Editing Computable number (section)

===Other properties===
The computable real numbers do not share all the properties of the real numbers used in analysis. For example, the least upper bound of a bounded increasing computable sequence of computable real numbers need not be a computable real number.{{sfnp|Bridges|Richman|1987|p=58}} A sequence with this property is known as a [[Specker sequence]], as the first construction is due to [[Ernst Specker]] in 1949.{{sfnp|Specker|1949}} Despite the existence of counterexamples such as these, parts of calculus and real analysis can be developed in the field of computable numbers, leading to the study of [[computable analysis]].

Every computable number is [[arithmetically definable number|arithmetically definable]], but not vice versa. There are many arithmetically definable, noncomputable real numbers, including:
*any number that encodes the solution of the [[halting problem]] (or any other [[undecidable problem]]) according to a chosen encoding scheme.
*[[Chaitin's constant]], <math>\Omega</math>, which is a type of real number that is [[Turing degree|Turing equivalent]] to the halting problem.
Both of these examples in fact define an infinite set of definable, uncomputable numbers, one for each [[universal Turing machine]].
A real number is computable if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable.

The set of computable real numbers (as well as every countable, [[densely ordered]] subset of computable reals without ends) is [[order-isomorphic]] to the set of rational numbers.