4 ideas
| 21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
| Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
| 21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
| Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: Formulated by Burali-Forti in 1897. |
| 12710 | As well as extension, bodies contain powers [Leibniz] |
| Full Idea: Over and above what can be deduced from extension, we must add and recognise in bodies certain notions or forms that are immaterial, so to speak, or independent of extension, which you can call powers [potentia], by which speed is adjusted to magnitude. | |
| From: Gottfried Leibniz (De Natura Corporis [1678], A6.4.1980), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
| A reaction: He boldly asserts that the powers are 'immaterial', but is then forced to qualify it (as he often does) with 'so to speak'. The notion that bodies just have extension (occupy space) comes from Descartes, and is firmly opposed by Leibniz. |
| 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? |