Continuum (set theory) explained

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by

ak{c} .^{[1] [2]} Georg Cantor proved that the cardinality ak{c} is larger than the smallest infinity, namely, \aleph _{0} . He also proved that ak{c} is equal to 2 ^{\aleph _{0}} , the cardinality of the power set of the natural numbers.

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,

\aleph _{0} , or alternatively, that ak{c} = \aleph _{1} .

Linear continuum

See main article: Linear continuum. According to Raymond Wilder (1965), there are four axioms that make a set C and the relation < into a linear continuum:

• C is simply ordered with respect to <.
• If [''A,B''] is a cut of C, then either A has a last element or B has a first element. (compare Dedekind cut)
• There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (separability axiom)
• C has no first element and no last element. (Unboundedness axiom)

These axioms characterize the order type of the real number line.

See also

• Aleph null
• Suslin's problem
• Transfinite number

References

1. Web site: Weisstein. Eric W.. Continuum. 2020-08-12. mathworld.wolfram.com. en.
2. Web site: Transfinite number mathematics. 2020-08-12. Encyclopedia Britannica. en.

Bibliography

• Raymond L. Wilder (1965) The Foundations of Mathematics, 2nd ed., page 150, John Wiley & Sons.