Set theory does not allow sets {paradoxical set} that are elements of themselves {Foundation axiom}.
Set theories {Zermelo-Fraenkel set theory} {ZF set theory} can be axiomatic systems. Zermelo set theory has no paradoxes but is not consistent.
The empty set exists {axiom of empty set}. The empty set exists, so at least one set exists {axiom of existence}. Sets with same elements are equal {axiom of extension} {axiom of extensionality}. For two sets, another set exists that contains all and only elements of the two sets {axiom of union}. For two sets, another set exists that has the two sets as only elements {axiom of pairing}. For a set, another set exists whose elements are the subsets of the original set {axiom of powers} {axiom of power set}. A set exists that has the empty set as an element and, if an element is in the set, the set that contains only that element is an element in the set {axiom of infinity}.
Non-empty sets contain at least one element, and the non-empty set and the set of any element are disjoint sets {axiom of regularity}. For any set and any mapping, a subset of the set exists that has as elements the domain of the mapping {axiom of separation}. For any set and any mapping, a set exists that has as elements the range of the mapping over the original set's elements (as domain of mapping) {axiom of replacement} {axiom of specification}. For any set, a mapping exists that chooses one element of each subset (axiom of choice).
For any set, a mapping exists that chooses one element of each subset {axiom of choice} {Zermelo's axiom}. Elements are not in any other non-empty set, even if number of non-empty subsets is infinite. Axiom of choice is independent of set theory. Zermelo set theory has no paradoxes but is not consistent. If Zermelo-Fraenkel set theory is consistent without axiom of choice, then ZF set theory is consistent with axiom of choice.
Axiom of choice leads to unexpected consequences if applied to sets with uncountably infinite members. In such set, object can divide into five pieces that can rotate, translate, and invert to make much greater volume {Banach-Tarski theorem}.
In partially ordered sets in which subsets have upper bounds, sets have greatest member {Zorn's lemma} {Zorn lemma}. Zorn's lemma is equivalent to axiom of choice.
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Date Modified: 2022.0225