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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' :
The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents). [1] [2] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. [3]
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 ...
The first publication of the von Neumann universe was by Ernst Zermelo in 1930. [7] Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory [16] and von Neumann's own set theory (which later developed into NBG set theory). [17]
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 ...
Zermelo worked with models of the form V κ where κ is a cardinal. The classes of the model are the subsets of V κ, and the model's ∈-relation is the standard ∈-relation. The sets of the model are the classes X such that X ∈ V κ. [j] Zermelo identified cardinals κ such that V κ satisfies: [12] Theorem 1. A class X is a set if and ...