Ivory: Zero, One, Many
┌─────────┬─────────────────┐
│ boolean │ even-square │
├─ ─ ─ ─ ─┴─ ─ ─┬─ ─ ─ ─ ─ ─┤
│ square │ quadruple │
│ ├─ ─ ─ ─ ─ ─┤
│ │ even │
├─ ─ ─ ─ ─ ─ ─ ─┴─ ─ ─ ─ ─ ─┤
│ natural │
└───────────────────────────┘TODO
- Describe zero, boolean, etc.
- Closed under addition, multiplication, max, min, gcd, lcm, etc.
- Defining boolean operations is easy, since there are so few; but need to be careful how they generalise to other numbers. For example, xor is often seen as addition mod 2, but that’s hard to generalise to rationals, etc. Cleaner to use ≠
- Describe thought process regarding
prime:- Would need
0, which is weird; but maybe a reasonable fudge (e.g. “primitive” or “primal” or something) - If we’re going to include
0then we should definitely include1, which puts it belowboolean - Would need
2, but can’t be above/beloweven - Could maybe introduce 2 in a set {0, 2}, but seems contrived
- Can’t introduce 2 with {0, 1, 2} since it would need to appear above
evenbut that doesn’t include1
- Would need
- Describe through process regarding powers of two:
- Would need
0, which is awkward but still closed under many operations - Would need to generalise
boolean, since 2⁰ = 1 - Again, can’t introduce 2 this way, since 1 isn’t in
even - Powers of 2 other than 1 and 2 are
quadruples - Would conflict with
even-square, since 16, 64, etc. are both
- Would need