For some reason I have managed to not encounter the notion of computable number (see this) as opposed to that of noncomputable number (see this). The reason is perhaps that I have been too lazy to take computationalism seriously enough.
Computable real number is a number, which can be produced to an arbitrary accuracy by a Turing computer, which by definition has a finite number of internal states, has input which is natural number and produces output which is natural numbers. Turing computer computes values of a function from natural numbers to itself by applying a recursive algorithm.
The following three formal definitions of the notion are equivalent.
 The number a is computable, if it can be expressed in terms of a computable function n→ f(n) from natural numbers to natural numbers characterized by the property
(f(n)1)/n≤ a≤ (f(n)+1)/n .
For rational a=q, f(n)= nq satisfies the conditions. Note that this definition does not work for padic numbers since they are not wellordered.
 The number a is computable if for an arbitrarily small rational number ε there exists a computable function producing a rational number r satisfying rx≤ ε. This definition works also for padic numbers since it involves only the padic norm which has values which are powers of p and is therefore real valued.
 The number a is computable if there exists a computable sequence of rational numbers r_{i} converging to a such that ar_{i} ≤ 2^{i} holds true. This definition works also for 2adic numbers and its variant obtained by replacing 2 with the padic prime p makes sense for padic numbers.
The set R_{c} of computable real numbers and the padic counterparts Q_{p,c} of R_{c}, have highly interesting properties.
 R_{c} is enumerable and therefore can be mapped to a subset of rationals: even the ordering can be preserved. Also Q_{p,c} is enumerable but now one cannot speak of ordering. As a consequence, most real (padic) numbers are noncomputable. Note that the pinary expansion of a rational is periodic after some pinary digit. For a padic transcendental this is not the case.
 Algebraic numbers are computable so that one can regard R_{c} as a kind of completion of algebraic numbers obtained by adding computable reals. For instance, π and e are computable. 2π can be computed by replacing the unit circle with a regular polygon with n sides and estimating the length as nL_{n}. L_{n} the length of the side. e can be computed from the standard formula. Interestingly, e^{p} is an ordinary padic number. An interesting question is whether there are other similar numbers. Certainly many algebraic numbers correspond to ordinary padic numbers.
 R_{c} (Q_{p,c}) is a number field since the arithmetic binary operations +, , ×, / are computable. Also differential and integral calculus can be constructed. The calculation of a derivative as a limit can be carried out by restricting the consideration to computable reals and there is always a computable real between two computable reals. Also Riemann sum can be evaluated as a limit involving only computable reals.
 An interesting distinction between real and padic numbers is that in the sum of real numbers the sum of arbitrarily high digits can affect even all lower digits so that it requires computational work to predict the outcome. For padic numbers memory digits affect only the higher digits. This is why padic numbers are tailor made for computational purposes. Canonical identification ∑ x_{n}p^{n} → ∑ x_{n}p^{n} used in padic mass calculations to map padic mass squared to its real counterpart (see this) maps padics to reals in a continuous manner. For integers this corresponds is 2to1 due to the fact that the padic numbers 1= (p1)/(1p) and 1/p are mapped to p.
 For computable numbers, one cannot define the relation =. One can only define equality in some resolution ε. The category theoretical view about equality is also effective and conforms with the physical view.
Also the relations ≤ and ≥ fail to have computable counterparts since only the absolute value xy can appear in the definition and one loses the information about the wellordered nature of reals. For padic numbers there is no wellordering so that nothing is lost. A restriction to nonequal pairs however makes order relation computable. For padic numbers the same is true.
 Computable number is obviously definable but there are also definanable numbers, which are not computable. Examples are Gödel numbers in a given coding scheme for statements, which are true but not provable. More generally, the Gödel numbers coding for undecidable problems such as the halting problem are uncomputable natural numbers in a given coding scheme. Chaitin's constant, which gives the probability that random Turing computation halts, is a noncomputable but definable real number.
 Computable numbers are arithmetic numbers, which are numbers definable in terms of first order logic using Peano's axioms. First order logic does not allow statements about statements and one has an entire hierarchy of statements about... about statements. The hierarchy of infinite primes defines an analogous hierarchy in the TGD framework and is formally similar to a hierarchy of second quantizations (see this).
See the chapter Evolution of Ideas about Hyperfinite Factors in TGD or the article MIP*= RE: What could this mean physically?.
