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Linear continuum, any ordered set that shares certain properties of the real line; Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real ...
The continuum hypothesis was advanced by Georg Cantor in 1878, [1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being ...
An abelian group with Ext 1 (A, Z) = 0 is called a Whitehead group; MA + ¬CH proves the existence of a non-free Whitehead group, while V = L proves that all Whitehead groups are free. In one of the earliest applications of proper forcing, Shelah constructed a model of ZFC + CH in which there is a non-free Whitehead group. [12] [13]
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .
This is a list of notable theorems.Lists of theorems and similar statements include: List of algebras; List of algorithms; List of axioms; List of conjectures
Continuum theory of specific heats of solids, see Debye model; Triune continuum, trinity of continual representations in general system modeling defined in the theory of triune continuum, used in the triune continuum paradigm; Continuous spectrum, referred to simply as the continuum in contrast to discrete spectral lines
The continuum hypothesis asserts that is also the second aleph number, . [2] In other words, the continuum hypothesis states that there is no set A {\displaystyle A} whose cardinality lies strictly between ℵ 0 {\displaystyle \aleph _{0}} and c {\displaystyle {\mathfrak {c}}}
Indeed, they had so much power available to them that two thirds was attenuated away to prevent damage to the fibre. The 50 kW coupled into the fibre emerged as a 12 kW continuum . Stokes lines were clearly visible up to 1.3 μm, at which point the continuum began to smooth out, except for a large loss due to water absorption at 1.38 μm.