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Zermelo—Fraenkel Problem solution paper thesis theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but they did not explicitly exclude the possibility of the existence of a set that belongs to itself.

In his doctoral thesis ofvon Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.

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 a method of demonstration, called the method of inner modelswhich later became an essential instrument in set theory.

Under the Zermelo—Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves.

In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set. 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.

Moreover, every consistent extension of these systems would necessarily remain incomplete.

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Von Neumann algebra Von Neumann introduced the study of rings of operators, through the von Neumann algebras. Murrayon the general study of factors classification of von Neumann algebras.

The six major papers in which he developed that theory between and "rank among the masterpieces of analysis in the twentieth century".

Lifting theory In measure theorythe "problem of measure" for an n-dimensional Euclidean space Rn may be stated as: 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.

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.

In mathematics, continuous geometry is a substitute of complex projective geometrywhere instead of the dimension of a subspace being in a discrete set 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 integersthe 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.

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.

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John von Neumann - Wikipedia