If most numbers are uncomputable, in what sense do they exist?

Do Uncomputable numbers exist?

You can show that there are uncountably many uncomputable real numbers more directly – there are only countably many Turing machines, so there are only countably many computable numbers, so there are uncountably many remaining real numbers that aren’t computable.

Are most numbers Uncomputable?

It turns out that almost every number is uncomputable. To understand this we first introduce the concept of a set being countable. A set is called countable if it can be put in one-to-one coorespondence with the integers. For instance, rational numbers are countable.

Why are Uncomputable numbers important?


Them this argument works for functions as well the functions that involve most things that we care about have a cardinality larger than the natural numbers.

Are all numbers computable?

Real numbers used in any explicit way in traditional mathematics are always computable in this sense. But as Turing pointed out, the overwhelming majority of all possible real numbers are not computable. For certainly there can be no more computable real numbers than there are possible Turing machines.

What is Uncomputable?

uncomputable (not comparable) Not computable; that cannot be computed.

Are all rational numbers constructible?

All rational numbers are constructible, and all constructible numbers are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins 1996, p. 136).

Are computable numbers transcendental?

Yes, every incomputable number is transcendental, or, differently said, every algebraic number is computable. (Because it is possible to compute an arbitrary close rational approximation to every algebraic number).

Is Uncomputable a word?

Uncomputable definition



Not computable; that cannot be computed.

Are integers computable?

No, there is not. If you consider an integer with the usual meaning, it is finite information. All finite information can be computed. For example, you can build a machine M and the proposition “M halts on input 0” can be impossible to prove in any known usual theory.

How do you prove something is Uncomputable?

Good yes the most natural way to prove something is computable is a construction if you can show how to make a machine that computes it there's an implied.

What makes a function computable?

Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output.

Why is busy beaver Uncomputable?

The answer is that the Busy Beaver numbers are co-recursively enumerable. This means that there is an algorithm that takes in a number N and returns False if N is not a Busy Beaver number, and runs forever otherwise.

Are complex numbers constructible?

A complex number is constructible if and only if it is algebraic and the field generated by its conjugates is a finite extension of Q whose degree is a power of 2. The remaining part is usually proved using Galois theory.

Are all algebraic numbers constructible?

Not all algebraic numbers are constructible. For example, the roots of a simple third degree polynomial equation x³ – 2 = 0 are not constructible. (It was proved by Gauss that to be constructible an algebraic number needs to be a root of an integer polynomial of degree which is a power of 2 and no less.)

Are the constructible numbers a field?

Algebraic properties



The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. Thus, the constructible numbers (defined in any of the above ways) form a field.

Are transcendental numbers constructible?

Computable Numbers. Crucially, transcendental numbers are not constructible geometrically nor algebraically

Which angle is constructible?

It means an angle is constructible if and only if its order is either a power of two, or a power of two times a set of Fermat primes. For example, 10 = 2*5, and 2 is a power of two and 5 is a fermat prime, thus you can make an angle of 360/10 = 36 degrees.

What are constructive numbers?

Numbers that follow each other continuously in the order from smallest to largest are called consecutive numbers. For example: 1, 2, 3, 4, 5, 6, and so on are consecutive numbers.

What are successive numbers?

One after the other. Examples: Monday, Tuesday and Wednesday are successive days. 5 and 6 are successive whole numbers. 5 and 7 are successive odd numbers.

What is consecutive composite numbers?

Consecutive composite number means numbers between them that have no prime numbers.