those who base their metaphysics of maths on sets. It defines zero, and hence is the starting poing for Peano's Postulates (Idea 5897). It might not have identity in itself, but you know where you have arrived after 2 - 2. |