Combining Texts

Ideas for 'Being and Nothingness', 'Guidebook to Wittgenstein's Tractatus' and 'A Tour through Mathematical Logic'

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4 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
     Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
     A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
23452 Discriminating things for counting implies concepts of identity and distinctness [Morris,M]
     Full Idea: The discrimination of things for counting needs to bring with it the notion of identity (and, correlatively, distinctness).
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5)
     A reaction: Morris is exploring how practices like counting might reveal necessary truths about the world.
23451 Counting needs to distinguish things, and also needs the concept of a successor in a series [Morris,M]
     Full Idea: Just distinguishing things is not enough for counting (and hence arithmetic). We need the crucial extra notion of the successor in a series of some kind.
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5)
     A reaction: This is a step towards the Peano Axioms of arithmetic. The successors could be fingers and toes, taken in a conventional order, and matched one-to-one to the objects. 'My right big toe of cows' means 16 cows (but non-verbally).
23460 To count, we must distinguish things, and have a series with successors in it [Morris,M]
     Full Idea: Distinguishing between things is not enough for counting. …We need the crucial extra notion of a successor in a series of a certain kind.
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro)
     A reaction: This is the thinking that led to the Dedekind-Peano axioms for arithmetic. E.g. each series member can only have one successor. There is an unformalisable assumption that the series can then be applied to the things.