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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?

Phillip Feldman
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  • You can drop empty set once you have infinity, since you can use infinity and extensionality to create an empty set. – Arturo Magidin May 05 '19 at 03:14
  • Can you show me such a construction please? My book defines infinity in terms of the empty set. – Phillip Feldman May 05 '19 at 03:37
  • 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. – Arturo Magidin May 05 '19 at 03:46
  • @ArturoMagidin: Actually, you can use Infinity and inference rules to prove the existence of the empty set. – Asaf Karagila May 05 '19 at 08:16
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    Philip, Infinity is defined in terms of the empty set, but the empty set is not part of the language of set theory. – Asaf Karagila May 05 '19 at 08:22
  • https://math.stackexchange.com/questions/916072/what-axioms-does-zf-have-exactly/ – Asaf Karagila May 05 '19 at 08:30
  • There have been a few other questions about removing power set, infinity, choice, replacement, etc. Where counterexamples of this are discussed. – Asaf Karagila May 05 '19 at 08:36
  • In some presentations we also have Existence: $\exists x,(x=x)$ because in some presentations the only other axiom that does not begin with "$\forall$" is Infinity. And if you want to study the consequences that do not rely on Infinity, and if all your other axioms start with "$\forall$" then they can't prove that anything exists. – DanielWainfleet May 27 '19 at 05:57

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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.

spaceisdarkgreen
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    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). – Arthur May 05 '19 at 03:05