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). 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 foundation and the notion of
class. 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 models, which became an essential demonstration instrument in set theory. The second approach to the problem of sets belonging to themselves took as its base the notion of
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. Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the
ordinal and
cardinal numbers as well as the first strict formulation of principles of definitions by the
transfinite induction".
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 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, von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving
affine transformations instead of translations and rotations. The result depended on finding
free groups of affine transformations, an important technique extended later by von Neumann in
his work on measure theory.
Proof theory With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its
consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger
axioms that could be used to prove a broader class of theorems. By 1927, von Neumann was involving himself in discussions in Göttingen on whether
elementary arithmetic followed from
Peano axioms. Building on the work of
Ackermann, he began attempting to prove (using the
finistic methods of
Hilbert's school) the consistency of
first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on
induction). He continued looking for a more general proof of the consistency of classical mathematics using methods from
proof theory. A strongly negative answer to whether it was definitive arrived in September 1930 at the
Second Conference on the Epistemology of the Exact Sciences, in which
Kurt Gödel announced his
first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete. At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers. Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency. Gödel replied that he had already discovered this consequence, now known as his
second incompleteness theorem, and that he would send a preprint of his article containing both results, which never appeared. Von Neumann acknowledged Gödel's priority in his next letter. However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did. With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the
foundations of mathematics and
metamathematics and instead spent time on problems connected with applications.
Ergodic theory In a series of papers published in 1932, von Neumann made foundational contributions to
ergodic theory, a branch of mathematics that involves the states of
dynamical systems with an
invariant measure. Of the 1932 papers on ergodic theory,
Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his articles on
operator theory, and the application of this work was instrumental in his
mean ergodic theorem. The theorem is about arbitrary
one-parameter unitary groups \mathit{t} \to \mathit{V_t} and states that for every vector \phi in the
Hilbert space, \lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} V_t(\phi) \, dt exists in the sense of the metric defined by the Hilbert norm and is a vector \psi which is such that V_t(\psi) = \psi for all t. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to
Boltzmann's ergodic hypothesis. He also pointed out that
ergodicity had not yet been achieved and isolated this for future work. Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic
measure preserving actions of the real line are the fundamental building blocks from which all measure-preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with
Paul Halmos have significant applications in other areas of mathematics.
Measure theory In
measure theory, the "problem of measure" for an -dimensional
Euclidean space may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of ?" The work of
Felix Hausdorff and
Stefan Banach had implied that the problem of measure has a positive solution if or and a negative solution (because of the
Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the
transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the
Euclidean group is a
solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." Around 1942 he told
Dorothy Maharam how to prove that every
complete σ-finite measure space has a multiplicative lifting; he did not publish this proof and she later came up with a new one. In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of
Haar regarding whether there existed an
algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". He proved this in the positive, and in later papers with
Stone discussed various generalizations and algebraic aspects of this problem. He also proved by new methods the existence of
disintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for
compact groups. He had to create entirely new techniques to apply this to
locally compact groups. He also gave a new, ingenious proof for the
Radon–Nikodym theorem. His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.
Topological groups Using his previous work on measure theory, von Neumann made several contributions to the theory of
topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended
Bohr's theory of
almost periodic functions to arbitrary
groups. He continued this work with another paper in conjunction with
Bochner that improved the theory of almost
periodicity to include
functions that took on elements of
linear spaces as values rather than numbers. In 1938, he was awarded the
Bôcher Memorial Prize for his work in
analysis in relation to these papers. In a 1933 paper, he used the newly discovered
Haar measure in the solution of
Hilbert's fifth problem for the case of
compact groups. The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of
linear transformations and found that closed
subgroups of a general
linear group are
Lie groups. This was later extended by
Cartan to arbitrary Lie groups in the form of the
closed-subgroup theorem.
Functional analysis Von Neumann was the first to axiomatically define an abstract
Hilbert space. He defined it as a
complex vector space with a
Hermitian scalar product, with the corresponding
norm being both separable and complete. In the same papers he also proved the general form of the
Cauchy–Schwarz inequality that had previously been known only in specific examples. He continued with the development of the
spectral theory of operators in Hilbert space in three seminal papers between 1929 and 1932. This work cumulated in his
Mathematical Foundations of Quantum Mechanics which alongside two other books by
Stone and
Banach in the same year were the first monographs on Hilbert space theory. Previous work by others showed that a theory of
weak topologies could not be obtained by using
sequences. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining
locally convex spaces and
topological vector spaces for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from
Hausdorff from Euclidean to Hilbert spaces) such as
boundness and
total boundness are still used today. For twenty years von Neumann was considered the 'undisputed master' of this area. These developments were primarily prompted by needs in
quantum mechanics where von Neumann realized the need to extend
the spectral theory of Hermitian operators from the bounded to the
unbounded case. Other major achievements in these papers include a complete elucidation of spectral theory for
normal operators, the first abstract presentation of the
trace of a
positive operator, a generalisation of
Riesz's presentation of
Hilbert's spectral theorems at the time, and the discovery of
Hermitian operators in a Hilbert space, as distinct from
self-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of
infinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory led to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of
operator algebras. His later work on rings of operators led to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces. Like in his work on measure theory he proved several theorems that he did not find time to publish. He told
Nachman Aronszajn and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the
invariant subspace problem. With
I. J. Schoenberg he wrote several items investigating
translation invariant Hilbertian
metrics on the
real number line which resulted in their complete classification. Their motivation lie in various questions related to embedding
metric spaces into Hilbert spaces. With
Pascual Jordan he wrote a short paper giving the first derivation of a given norm from an
inner product by means of the
parallelogram identity. His
trace inequality is a key result of matrix theory used in matrix approximation problems. He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric absolute norms). This paper leads naturally to the study of symmetric
operator ideals and is the beginning point for modern studies of symmetric
operator spaces. Later with
Robert Schatten he initiated the study of
nuclear operators on Hilbert spaces,
tensor products of Banach spaces, introduced and studied
trace class operators, their
ideals, and their
duality with
compact operators, and
preduality with
bounded operators. The generalization of this topic to the study of
nuclear operators on Banach spaces was among the first achievements of
Alexander Grothendieck. Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on \textit{l}\,_2^n\otimes\textit{l}\,_2^n and proving several other results on what are now known as Schatten–von Neumann ideals.
