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

In a two-tape Turing machine endowed with an extra binary flag,  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  (when the flag is set)  is a  Cauchy sequence.

That  validity  flag isn't absolutely necessary, but it makes the details of bitwise updates to the first tape utterly irrelevant.

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

Computable numbers   |   Chaitin's constant   |   Gregory Chaitin (1947-)

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

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 thing.

Halting problem