Ivory: Zero, One, Many

natural

Wheels within wheels
In a spiral array
A pattern so grand
And complex
  • Rush, Natural Science

The next level from zero is natural, which contains all of the “counting” numbers: 0, 1, 2, 3, etc. There is rich structure in the natural numbers, but our requirement for levels to follow a subset relation constrains how much we’re able to represent.

Ivory exposes the following subsets as mezzanines within the overall natural level:

┌─────────┬───────────────────┐
│ boolean │    even-square    │
├─ ─ ─ ─ ─┴─ ─ ─┬─ ─ ─ ─ ─ ─ ─┤
│     square    │ doubly-even │
│               ├─ ─ ─ ─ ─ ─ ─┤
│               │     even    │
├─ ─ ─ ─ ─ ─ ─ ─┴─ ─ ─ ─ ─ ─ ─┤
│            natural          │
└─────────────────────────────┘

boolean

This is home to the number 1, which (along with 0 inherited from zero) forms a simple but important numerical system. The boolean numbers are closed under ×, max, min, gcd, lcm and / (excluding division by 0).

There are many operations which make sense for boolean but do not generalise well to other levels. For example, the xor operation is often seen as a form of addition (modulo 2), but that perspective doesn’t make much sense for supersets like rational. If we instead consider xor to be a specialised form of ≠ then it generalises naturally.

even-square

This subset contains even square numbers, of the form (^ (× 2 n) 2) for any natural n: this reduces to 0 in the case that (= n 0). The utility of this mezzanine is limited on its own, but its presence allows the square and even subsets to coexist.

square

This extends the contents of even-square to all square numbers of the form (^ n 2); with (= (^ 1 2) 1) inherited from boolean.

doubly-even

The doubly-even numbers numbers have the form (× 4 n) for some natural n, i.e. multiples of 4, or natural numbers with 4 as a factor. This is a strict extension of even-square, since:

(= (^ (× 2 n) 2)
   (× (× 2 n) (× 2 n))
   (× 2 2 n n)
   (× 4 n n)
   (× 4 (^ n 2)))

This set is closed under +, ×, ^, max, min, gcd, lcm, etc. and is useful in certain number-theoretic contexts.

even

The even numbers are multiples of 2, of the form (× 2 n) for some natural n. When n is itself even, the result is doubly-even; when n is doubly-even the result is triply-even, etc. This set is closed under +, ×, ^, max, min, gcd, lcm, etc.

Leftovers

The bottom of the natural level fills in any remaining gaps: i.e. odd numbers which are not square. Many of these remaining numbers will be odd primes, which would be useful to represent, but doesn’t fit as a mezzanine since it’s not a superset of zero.

Code

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