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...

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...

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...

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...

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...

century by Georg Cantor with his theory of infinite sets, and was later formalized into Zermelo–Fraenkel set theory. This theory, which is presently commonly...

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

or even other sets. A set may be finite or infinite, depending whether the number of its elements is finite or not. There is a unique set with no elements...

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....

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

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 ,...

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...

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...

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...

} 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...

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...

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}}...

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...

are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the...

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...

ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). Every net valued in a singleton...

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...

mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set D of natural numbers...

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...

science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here...

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

count from zero. For an infinite set, the order type determines the cardinality, but not conversely: sets of a particular infinite cardinality can have well-orders...