Search results
Results from the WOW.Com Content Network
Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets.
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]
The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be ...
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results.
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 ...
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 ...
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 ...