9 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 |
| 490 | Everything happens by reason and necessity [Leucippus] |
| Full Idea: Nothing happens at random; everything happens out of reason and by necessity. | |
| From: Leucippus (fragments/reports [c.435 BCE], B002), quoted by (who?) - where? |
| 8724 | The meaning of 'know' does not change from courtroom to living room [Unger] |
| Full Idea: There is no reason to suppose that the meaning of 'know' changes from the courtroom to the living room and back again; no more than for supposing that 'vacuum' changes from the laboratory to the cannery. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], 2.1) | |
| A reaction: I disagree. Lots of words change their meaning (or reference) according to context. Flat, fast, tall, clever. She 'knows a lot' certainly requires a context. The bar of justification goes up and down, and 'knowledge' changes accordingly. |
| 8722 | No one knows anything, and no one is ever justified or reasonable [Unger] |
| Full Idea: I argue for the thesis that no one ever knows about anything, ...and that consequently no one is ever justified or at all reasonable in anything. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], Intro) | |
| A reaction: The premiss of his book seems to be that knowledge is assumed to require certainty, and is therefore impossible. Unger has helped push us to a more relaxed and fallibilist attitude to knowledge. 'No one is reasonable' is daft! |
| 8723 | An evil scientist may give you a momentary life, with totally false memories [Unger] |
| Full Idea: The evil scientist might not only be deceiving you with his electrodes; maybe he has just created you with your ostensible memory beliefs and experiences, and for good measure he will immediately destroy you, so in the next moment you no longer exist. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], 1.12) | |
| A reaction: This is based on Russell's scepticism about memory (Idea 2792). Even this very train of thought may not exist, if the first half of it was implanted, rather than being developed by you. I cannot see how to dispute this possibility. |