Definable number

In Data-processing and Algorithmic, a definable real number is a real has for which there exists a φ proposal that one can write within the framework of the Set theory comprising a free Variable, such as has is single reality checking φ (a).

It is shown that the whole of definable realities is a countable body containing the whole of all the calculable numbers. It includes any reality which one is able to express (0, 1, π, E, $\ zeta (sin (log (\ pi)))$ ).

There exist noncalculable definable realities, one of most famous being the constant Oméga of Chaitin.

Example of definable numbers

Let us note here $\ mathbb T$ (in homage to Alan Turing; attention, this notation is not official, it is introduced to clarify the presentation) the body of the calculable numbers. It is known that the definable numbers form a surcorps $\ mathbb T$ and include Ω (Oméga of Chaitin, referred to above). $\ mathbb T (\ Omega)$ is thus an example of subfield of the body of the definable numbers (the whole of the values taken by the rational fractions with calculable coefficients in Ω).

It is noticed that the body of the definable numbers is thus " much more grand" that the body of the calculable numbers to the direction which the definable numbers include a $\ mathbb T$ - vector Space of infinite size. On the other hand, there is " autant" definable numbers that calculable numbers with the direction which these bodies are both in Bijection with $\ mathbb N$ .

What brings to the rather unexpected result: there is " infinitely plus" numbers in the triadic Ensemble of Cantor, however so thin, that there do not exist numbers which we are able to seize.

Definable complex number

By extension, one calls definable complex number a complex number of which the parts real and imaginary are simultaneously definable.

External bonds

the hidden side of the numbers

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