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power_set [2015/10/08 14:24] nikolaj |
power_set [2015/10/08 20:35] nikolaj |
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| - | === Discussion === | ||
| Here we define | Here we define | ||
| $\mathcal{P}(X) \equiv \{Y\mid Y\subseteq X\}$ | $\mathcal{P}(X) \equiv \{Y\mid Y\subseteq X\}$ | ||
| - | and want to claim that for each $X$, we have | + | which is sensible in our set theory if, for each set $X$, we have |
| $\exists! P.\,P = \{Y\mid Y\subseteq X\}$ | $\exists! P.\,P = \{Y\mid Y\subseteq X\}$ | ||
| - | or more formally | + | or, more formally, |
| $\forall X.\,\exists! P.\,P = \{Y\mid Y\subseteq X\}$ | $\forall X.\,\exists! P.\,P = \{Y\mid Y\subseteq X\}$ | ||
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| $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow Y\subseteq X\right)$ | $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow Y\subseteq X\right)$ | ||
| - | which is short for | + | === Discussion === |
| + | The above is short for | ||
| $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | ||
| - | The existence is granted from the [[https://en.wikipedia.org/wiki/Axiom_of_power_set|Axiom of power set]] | + | and this is, apart from the exclamation mark,exactly the [[https://en.wikipedia.org/wiki/Axiom_of_power_set|Axiom of power set]]. |
| - | + | ||
| - | $\forall X.\,\exists P.\,\forall Y.\,\left(Y\in P\Leftrightarrow \forall Z.\,(Z\in Y\implies Z\in X)\right)$ | + | |
| - | and uniqueness follows from extensionality. | + | Like in the case of the [[empty set]], uniqueness follows from extensionality. |
| == Examples == | == Examples == | ||
