#
*“On the shoulders of giant numbers”*

http://www.iteror.org/big/book/ch2/ch2_3.html
big**Ψ**

http://www.iteror.org/big/book/ch2/ch2_3.html

## Ψ.2. Up to the Ackermann numbers

#####
chapter 2.3, *edit 0.2.9*

published: *2011-01-07*

updated: *2011-12-30*

### # 2.3. Deeper than counting

#### §2.3.1. Transcendence of numbers

*Leonhard Euler*

Berlin 1753

Berlin 1753

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 `a`

and an unknown variable _{i}`x`

.

The right
meta-statement
repeats the left, growing the `x**n`

exponents.

_{i}*x..).:.

`{*x#i #n}`= a

_{0}

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.

**½**)^2 = π = 11*(11**-)!**11

*= (2*½!)^2*

**= ½!*(-½)!*-½***= 3.141592653589..*
Why for other fractions like
`Γ(`

and **1/3**)`Γ(`

there is no known definition, as Julian Havil notes in his giddy monograph
"Gamma",
we hope you can tell.**1/4**)

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.

`where`

`î = -**11**-`

*= -1^½*

`and`

ê = 1(a**-)**a

`{a↑ω}`

*= lim (1+1/a)^a*

`{a↑ω}`= 2.718281828459..
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

*Gregory Chaitin*

Buenos Aires 2009

Buenos Aires 2009

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!