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In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967. [ 12 ]
The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the ...
The 3-adic integers, with selected corresponding characters on their Pontryagin dual group. In number theory, given a prime number p, [note 1] the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime ...
In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . [ 1 ] [ 2 ] Georg Cantor proved that the cardinality c {\displaystyle {\mathfrak {c}}} is larger than the smallest infinity, namely, ℵ 0 {\displaystyle \aleph _{0}} .
Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, 21, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, [a] 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.
The cardinality of the set of real numbers (cardinality of the continuum) is 2. It cannot be determined from ZFC ( Zermelo–Fraenkel set theory augmented with the axiom of choice ) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity
Cantor's diagonal argument shows that is strictly greater than , but it does not specify whether it is the least cardinal greater than (that is, ).Indeed the assumption that = is the well-known Continuum Hypothesis, which was shown to be consistent with the standard ZFC axioms for set theory by Kurt Gödel and to be independent of it by Paul Cohen.
The continuum hypothesis is equivalent to =. The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., = for all ordinals .