7 ideas
13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: [see Hart 73-74 for an explication of this] |
12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
7634 | Icons resemble their subject, an index is a natural sign, and symbols are conventional [Peirce, by Maund] |
Full Idea: For Peirce there are three different kinds of sign, which are different kinds of representation, built on different relationships: an 'icon' represents what it resembles, an 'index' is a natural sign, and a 'symbol' is a conventional sign. | |
From: report of Charles Sanders Peirce (Logic as Semiotic: Theory of Signs [1897]) by Barry Maund - Perception Ch.4 | |
A reaction: Maund makes use of natural signs (like footprints) to explain representative perception. Peirce's distinctions seem useful in philosophy of mind generally, if the brain somehow represents what it experiences. How subjective are signs? |
9425 | Lewis later proposed the axioms at the intersection of the best theories (which may be few) [Mumford on Lewis] |
Full Idea: Later Lewis said we must choose between the intersection of the axioms of the tied best systems. He chose for laws the axioms that are in all the tied systems (but then there may be few or no axioms in the intersection). | |
From: comment on David Lewis (Subjectivist's Guide to Objective Chance [1980], p.124) by Stephen Mumford - Laws in Nature |