Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
|
empty_set [2015/10/08 20:32] nikolaj |
empty_set [2015/10/09 15:39] nikolaj |
||
|---|---|---|---|
| Line 15: | Line 15: | ||
| $\exists! y.\,y=\{x\mid \bot\}$ | $\exists! y.\,y=\{x\mid \bot\}$ | ||
| - | holds, really is an abbreviation for | + | holds, which really is an abbreviation for |
| $\exists! y.\,\forall x.\,\left(x\in y\Leftrightarrow \bot\right)$ | $\exists! y.\,\forall x.\,\left(x\in y\Leftrightarrow \bot\right)$ | ||
| Line 26: | Line 26: | ||
| So is it true? Does our set theory permit the existence of such a set $y$? | So is it true? Does our set theory permit the existence of such a set $y$? | ||
| - | Existence is granted by the [[http://en.wikipedia.org/wiki/Axiom_of_empty_set|axiom of empty set (Wikipedia)]]: | + | As $\bot\implies P$ is $\top$ for any $P$ and as $Q\land\top$ is logically equivalent to $Q$, the above is logically equivalent to |
| - | $\exists y.\,\nexists x.\,x\in y$ | + | $\exists! y.\,\forall x.\,\left(x\in y\implies \bot\right)$ |
| - | which is equivalent to | + | which is |
| - | $\exists y.\,\forall x.\,\neg(x\in y)$ | + | $\exists! y.\,\forall x.\,\neg(x\in y)$ |
| - | which is short for | + | or |
| - | $\exists y.\,\forall x.\,\left(x\in y\implies\bot\right)$ | + | $\exists! y.\,\nexists x.\,x\in y$ |
| - | As $\bot\implies P$ is $\top$ for any $P$ and as $Q\land\top$ is logically equivalent to $Q$, we know | + | Apart from the exclamation mark, this is exactly the [[http://en.wikipedia.org/wiki/Axiom_of_empty_set|axiom of empty set (Wikipedia)]]. |
| - | + | ||
| - | $\exists y.\,\forall x.\,\left((x\in y\implies \bot)\land(\bot\implies x\in y)\right)$ | + | |
| - | + | ||
| - | which is also written | + | |
| - | + | ||
| - | $\exists y.\,\forall x.\,\left(x\in y\Leftrightarrow \bot\right)$ | + | |
| - | + | ||
| - | which is what we wanted. | + | |
| Uniqueness is discussed, for example, in | Uniqueness is discussed, for example, in | ||
| - | [[https://proofwiki.org/wiki/Empty_Set_is_Unique|Empty_Set_is_Unique (Proof Wiki)]]. | + | [[https://proofwiki.org/wiki/Empty_Set_is_Unique|Empty Set is Unique (Proof Wiki)]]. |
| It requires judgements of equality of sets, which is proven via the axiom of extensionality. | It requires judgements of equality of sets, which is proven via the axiom of extensionality. | ||
| Line 61: | Line 53: | ||
| Proof Wiki: | Proof Wiki: | ||
| - | [[https://proofwiki.org/wiki/Empty_Set_is_Unique|Empty_Set_is_Unique (Proof Wiki)]] | + | [[https://proofwiki.org/wiki/Empty_Set_is_Unique|Empty Set is Unique (Proof Wiki)]] |
| ----- | ----- | ||
| === Subset of === | === Subset of === | ||
| [[First infinite von Neumann ordinal]] | [[First infinite von Neumann ordinal]] | ||
