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The power set of an infinite set is infinite. [3] Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The ...
The set of real algebraic numbers is countably infinite (assign to each formula its Gödel number.) So the cardinality of the real algebraic numbers is ℵ 0 {\displaystyle \aleph _{0}} . Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is R {\displaystyle \mathbb {R} } .
Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number. In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. [1]
Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.
From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing. [7]
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 infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets.
However, when dealing with infinite sets, it is essential to distinguish between the two, since the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers , while the size aspect is generalized by the cardinal numbers described here.