ONT Re: Set Theory
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Note 12
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| Elementary Set Theory
|
| Existence of Sets (cont.)
|
| 36. Definition. 2^x = {y : y c x}.
|
| 37. Theorem. $U$ = 2^$U$.
|
| Proof. Every member of 2^$U$ is a set
| and consequently belongs to $U$.
| Every member of $U$ is a set and
| is contained in $U$ (theorem 26)
| and hence belongs to 2^$U$. þ
|
| 38. Theorem.
|
| If x is a set, then 2^x is a set,
| and for each y, y c x iff y in 2^x.
|
| It is interesting to notice that the existence of sets is
| not provable on the basis of the axioms so far enunciated,
| but it is possible to prove that there is a class which is
| not a set. Letting R be {x : x ~in x}, by the classifier
| axiom R in R iff R ~in R and R is a set. It follows that
| R is not a set. Observe that, if the classifier axiom did
| not contain the "is a set" qualification, then an outright
| contradiction, R in R iff R ~in R, would result. This is
| the Russell paradox. A consequence of this argument is
| that $U$ is not a set, because R c $U$ and 33 applies.
| (The regularity axiom will imply that R = $U$; this
| axiom also yields a different proof that $U$ is not
| a set.)
|
| 39. Theorem. $U$ is not a set.
|
| JLK, Gen Top, page 257.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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