Continuum (set theory)
The real numbers or their cardinality
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
is larger than the smallest infinity, namely,
. He also proved that
is equal to
, 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
,
, or alternatively, that
.
[1]
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
References
- 1 2 Weisstein, Eric W. "Continuum" . mathworld.wolfram.com . Retrieved 2020-08-12 .
- ↑ "Transfinite number | mathematics" . Encyclopedia Britannica . Retrieved 2020-08-12 .
Bibliography
- Raymond L. Wilder (1965) The Foundations of Mathematics , 2nd ed., page 150, John Wiley & Sons .
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