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I am studying enderton elements of set theory and trying to figure out what combinations of axioms are redundant. Given: extensionality, empty set, pair set, union, power set, SUBSETS, AOC, replacement, infinity, regularity. My most pertinent question: I know I can delete subset and pairing. Can I instead delete subset and power set?

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  • $\begingroup$ You can drop empty set once you have infinity, since you can use infinity and extensionality to create an empty set. $\endgroup$ – Arturo Magidin May 5 at 3:14
  • $\begingroup$ Can you show me such a construction please? My book defines infinity in terms of the empty set. $\endgroup$ – Phillip Feldman May 5 at 3:37
  • $\begingroup$ Depends on the precise definition; but Infinity tells you there is an inductive set; then from separation you can deduce the existence of an empty set by letting $A$ be any inductive set and then considering $\{x\in A\mid x\neq x\}$. Of course, if your notion of “inductive set” requires the existence of a set (so that you can deduce the existence of an empty set, etc) then this does not work. $\endgroup$ – Arturo Magidin May 5 at 3:46
  • $\begingroup$ @ArturoMagidin: Actually, you can use Infinity and inference rules to prove the existence of the empty set. $\endgroup$ – Asaf Karagila May 5 at 8:16
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    $\begingroup$ Philip, Infinity is defined in terms of the empty set, but the empty set is not part of the language of set theory. $\endgroup$ – Asaf Karagila May 5 at 8:22
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No, the power set axiom adds a lot of strength to the theory. In fact, without it, you cannot even prove there are uncountable sets. The set of hereditarily countable sets is a model of ZFC minus power set.

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  • $\begingroup$ Power set is the only axiom which lets you take an infinite set and make a larger set (in the sense of cardinality / existence of injections). $\endgroup$ – Arthur May 5 at 3:05

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