Operator algebras Von Neumann founded the study of rings of operators, through the
von Neumann algebras (originally called W*-algebras). While his original ideas for
rings of
operators existed already in 1930, he did not begin studying them in depth until he met
F. J. Murray several years later. A von Neumann algebra is a
*-algebra of bounded operators on a
Hilbert space that is closed in the
weak operator topology and contains the
identity operator. The
von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the
bicommutant. After elucidating the study of the
commutative algebra case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the
noncommutative case, the general study of
factors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century"; they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of
factors. In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949. Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out. Another important result on
polar decomposition was published in 1932.
Lattice theory Between 1935 and 1937, von Neumann worked on
lattice theory, the theory of
partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. As
Garrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann combined traditional projective geometry with modern algebra (
linear algebra,
ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general
modules over rings. His work laid the foundations for some of the modern work in projective geometry. His biggest contribution was founding the field of
continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex
projective geometry, where instead of the
dimension of a
subspace being in a discrete set 0, 1, ..., \mathit{n} it can be an element of the
unit interval [0,1]. Earlier,
Menger and Birkhoff had axiomatized
complex projective geometry in terms of the properties of its
lattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened those
axioms to describe a broader class of lattices, the continuous geometries. While the dimensions of the subspaces of projective geometries are a discrete set (the
non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval [0,1]. Von Neumann was motivated by his discovery of
von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the
projections of the
hyperfinite type II factor. In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of \mathit{CG(F)} (continuous-dimensional projective geometry over an arbitrary
division ring \mathit{F}\,) in abstract language of lattice theory. Von Neumann provided an abstract exploration of dimension in completed
complemented modular topological lattices (properties that arise in the
lattices of subspaces of
inner product spaces): Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity. For any integer n > 3 every \mathit{n}-dimensional abstract projective geometry is
isomorphic to the subspace-lattice of an \mathit{n}-dimensional
vector space V_n(F) over a (unique) corresponding division ring F. This is known as the
Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case. This
coordinatization theorem stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques. Birkhoff described this theorem as follows: Any complemented modular lattice having a "basis" of pairwise perspective elements, is isomorphic with the lattice of all principal
right-ideals of a suitable
regular ring . This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe. This work required the creation of
regular rings. A von Neumann regular ring is a
ring where for every a, an element x exists such that axa = a. These rings came from and have connections to his work on von Neumann algebras, as well as
AW*-algebras and various kinds of
C*-algebras. Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding
distributivity (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of
metric lattices. Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication.
Mathematical statistics Von Neumann made fundamental contributions to
mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically
normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the
Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order
autoregression.
Other work In his early years, von Neumann published several papers related to set-theoretical real analysis and number theory. In a paper from 1925, he proved that for any dense sequence of points in [0,1], there existed a rearrangement of those points that is
uniformly distributed. In 1926 his sole publication was on
Prüfer's theory of
ideal algebraic numbers where he found a new way of constructing them, thus extending Prüfer's theory to the
field of all
algebraic numbers, and clarified their relation to
p-adic numbers. In 1928 he published two additional papers continuing with these themes. The first dealt with
partitioning an
interval into
countably many
congruent subsets. It solved a problem of
Hugo Steinhaus asking whether an interval is \aleph_0-divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are \aleph_0-divisible by translations (i.e. that these intervals can be decomposed into \aleph_0 subsets that are congruent by translation). His next paper dealt with giving a
constructive proof without the
axiom of choice that 2^{\aleph_0}
algebraically independent reals exist. He proved that A_r = \textstyle\sum_{n=0}^{\infty} 2^{2^{[nr]}}\! \big/ \, 2^{2^{n^2}} are algebraically independent for r > 0. Consequently, there exists a perfect algebraically independent set of reals the size of the
continuum. Other minor results from his early career include a proof of a
maximum principle for the gradient of a minimizing function in the field of
calculus of variations,u: \mathbb{R}^n \rightarrow \mathbb{R} be a
Lipschitz function with constant K, and \Omega an open and bounded set in \mathbb{R}^n. If u is a minimum for F in Lip_K(\Omega), then \sup_{x \in \Omega, y \in \delta\Omega} \frac = \sup_{x \neq y \in \Omega} \frac (unnecessary detail for a minor result) --> and a small simplification of
Hermann Minkowski's theorem for linear forms in
geometric number theory. Later in his career together with
Pascual Jordan and
Eugene Wigner he wrote a foundational paper classifying all
finite-dimensional formally real Jordan algebras and discovering the
Albert algebras while attempting to look for a better
mathematical formalism for quantum theory. In 1936 he attempted to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras in a paper investigating the infinite-dimensional case; he planned to write at least one further paper on the topic but never did. Nevertheless, these axioms formed the basis for further investigations of algebraic quantum mechanics started by
Irving Segal. ==Physics==