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Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit ...
Zermelo's paper may be the first to mention the name "Cantor's theorem". Cantor's theorem: "If M is an arbitrary set, then always M < P(M) [the power set of M]. Every set is of lower cardinality than the set of its subsets". Zermelo proves this by considering a function φ: M → P(M). By Axiom III this defines the following set M' :
In second-order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem. [7] There is a well-known joke about the three statements, and their relative amenability to intuition:
Ernst Friedrich Ferdinand Zermelo (/ z ɜːr ˈ m ɛ l oʊ /, German: [tsɛɐ̯ˈmeːlo]; 27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering ...
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: =, or even shorter with beth numbers: =. The continuum hypothesis was advanced by Georg Cantor in 1878, [ 1 ] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900.
An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.. The cumulative hierarchy is a collection of sets V α indexed by the class of ordinal numbers; in particular, V α is the set of all sets having ranks less than α.
An abelian group with Ext 1 (A, Z) = 0 is called a Whitehead group; MA + ¬CH proves the existence of a non-free Whitehead group, while V = L proves that all Whitehead groups are free. In one of the earliest applications of proper forcing , Shelah constructed a model of ZFC + CH in which there is a non-free Whitehead group.
The phrase “Zermelo-Fraenkel set theory” was first used in print by von Neumann in 1928. [8] Zermelo and Fraenkel had corresponded heavily in 1921; the axiom of replacement was a major topic of this exchange. [7] Fraenkel initiated correspondence with Zermelo sometime in March 1921. However, his letters before the one dated 6 May 1921 are ...