home | index | units | counting | geometry | algebra | trigonometry & functions | calculus
analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Subreal Numbers
The Field of Computable Numbers

 Alan Turing
Science is a differential equation.
Religion is a boundary condition.

 Alan Turing  (1912-1954)
 border  border

On this site, see also:

Related Links (Outside this Site)

On Computable Numbers  by  Alan M. Turing  (1936).
How Quantum Computers Work
There's Plenty of Room at the Bottom  by  Richard P. Feynman (Dec. 1959).
Wikipedia :   Quantum computing   |   Quantum information   |   Qubit   |   Shor's algorithm

Subreal Numbers   =   Computable Numbers

 Gerard Michon (2016-07-10  4:45 PDT)   Subreal Numbers
Real numbers specified by  converging  computer programs.

In a two-tape Turing machine we dedicate the first tape strictly to successive approximations of numerical result  (expressed in some kind of binary format.  Such a Turing machine is said to be  converging  if the sequence of numbers appearing on the first tape as time goes on is a  Cauchy sequence.

 Come back later, we're
 still working on this one...

Subreal numbers are a countable extension of rational and algebraic numbers which contains all transcendental numbers which can be named in a constructive way.

Wikipedia :  

(2016-07-10)   Subreal Numbers are Countable
The reason is that there are countably many computer programs.

 Come back later, we're
 still working on this one...

Countable sets

(2016-07-14)   The set of subreal numbers is not complete
Some Cauchy sequences of  subreal numbers  have no subreal limit.

Almost all real numbers are not subreal  (since real numbers are uncountable whereas subreal numbers are countable).

Let  x  be a real number which is not subreal.  It is the limit of a sequence of rationals.  That sequence is a  Cauchy sequence  of subreals which doesn't have a subreal limit.

Complete sets

(2016-07-12)   Equality of Subreal Numbers
The equivalence of two subreal representations isn't a computable property.

 Come back later, we're
 still working on this one...

Halting problem

visits since July 10, 2016
 (c) Copyright 2000-2018, Gerard P. Michon, Ph.D.