Continuum
Robert P. Munafo, 2002 May 7.
Roughly speaking, a continuum is a type of connected set that can be divided into smaller and smaller pieces infinitely many times and any such pieces, if they are obtained after a finite number of steps, have the same order as the original set. Examples of continuums are a straight line, a plane, a circle, a disc, the set of real numbers, and the set of complex numbers. It can be shown that all continuums have the same order.
The term "continuum" is also used to refer to an infinite quantity, equal to the order of any continuum. In other words, "continuum" can be used to mean "the number of points on a line" instead of meaning "a line".
It was proven by Cantor in the late 1800's that the power set of the integers (or of any other set of order aleph0) has the same order as the set of reals or any other continuum.
The Continuum Hypothesis states that there is no infinity between Aleph-0 and the order of a continuum, which would mean that the order of the continuum is Aleph-1. Although it is called a "hypothesis", the truth or falsehood of the Continuum Hypothesis has been shown (by Godel and Paul Cohen) to be an axiomatic issue, like the parallel postulate in geometry, if one is working within Zermelo-Fraenkel set theory with the Axiom of Choice. Different systems of set theory and of transfinite quantities, each consistent within itself, can be constructed on the basis of whether or not the Continuum Hypothesis is taken to be true, false, or undetermined.
The Generalized Continuum Hypothesis states that if N is the order of set S and M is the order of the power set of S, there exist no sets that have more elements than N and fewer elements than M. This would mean that the order of a power set of a continuum is Aleph-2.
By the late 1990's, the community of specialists in set theory had leaned in favor of taking the continuum hypothesis as false. If so, then the continuum is greater than aleph1, and anything else (such as whether it is equal to aleph2) is indeterminate.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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