6 ideas
3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
9464 | One of their own prophets said that Cretans are always liars [Anon (Titus)] |
Full Idea: One of themselves, even a prophet of their own, said, the Cretians are always liars, evil beasts, slow bellies. This witness is true. | |
From: Anon (Titus) (17: Epistle to Titus [c.115], I.12) | |
A reaction: The classic statement of the paradox, the word 'always' being the source of the problem. |
22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
Full Idea: Archelaus was the first person to say that the universe is boundless. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |