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The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved by Andreas Blass and Saharon Shelah in 1987. [ 22 ] Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ 1 , elements" is independent of ZFC.
Group (mathematics) Halting problem. insolubility of the halting problem; Harmonic series (mathematics) divergence of the (standard) harmonic series; Highly composite number; Area of hyperbolic sector, basis of hyperbolic angle; Infinite series. convergence of the geometric series with first term 1 and ratio 1/2; Integer partition; Irrational ...
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.
For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using ...
The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, [2] [3] [4] along with the accepted rules of inference.
In the field of geometric topology, a two-dimensional sphere is characterized by the fact that it is the only closed and simply-connected two-dimensional surface. In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as the Poincaré conjecture, the precise ...
One of many examples from algebraic geometry in the first half of the 20th century: Severi (1946) claimed that a degree-n surface in 3-dimensional projective space has at most (n+2 3 )−4 nodes, B. Segre pointed out that this was wrong; for example, for degree 6 the maximum number of nodes is 65, achieved by the Barth sextic , which is more ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.