In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

uncountable sets, that is, sets that are not countable; for example the set of the real numbers. Although the terms "countable" and "countably infinite" as defined...

infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its...

In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...

B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is: A = {x is an even integer larger than 1} B = {x is an odd integer...

(colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated...

the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accepts infinite sets, then there are different sizes...

number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite: { 1 , 2 , 3 ,...

one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than...

of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908....

of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for...

} where Z 0 {\displaystyle Z_{0}} is any infinite set and P {\displaystyle {\mathcal {P}}} is the power set operation. Moreover, one of Zermelo's axioms...

a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the...

countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real...

contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be...

quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite...

cardinality of set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number...

property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The cardinal number corresponding to a set A {\displaystyle...

particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced...

Cantor ternary set contains all points in the interval [ 0 , 1 ] {\displaystyle [0,1]} that are not deleted at any step in this infinite process. The same...

Look up infinite in Wiktionary, the free dictionary. Infinite may refer to: Infinite set, a set that is not a finite set Infinity, an abstract concept...

two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along...

The infinite monkey theorem states that a monkey hitting keys independently and at random on a typewriter keyboard for an infinite amount of time will...

infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, N 0 {\displaystyle \mathbb {N} _{0}}...

! {\displaystyle n!} . Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications...

denumerably infinite. This is not, in fact, a theorem of mathematics. But, according to Hartman, people are capable of a denumerably infinite set of predicates...

In set theory, an amorphous set is an infinite set which is not the disjoint union of two infinite subsets. Amorphous sets cannot exist if the axiom of...