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we can form any subset of a set using axiom of specification. then we can form collection of subsets using axiom of pairing repeatedly.

we have for every condition S , (A={x belongs to X & S(x)} => (A is subset of X and {A} exists ) then defining the union over the collection of singletons {A} as P(X).

the problem is the collection of singletons of each subsets , done using axiom of pairing. So can this be done?

Asaf Karagila
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Arjun
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    Is your collection of singletons actually a set? You can only take unions over sets. – Zhen Lin May 25 '15 at 11:25
  • by axiom of pairing we can form set containing the subsets – Arjun May 25 '15 at 11:32
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    Comprehension gives you only countably many subsets. – Brian M. Scott May 25 '15 at 12:22
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    It seems almost as if you're asking whether or not the collection of all singletons of a set is also a set without assuming the power set axiom. But it's very unclear. – Asaf Karagila May 25 '15 at 21:19
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    Your question is unclear, but this might be related (or even a duplicate). – Milo Brandt May 25 '15 at 21:55
  • Yes, my question was if its possible to define power set instead of having as an axiom. – Arjun May 25 '15 at 23:12
  • Define the power set, or define the set of singletons? – Asaf Karagila May 26 '15 at 04:22
  • for finite number of singletons we can tell that the collection of singletons exists, that is by using axiom of pairing and axiom of union and induction, so for this we can say power set is same as the union over the collection of singletons of each subsets. but we cant define power set if there are infinite subsets right? – Arjun May 27 '15 at 06:43

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