Editing John von Neumann (section)

=== Set theory ===
{{See also|Von Neumann–Bernays–Gödel set theory}}
[[File:NBG Evolution svg.svg|thumb|upright=1.5|History of approaches that led to NBG set theory]]
At the beginning of the 20th century, efforts to base mathematics on [[naive set theory]] suffered a setback due to [[Russell's paradox]] (on the set of all sets that do not belong to themselves).{{sfn|Macrae|1992|pp=104–105}} The problem of an adequate axiomatization of [[set theory]] was resolved implicitly about twenty years later by [[Ernst Zermelo]] and [[Abraham Fraenkel]]. [[Zermelo–Fraenkel set theory]] provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—the ''[[axiom of regularity|axiom of foundation]]'' and the notion of ''[[Class (set theory)|class]].''<ref name=vanheijenoort>{{cite book |last=Van Heijenoort |first=Jean |author-link=Jean van Heijenoort |year=1967 |title=From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931 |url=https://archive.org/details/fromfregetogodel0000vanh |url-access=registration |location=Cambridge, Massachusetts |publisher=Harvard University Press |isbn=978-0-674-32450-3|oclc=523838 }}</ref>

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced the ''method of [[inner model]]s'', which became an essential demonstration instrument in set theory.{{r|vanheijenoort}}

The second approach to the problem of sets belonging to themselves took as its base the notion of [[Class (set theory)|class]], and defines a set as a class that belongs to other classes, while a ''proper class'' is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a ''proper class'', not a set.{{r|vanheijenoort}}

Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the [[Ordinal number|ordinal]] and [[cardinal number]]s as well as the first strict formulation of principles of definitions by the [[transfinite induction]]".{{sfn|Murawski|2010|p=196}}

====Von Neumann paradox====
{{main|Von Neumann paradox}}
Building on the [[Hausdorff paradox]] of [[Felix Hausdorff]] (1914), [[Stefan Banach]] and [[Alfred Tarski]] in 1924 showed how to subdivide a three-dimensional [[ball (mathematics)|ball]] into [[disjoint sets]], then translate and rotate these sets to form two identical copies of the same ball; this is the [[Banach–Tarski paradox]]. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929,<ref>{{citation | first=J. | last=von Neumann | author-link=John von Neumann | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm13/fm1316.pdf | title=Zur allgemeinen Theorie des Masses |trans-title=On the general theory of mass |language=de | journal=[[Fundamenta Mathematicae]] | volume=13 | pages=73–116 | year=1929 | doi=10.4064/fm-13-1-73-116 | doi-access=free }}</ref> von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving [[affine transformation]]s instead of translations and rotations. The result depended on finding [[free group]]s of affine transformations, an important technique extended later by von Neumann in [[#Measure theory|his work on measure theory]].{{sfn|Ulam|1958|pages=14–15}}