In the mathematical field of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, the continuum means the
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
, or the corresponding (infinite)
cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
, denoted by
.
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
proved that the cardinality
is larger than the smallest infinity, namely,
. He also proved that
is equal to
, the cardinality of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of the
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
.
The ''
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \ ...
'' is the
size
Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or volume. Length can be generalized ...
of the set of real numbers. The
continuum hypothesis is sometimes stated by saying that no
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
lies between that of the continuum and that of the
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
,
, or alternatively, that
.
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
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
)
* There exists a non-empty,
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
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
In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of i ...
References
Bibliography
* Raymond L. Wilder (1965) ''The Foundations of Mathematics'', 2nd ed., page 150,
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...
.
Set theory
Infinity
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