Hacker News Reader

Comment by interroboink | Hacker News Reader

Yeah I didn't get that comment, either.

At least in my education, you first define the natural numbers (where 0 and 1 are special, and the rest are defined in terms of that (ie 2 = 1+1)), then you define negation, which gives you the negative numbers. Then you layer on multiplication and division, which gives you rational numbers, and so on.

So, it's the same "0 and 1" definition all the way through, just with additional operations being added to the mix.

Though maybe other approaches do it differently.

View on HN

The usual formal definition for the rational numbers is equivalence classes of pairs of integers. The zero is then the equivalence class of (0, 1) which is not the same as the integer 0.

You could certainly somehow get it to work by starting with the closure of the division operation but would introduce a lot of unnecessary headache along the way.

I think you may have not gotten far enough to get a formal explanation of what 0 and 1 are.

The definition taught to most mathematicians is that for the natural numbers,

  0 = {}
  1 = {{}}

In general

  S(N) = N ∪ {N}

But this is not how the real numbers are constructed. You either use Dedekind cuts, Cauchy sequences, or something else.

Complex numbers are just pairs (real and imaginary) of Cauchy sequences of pairs (numerator and denominator) of pairs (positive and negative) of nested versions of the empty set. (modulo al the necesary equivalence to make this work).

So the natural number {{}} is canonical included in the complex numbers as [something].

Natural {{}}

Integer ({{}},{})

Rational (({{}},{}),({{}},{}))

Real (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), ...

Complex ((({{}},{}),({{}},{})), (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), ... , (({},{}),({{}},{})), (({},{}),({{}},{})), (({},{}),({{}},{})), ... )

(And there are a few alternative definition, but I think they are even more messy.)

Reply on news.ycombinator.com