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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST

[useful simple theorems derived within set theory]

4 ideas
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).
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
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)
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)