4 ideas
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 12735 | Everything has a fixed power, as required by God, and by the possibility of reasoning [Leibniz] |
| Full Idea: It follows from the nature of God that there is a fixed power of a definite magnitude [non vagam] in anything whatsoever, otherwise there would be no reasonings about those things. | |
| From: Gottfried Leibniz (De aequopollentia causae et effectus [1679], A6.4.1964), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6 | |
| A reaction: This is double-edged. On the one hand there is the grand claim that the principle derives from divine nature, but on the other it derives from our capacity to reason and explain. No one doubts that powers are 'fixed'. |
| 1743 | The greatest deterrence for injustice is if uninjured parties feel as much indignation as those who are injured [Solon, by Diog. Laertius] |
| Full Idea: Men can be most effectively deterred from committing injustice if those who are not injured feel as much indignation as those who are. | |
| From: report of Solon (reports [c.600 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.So.10 |