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A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity.. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that [1]
The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself. The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω .
The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the ...
Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set of natural numbers has an infimum, say .
Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbers less than a given natural number is computable. The subset of prime numbers is computable. A recursive language is a computable subset of a formal language. The set of Gödel numbers of arithmetic proofs described in Kurt Gödel's paper ...
As an example, let be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from together with the set of integers and the set of positive real numbers +, ordered by subset inclusion as above.
The set of natural numbers is a subset of , which in turn is a subset of the set of all rational numbers, itself a subset of the real numbers. [ a ] Like the set of natural numbers, the set of integers Z {\displaystyle \mathbb {Z} } is countably infinite .
First, define a function over the natural numbers (that is, over the finite ordinals) f : N → Power(Power(X)), so that for every natural number n, f(n) is the set of finite subsets of X of size n (i.e. that have a bijection with the finite ordinal n). f(n) is never empty, or otherwise X would be finite (as can be proven by induction on n).