Zermelo-Fraenkel set theory

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Zermelo-Fraenkel set theory

mathematics

Also known as: ZF, ZFC, Zermelo-Fraenkel-Skolem set theory

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July 15, 2026

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Zermelo-Fraenkel set theory (ZF) is an axiomatic system used to describe sets, which are collections of objects. It was developed to avoid paradoxes found in earlier, less formal set theories, such as Russell's paradox.12

ZF is built upon a set of axioms that define how sets can be formed and what properties they possess. Key axioms include the axiom of extensionality (sets with the same members are identical), the axiom of elementary sets (existence of the empty set and singleton sets), and the axiom of separation (allowing the creation of subsets based on properties). The theory also includes axioms for power sets, unions, and infinity, ensuring the existence of infinite sets necessary for much of mathematics.13

The Zermelo-Fraenkel axioms, often with the addition of the axiom of choice (ZFC), are now the standard foundation for most mathematical work. This system provides a rigorous framework for discussing concepts like numbers, functions, and relations.4536

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history of logic
Zermelo-Fraenkel set theory (ZF)
Contradictions like Russell’s paradox arose from what was later called the unrestricted comprehension principle: the assumption that, for any property p, there is a set that contains all and only those sets that have p. In Zermelo’s system, the comprehension principle…
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Contradictions like Russell’s paradox arose from what was later called the unrestricted comprehension principle: the assumption that, for any property p, there is a set that contains all and only those sets that have p. In Zermelo’s system, the comprehension principle…
Read More

set theory
The Zermelo-Fraenkel axioms
The first axiomatization of set theory was given in 1908 by German mathematician Ernst Zermelo. From his analysis of the paradoxes described above in the section Cardinality and transfinite numbers, he concluded that they are associated with sets that are “too big,” such…
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The first axiomatization of set theory was given in 1908 by German mathematician Ernst Zermelo. From his analysis of the paradoxes described above in the section Cardinality and transfinite numbers, he concluded that they are associated with sets that are “too big,” such…
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continuum hypothesis
…that, if the other standard Zermelo-Fraenkel axioms (ZF; see the Zermelo-Fraenkel axiomsZermelo-Fraenkel axioms.Encyclopædia Britannica, Inc.table) are consistent, then they do not
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…that, if the other standard Zermelo-Fraenkel axioms (ZF; see the Zermelo-Fraenkel axiomsZermelo-Fraenkel axioms.Encyclopædia Britannica, Inc.table) are consistent, then they do not
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foundations of mathematics
Set theoretic beginnings
…essentially equivalent first-order language, the Zermelo-Fraenkel axioms, which allow one to construct new sets only as subsets of given old sets. Mention should also be made of the system of the American philosopher Willard Van Orman Quine (1908–2000), which admits a universal set. (Cantor had not allowed such a “biggest”…
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…essentially equivalent first-order language, the Zermelo-Fraenkel axioms, which allow one to construct new sets only as subsets of given old sets. Mention should also be made of the system of the American philosopher Willard Van Orman Quine (1908–2000), which admits a universal set. (Cantor had not allowed such a “biggest”…
Read More

infinity
Mathematical infinities
…as ZFC, which stands for Zermelo-Fraenkel set theory with the axiom of choice. CH is known to be undecidable on the basis of the axioms in ZFC. In 1940 the Austrian-born logician Kurt Gödel was able to show that ZFC cannot disprove CH, and in 1963 the American mathematician Paul…
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…as ZFC, which stands for Zermelo-Fraenkel set theory with the axiom of choice. CH is known to be undecidable on the basis of the axioms in ZFC. In 1940 the Austrian-born logician Kurt Gödel was able to show that ZFC cannot disprove CH, and in 1963 the American mathematician Paul…
Read More