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In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model
In fact, the standard ordering on the reals, extending the ordering of the rational numbers, is not necessarily decidable either. Neither are most properties of interesting classes of functions decidable, by Rice's theorem, i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail to be countable. The ...
The set of all rational numbers is countable, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a square lattice as in a Cartesian coordinate system, such that any grid point corresponds to a rational number. This method, however ...
Within the rational numbers, certain subsets are also countable, unbounded, and dense. The rational numbers in the open unit interval are an example. Another example is the set of dyadic rational numbers, the numbers that can be expressed as a fraction with an integer numerator and a power of two as the denominator. By Cantor's isomorphism ...
The set of rationals is an F σ set in . More generally, any countable set in a T 1 space is an F σ set, because every singleton { x } {\displaystyle \{x\}} is closed. The set R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} } of irrationals is not an F σ set.
Firstly, the order type of the set of natural numbers is ω. Any other model of Peano arithmetic, that is any non-standard model, starts with a segment isomorphic to ω but then adds extra numbers. For example, any countable such model has order type ω + (ω* + ω) ⋅ η. Secondly, consider the set V of even ordinals less than ω ⋅ 2 + 7:
Intuitively however, the set of rational numbers is a "small" set, as it is countable, and it should have "size" zero. That is indeed true, but only if one replaces the Jordan measure with the Lebesgue measure. The Lebesgue measure of a set is the same as its Jordan measure as long as that set has a Jordan measure.