Update

number.md

@@ -127,6 +127,8 @@ Here patterns start to show, for example the level one of the tree are all prime
 There are different types of numbers, in mathematics we classify them into [sets](set.md) (if we further also consider the operations we can perform with numbers we also sort them into algebras and structures like [groups](group.md), [fields](field.md) or [rings](ring.md)). Though we can talk about finite sets of numbers perfectly well (e.g. [modulo](mod.md) arithmetic, [Boolean algebra](boolean_algebra.md) etc.), we are firstly considering [infinite](infinity.md) sets (curiously some of these infinite sets can still be considered "bigger" than other infinite sets, e.g. by certain logic there is more real numbers than rational numbers, i.e. "fractions"). Some of these sets are subsets of others, some overlap and so forth. Here are some notable number sets (note that a list can potentially not capture all relationships between the sets):
 
 - **all**: Anything conceivable as a number, even by stretch. E.g. [zero](zero.md), minus [infinity](infinity.md) or aleph one.
+  - **[noncomputable](computability.md)**: Cannot be computed, i.e. any such number has no [Turing machine](turing_machine.md) which when passed *N* on input would output *N*th digit of the number in finite time. E.g. Chaitin's constant (probability that a randomly generated program will halt).
+  - **[unknowable](knowability.md)**: Cannot be known for some reason, e.g. being non-computable or requiring more energy for their computation than will ever be present in our [Universe](universe.md).
   - **[transfinite (infinite) numbers](transfinite_number.md)**: Numbers that are in a sense "infinite", used to compare objects that are infinite in size (e.g. number sets themselves). E.g. omega, beth two or aleph one. 
   - **[surreal numbers](surreal_number.md)**, **\*R: hyperreal numbers**, **superreal numbers**, ...: Various extensions of real numbers, include also infinitesimals and some transfinite numbers.
     - **[infinitesimals](infinitesimal.md)**: Are closer to zero than any real number without actually being zero, i.e. "infinitely small" numbers, play big role in [calculus](calculus.md). E.g. 0.000...1 (with infinitely many 0 digits before the 1).
@@ -163,6 +165,8 @@ Of course there are countless other number sets, especially those induced by var
 
 There is a famous [encyclopedia](encyclopedia.md) of integer sequences at https://oeis.org/, made by number theorists -- it's quite [minimalist](minimalism.md), now also [free licensed](free_culture.md) (used to be [proprietary](proprietary.md), they seem to enjoy license hopping). At the moment it contains more than 370000 sequences; by browsing it you can get a glimpse of how deep the study of numbers goes. These people are also [funny](fun.md), they give numbers entertaining names like *happy numbers* (adding its squared digits eventually gives 1), *polite numbers*, *friendly numbers*, *cake numbers*, *lucky numbers* or *weird numbers*.
 
+**Some numbers cannot be computed**, i.e. there exist [noncomputable](computability.md) numbers. This follows from the existence of noncomputable functions (such as that representing the [halting problem](halting_problem.md)). For example let's say we have a real number *x*, written in binary as *0. d0 d1 d2 d3 ...*, where *dn* is *n*th digit (1 or 0) after the radix point. We can define the number so that *dn* is 1 if and only if a [Turing machine](turing_machine.md) represented by number *n* halts. Number *x* is noncomputable because to compute the digits to any arbitrary precision would require being able to solve the unsolvable halting problem.
+
 **All [natural numbers](natural_number.md) are [interesting](interesting.md)**: there is a [fun](fun.md) [proof](proof.md) by contradiction of this. Suppose there exists a set of uninteresting numbers which is a subset of natural numbers; then the smallest of these numbers is interesting by being the smallest uninteresting number -- we've arrived at contradiction, therefore a set of uninteresting numbers cannot exist.
 
 TODO: what is the best number? maybe top 10? would 10 be in top 10?