* ZF – Zermelo – Fraenkel set theory omitting the Axiom of Choice.

* ZFC – Zermelo – Fraenkel set theory, extended to include the Axiom of Choice.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

Informal applications of set theory in other fields are sometimes referred to as applications of " naive set theory ", but usually are understood to be justifiable in terms of an axiomatic system ( normally the Zermelo – Fraenkel set theory ).

It can be done by systematically making explicit all the axioms, as in the case of the well-known book Naive Set Theory by Paul Halmos, which is actually a somewhat ( not all that ) informal presentation of the usual axiomatic Zermelo – Fraenkel set theory.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Many common axiomatic systems, such as first-order Peano arithmetic and axiomatic set theory, including the canonical Zermelo – Fraenkel set theory ( ZF ), can be formalized as first-order theories.

Weak König's lemma is provable in ZF, the system of Zermelo – Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF.

The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over Zermelo – Fraenkel set theory.

In work on Zermelo – Fraenkel set theory, the notion of class is informal, whereas other set theories, such as Von Neumann – Bernays – Gödel set theory, axiomatize the notion of " class ", e. g., as entities that are not members of another entity.

A class that is not a set ( informally in Zermelo – Fraenkel ) is called a proper class, and a class that is a set is sometimes called a small class.

While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo – Fraenkel set theory and gives correct and rigorous definitions for basic objects.

Paul Joseph Cohen ( April 2, 1934 — March 23, 2007 ) was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo – Fraenkel set theory, the most widely accepted axiomatization of set theory.

As discussed below, the definition given above turned out to be inadequate for formal mathematics ; instead, the notion of a " set " is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo – Fraenkel axioms.

It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo – Fraenkel axioms is sufficient to prove the other, in first order logic.

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory.

In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation ( Aussonderung ).

For his part, Ernst Zermelo in his ( 1908 ) A new proof of the possibility of a well-ordering ( published at the same time he published " the first axiomatic set theory ") laid claim to prior discovery of the antinomy in Cantor's naive set theory.

The Zermelo set theory of 1908 included urelements.

When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.

Ernst Zermelo in his 1908 A new proof of the possibility of a well-ordering presents an entire section " b. Objection concerning nonpredicative definition " where he argued against " Poincaré ( 1906, p. 307 ) states that a definition is ' predicative ' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it ".

The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

After reading Gibbs's textbook ( which was translated into German by Ernst Zermelo in 1905 ), Einstein declared that Gibbs's treatment was superior to his own and explained that he would not have written those papers if he had known Gibbs's work.

The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC.

Through the work of Zermelo and others, especially John von Neumann, the structure of what some see as the " natural " objects described by ZFC eventually became clear ; they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation.

van Heijenoort in his commentary before Russell's 1902 Letter to Frege states that Zermelo " had discovered the paradox independently of Russell and communicated it to Hilbert, among others, prior to its publication by Russell ".

This collection, which is formalized by Zermelo – Fraenkel set theory ( ZFC ), is often used to provide an interpretation or motivation of the axioms of ZFC.

Among the many famous professors he was taught by, he could count Eötvös Loránd, Kürschák, Carathéodory, Hilbert, Klein and Zermelo.

In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo – Fraenkel set theory in the absence of the Axiom of Choice, by showing that ( assuming the consistency of an inaccessible cardinal ) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

An error in Kőnig's proof was discovered by Ernst Zermelo soon thereafter.

Of a totally different orientation < nowiki > the " Old Formalist School " of Richard Dedekind | Dedekind, Georg Cantor | Cantor, Giuseppe Peano | Peano, Ernst Zermelo | Zermelo, and Louis Couturat | Couturat, etc .< nowiki ></ nowiki > was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue.

Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community.

Among the students of Hilbert were: Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel.

Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God.

Ernst Zermelo introduced the Axiom of Choice as an " unobjectionable logical principle " to prove the well-ordering theorem.

Ernst Zermelo was one of the major advocates of such a view ; he also developed much of axiomatic set theory.

In mathematics, Zermelo – Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory such as Russell's paradox.

Ernst Friedrich Ferdinand Zermelo (; 1871 – 1953 )

# redirect Ernst Zermelo

* Ernst Zermelo formulates the axiom of choice.

* Ernst Zermelo axiomizes set theory, thus avoiding Cantor's contradictions.

* Ernst Zermelo

He is known for his contributions to axiomatic set theory, especially his addition to Ernst Zermelo's axioms which resulted in Zermelo – Fraenkel axioms.

# REDIRECT Ernst Zermelo

Ernst Zermelo, the later editor of Cantor's collected works, found the error already the next day.