18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
10537 | The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett] |
Full Idea: A classic reduction is the class of ordered pairs <x,y> being reduced to the class of sets of the form {{x},{x,y}}. | |
From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14) |
9680 | The empty set Φ is a subset of every set (including itself) [Priest,G] |
Full Idea: The empty set Φ is a subset of every set (including itself). | |
From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6) |