### Summary

\$1. Foundations \$1.1. Counting numbers To count an unnumbered collection of items, single out one item and label it with the number 1 . Then for each next item increment this counting number with a unit 1 , and continue until the whole set is counted. Counting can be problematic in practice, but in theory the elements |No|1..| of a set of size n have a one-to-one correspondence (paired, indexed, named or counted) with the subset from 1 to n of the natural numbers. Use no numbers twice, and leave no numbers uncounted, so this subset is unique, no matter in what order the elements are counted. An empty set |{}| of size 0 has no elements, we count it without   unit one. Counting without halt |1...| equals the size 'omega' ω of the set of natural numbers (unlike counting without numbers). The assumption the whole ω of this set exists, is called the axiom of infinity. Even an unbounded number like ω can be counted upon in a next room, then counting ω rooms, etcetera, until another axiom knocks at the door... Mathematical expressions are essentially lines of text, compositions of signs in one discrete dimension, associated 'ltr', in left to right direction. Let uppercase letters from R to Z stand for text variables in rules, to express any permissible input sequence. Variables are possibly empty, but only if the context allows that. Write number variables or 'var' with a lowercase letter. For an empty 'var', we can put or add 0 in its place. In some array notations all entries must have value n>0 and empty positions and structures are removed first. While our notation for "Big numbers" evolves, smaller concepts appear first. We like to index these earlier constructs on the left, so that the structure of our array functions will expand to the right. Our output is always a string of units 1 with length x , which can be a natural number, some non-standard limit or a type of infinity. Arrays contain rows of signs that can be repeated, nested, renested, etc. These expanding structures equal the general recursive functions. We see only the "pretty Big numbers", most numbers are unpretty and remain hidden, given that computational resources are final. Take the universe mass of 2E93 standard kilogram photons, with a star spangled lifespan of 5E64 Planck times. In quantum superposition these represent 2^1E158 numbers, so most numbers smaller than 2^^6 will never be expressed. We can define a "Bigger number" or non-standard limit ψ , that lies right after any numbers expressible within our standard array system for Big numbers, given our physical resources. If we input this initial unit ψ in our system, the resulting output |ψ..| is definitively topped by the next Bigger number ψ_1 with index 1 . A succession of Bigger numbers ψ_k can well be expressed within the same standard array structure. Later such system revamps become enumerable in their own right. All Bigger numbers are still finite, but the same functionality applies to the ω -type numbers in the higher infinite. Just substitute the ordinal ω_k for ψ_k . In general, if a mathematical construction can be repeated, this construct in turn is countable by an index entry within a larger system. Note that numbers that enumerate sets, function as members of sets. This type of self-reference is a source of inconsistency, when Gödel numbering statements about the whole of arithmetic. Intuitively, to encode alphabets to maximally exploit signs is a perilous method, but does it help to get Big...?