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--- set_theory [2014/12/08 09:56]
nikolaj
+++ set_theory [2015/04/12 17:08]
nikolaj
@@ Line 110: / Line 110: @@
 Moreover, we have "$d=\{a,b,c\}$", expressing a similar statement about //three// mathematical object and so on. The set $\{a\}\cup\{b,c\}$, which can for example be build by taking the union of the pair of pairs $\{\{a,a\}\cup\{b,c\}\}$ then also happens to fulfill exactly that defining formula, i.e.  $\{a\}\cup\{b,c\}=\{a,b,c\}$ etc.. Terms which are estabilished equal in the "$=$"-sense can be replaced for each other in a deduction.
-The axioms replacement/comprehension/collection/specification tell us to which extend the deduction of "$x\in X\Leftrightarrow P(x)$" and the associated set construction is possible within our theory. If we can specify a set by a predicate $P$, then a like sentence $X=\{x|P(x)\}$ just denotes that $X$ is the set containing the sets (in the whole domain of discourse) for which $P$ is true:
+The axioms replacement/comprehension/collection/specification tell us to which extent the deduction of "$x\in X\Leftrightarrow P(x)$" and the associated set construction is possible within our theory. If we can specify a set by a predicate $P$, then a like sentence $X=\{x|P(x)\}$ just denotes that $X$ is the set containing the sets (in the whole domain of discourse) for which $P$ is true:
 | @#EEEE55: predicate   | @#EEEE55: $X=\{x|P(x)\} \equiv \forall x.\ (x\in X\Leftrightarrow P(x))$ |