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successor_set [2013/09/16 10:48] nikolaj |
successor_set [2014/12/05 10:51] nikolaj |
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===== Successor set ===== | ===== Successor set ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
- | + | | @#FFBB00: definiendum | @#FFBB00: $ {\mathrm{succ}}\ X \equiv X \cup \{X\} $ | | |
- | | @#FFBB00: $ S(X) \equiv X \cup \{X\} $ | | + | |
==== Discussion ==== | ==== Discussion ==== | ||
=== Theorems === | === Theorems === | ||
- | ^ $ X\in S(X) $ ^ | + | ^ $ X\in {\mathrm{succ}}\ X $ ^ |
- | ^ $ (S(X)=S(Y))\implies (X=Y) $ ^ | + | ^ $ ({\mathrm{succ}}\ X={\mathrm{succ}}\ Y)\Rightarrow (X=Y) $ ^ |
- | ^ $ (Y\in S(X))\iff (Y=X\lor Y=\{X\}) $ ^ | + | ^ $ (Y\in {\mathrm{succ}}\ X)\Leftrightarrow (Y=X\lor Y=\{X\}) $ ^ |
- | ^ $ X\neq S(X) $ ^ | + | ^ $ X\neq {\mathrm{succ}}\ X $ ^ |
=== Reference === | === Reference === | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Successor_ordinal|Successor ordinal]] | Wikipedia: [[http://en.wikipedia.org/wiki/Successor_ordinal|Successor ordinal]] |