Differences
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power_set [2015/10/08 14:22] nikolaj |
power_set [2015/10/08 14:24] nikolaj |
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| Here we define | Here we define | ||
| - | $\mathcal{P}(X) := \{Y\mid Y\subseteq X\}$ | + | $\mathcal{P}(X) \equiv \{Y\mid Y\subseteq X\}$ |
| and want to claim that for each $X$, we have | and want to claim that for each $X$, we have | ||
| Line 27: | Line 27: | ||
| $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | ||
| - | The existence is granted from the [[https://en.wikipedia.org/wiki/Axiom_of_power_set|Axiom of power set]] | + | Without the exclamation mark, this is exactly the [[https://en.wikipedia.org/wiki/Axiom_of_power_set|Axiom of power set]]. |
| - | + | Uniqueness follows from extensionality. | |
| - | $\forall X.\,\exists P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | + | |
| - | + | ||
| - | and uniqueness follows from extensionality. | + | |
| == Examples == | == Examples == | ||
