“On the shoulders of giant numbers”
 http://www.allergrootste.com/big/book/ch2/ch2_3.html  bigΨ

§2.3.1. Transcendence of numbers

Transcendental numbers are defined to be those real numbers, which are not algebraic. Algebraic numbers give the solutions (roots x) to polynomial equations, which are composed of addition, repeated multiplication, constants ai and an unknown variable x.
The right meta-statement repeats the left, growing the x**n exponents.

All polynomials with exponents 0<n<5 are solvable by radicals, but most higher polynomials n≥5 cannot be simplified, as Galois (1832) showed. For example, the quintic 2×q^5 + 5 = 10×q is not solvable (Abel 1824), though it has 5 roots.
While the defining polynomial of an algebraic number is enumerable, often that equation cannot be reduced to a simpler expression. Of course the unknown x can always be approximated by iterating over some continuous algorithm…

Generally, for any class of functions there are numbers which cannot be expressed by recursion alone, but can be created by mixing and inverting existing operations. And then there are the higher transcendent numbers that escape any such definition and can only be approached by adopting new rules or by infinite iteration of existing rules.

With the expansion of the function context many formerly transcendental numbers run the chance of finding expression. When a number can be defined by a operation of superpower type, it is not transcendental any more in the context of primitive recursive functions.
This may easily happen to pi π, which lost its functional transcendence in the year 1730, when Leonhard Euler conjured pi from his Gamma function, essentially a hybrid of the factorial and exponential functions.

Why for other fractions like Γ(1/3) and Γ(1/4) there is no known definition, as Julian Havil notes in his giddy monograph "Gamma", we hope you can tell.

In the context of a factorial algorithm halves n*(11**-) are still transcendent constants. We like to use the term transcendent for numbers which cannot be defined within the elementary class of functions under inverse composition.
Compare the algebraic polynomials, which cover the inversion of the function class up to multiplication totally.

Together with π a host of connected constants may fall from transcendental grace – foremost ê the base of natural logarithms, which is connected to π via Euler's famous theorem.

The ultimate criterion is countability in the sense of Cantor's countable sets. Because enumerable functions with countable parameter values always create enumerable numbers, the natural numbers can be used to list all those numbers which are defined by hybrid functions – finite combinations of finite sets of non-transcendental functions (strictly countable by a single natural number iterator).
This applies to the number π above, where it is defined by the value ½ in Euler's gamma function – essentially a hybrid of two operatorials (even though π can only be generated by an infinite series of one of these algorithms).

§2.3.2. A realm of random reals

Of course there are many algorithms that will approximate pi ever more closely and that can easily be implemented in a short computer program. Such a list of commands can be counted, and in practice always expresses some binary number, making pi enumerable in many, many different ways.
Our recursive operatorials travel well beyond the algebraic realm, so the inverted transcendent numbers that escape the net are a far more elusive set than the common transcendental. But both types of numbers are still part of the set of countable reals – they can be finitely generated.

Random real numbers are supposed to be based on ω alone, for they can only be produced by infinite iteration, and not by finite combinations of any mathematical algorithm.
We assume the majority of all numbers to be really uncountably deep, but appear helpless in constructing one…

After that last assumption was put online we were lucky to come across a book by Gregory Chaitin written for amateur nerds just like us. In the book (and in many of his lectures available on YouTube) Chaitin explains his theorem that the halting probability he calls Omega is just the right example of such a random real.
He would rather call his algorithmically incompressible numbers irreducibles, but keeps the term random for historical reasons. Just as well, because there can theoretically be a higher class Turing machine, which predict if previous machines' programs will halt or not, so irreducible is a relative concept in the hyperarithmetical context.

When we allow the complexity of our algorithms to move beyond exponentiation, whether any real units can ever be established as not reducible to algorithmic combinations of the primitive units 1 and - remains unclear. At the moment we don't even know if the super-root a///b of tetration can sufficiently be manipulated mathematically to reveal such discrete recipes for hitherto transcendent numbers.
There's always hope to express random real constants by clever use of finite recursive algorithms in the future.

Yet the more advanced our mathematical functions, the more elusive the transcendent numbers that can be generated by application of these functions to infinite series.
So if transcendence isn't here to stay, it seems the continuous generation of algorithms and quasi-random numbers that become ever more real will never stop …flowing in wider and wilder mathematical streams towards finite solutions on next higher levels. A truly infinite and uncomputable phenomenon!