In the beginning was the primordial distinction between darkness and light, nothing and something, before and after. Its name was the Unit, its faces named zero and one.
And with the Unit came the Successor, the primordial operation, for what is cleaved may ever be joined. From one comes two, from two comes three, and ever on until forever and always, with each successor given a name as a number.
And with these numbers came a set comprising them, and the set was Natural and good, and from it came many wondrous things.
For from repetition of the Successor came Addition, and the set was closed under Addition, and the Counter saw that this was good.
And from repetition of Addition came Multiplication, and the set was closed under Multiplication, and the Counter saw that this was good.
And from repetition of Multiplication came Exponentiation, and the set was closed under Exponentiation, and the Counter saw that this was good.
But if a thing can be done it can be undone. What is given can be taken away. If there is Addition there must be Subtraction. A shadow fell over the face of the Counter for under Subtraction the set was not closed.
Yet the set of Natural numbers had its closure under Subtraction, and this closure was another set named Integers, and the Counter saw that the Integers were good.
But if Addition of a Natural number has an inverse, so too must Multiplication by a Natural number, and this inverse was Division. A shadow passed again across the face of the Counter for under Division by a Natural number the set of Integers was not closed.
Yet the set of Integers had its closure under Division by a Natural number, and the closure was another set named Rational numbers, and the Counter saw that the Rational numbers were good and rejoiced at their scope, for between any two Rational numbers was an infinity of other Rational numbers, each with its own name.
But if Addition and Multiplication by Natural numbers have inverses, so too must Exponentiation, and indeed, so must the combination of Addition, Multiplication, and Exponentiation in a polynomial with Integer coefficients, and this inverse was the finding of Roots. A shadow passed again across the face of the Counter for almost never were the Roots of polynomials Rational.
Yet the Roots of polynomials with Integer coefficients gave rise to a new set, the set of Algebraic numbers, and the Counter saw that the Algebraic numbers were good and rejoiced at their scope, for the Algebraic numbers have complexities that delight and amaze, and each has its own name.
And yet.
Almost no number is Algebraic.
Almost every number belongs instead to a Transcendental realm where there are many terrors and almost nothing can be named.
And with the Unit came the Successor, the primordial operation, for what is cleaved may ever be joined. From one comes two, from two comes three, and ever on until forever and always, with each successor given a name as a number.
And with these numbers came a set comprising them, and the set was Natural and good, and from it came many wondrous things.
For from repetition of the Successor came Addition, and the set was closed under Addition, and the Counter saw that this was good.
And from repetition of Addition came Multiplication, and the set was closed under Multiplication, and the Counter saw that this was good.
And from repetition of Multiplication came Exponentiation, and the set was closed under Exponentiation, and the Counter saw that this was good.
But if a thing can be done it can be undone. What is given can be taken away. If there is Addition there must be Subtraction. A shadow fell over the face of the Counter for under Subtraction the set was not closed.
Yet the set of Natural numbers had its closure under Subtraction, and this closure was another set named Integers, and the Counter saw that the Integers were good.
But if Addition of a Natural number has an inverse, so too must Multiplication by a Natural number, and this inverse was Division. A shadow passed again across the face of the Counter for under Division by a Natural number the set of Integers was not closed.
Yet the set of Integers had its closure under Division by a Natural number, and the closure was another set named Rational numbers, and the Counter saw that the Rational numbers were good and rejoiced at their scope, for between any two Rational numbers was an infinity of other Rational numbers, each with its own name.
But if Addition and Multiplication by Natural numbers have inverses, so too must Exponentiation, and indeed, so must the combination of Addition, Multiplication, and Exponentiation in a polynomial with Integer coefficients, and this inverse was the finding of Roots. A shadow passed again across the face of the Counter for almost never were the Roots of polynomials Rational.
Yet the Roots of polynomials with Integer coefficients gave rise to a new set, the set of Algebraic numbers, and the Counter saw that the Algebraic numbers were good and rejoiced at their scope, for the Algebraic numbers have complexities that delight and amaze, and each has its own name.
And yet.
Almost no number is Algebraic.
Almost every number belongs instead to a Transcendental realm where there are many terrors and almost nothing can be named.