bigΨ.I. Number operations

bigΨ · part 1
edit 0.2.9

§I.0. Prelude

Finite beings can only build finite systems. In mathematics we often assume otherwise, but to reasonably refer to infinite items we need to keep our conversation finite, else nobody could hear us out.
To paraphrase the great Dedekind – is he listening? – a mathematical system is Big when it is similar to a previous part of itself. Something as simple as adding an extra counter will expand upon an algorithm for Big numbers to create mind-bogglingly Bigger ones…
In our own number constructions the initial operators are made to nearly match the later operatorial functions. As often, this requires a jump out of the box and into the ocean. When jump and match satisfy, previous concepts are ready to be counted in a new and higher, more general system.

Each number finds expression in some system of words and belongs to one of six different temperaments – four of man and two of the divine.

Address numbers
to point to quantities or chances in the physical world
Big numbers
expressions reduced by an algorithm to a series of units
Continuous numbers
the algorithm never halts, projected to the real or infinite
Diametric numbers
records that mark the jump to hitherto disparate levels
Endless numbers
various types of infinity derived by infinite functions
Fantastic numbers
these can only be understood by a philosopher

This on-line book is about the quest for the Biggest numbers and their construction tools – the operatorials bigO. Functions which produce theoretically countable, but practically uncountable natural numbers 1..
The sign ψ (psi) was used by the recursion theorist Wilhelm Ackermann to mark a higher type of function, iterating over a known family of functions. Therefore our prodigious paper is delivered with the tantalizing title bigΨ.
Also this ψ we conjure is kind of a running gag throughout the book – a number too ... to fantasize about.

We wish all readers a giant leap and an enchanting journey, into a boundless land of numbers untold of before!

Diophantine riddle

Asamkhyeya dâsa takes the liberty to start with a numerological riddle:
The background-coloured digits in the following number point to a diophantine equation. What is its formula? {click→}

Prelude> 6^^3 = 6^6^6
Average number of solutions per number base - for digit places 2,3,0,4,5

{hint→} 666 does seem to be the last solution in number base 10, and there are probably no diophantine solutions with more than 5 digits in any other radix. But beware! As foretold in a secret Sasataansanic saga, there might be a Big number solution with 6 digits in the true number basket of Kabbalah, where Tetragrammaton and Beast Nº.2*333 rest asleep in a sea of glass mixed with fire – humånity hanging upside down from Yggdrasil, the mångrove Church of Life, like a Lemån — whose diophantine equation is… (a+b)*(a-b)+a = b*(r^p-1)/(r-1) {b<r}