naturals

Another way to approach the problem is to talk about the reals and the naturals: the cardinality of the reals is greater than the cardinality of the naturals.

For example in the Exact and Naturals Science Faculty, the 30% of the Math professors are females.

You'd need to decide how to truncate the Naturals though; do you take min(x, 2^32 - 1)?

This > Naturals are not closed under subtraction or negation, and therefore you should never negate or subtract a length.

Also, it would be interesting to read about comparisons between programming languages and naturals ones.

The final expansion is worth writing, however, since it notes how there are unboundedly many naturals, each a different component in this infinite disjoint sum.

This is of a different type than the naturals (digital lengths).

But there are just as many people, if not more, who put in tremendous effort to become experts at which point they may appear as naturals.

I wonder's what's been used in real world code more, quaternions or Peano naturals.

The fact that the naturals have a different cardinality than the reals can be expressed by ({}, build_naturals) and ({}, build_reals) being in different equivalence classes.

Now univalence would have to implement the binary tree analogue for every function on naturals, in a way that preserves semantics with respect to our conversion function.

You can always write your music with 7 naturals.

Or how "naturals" compare to people who weren't very good at first, but kept practicing.

"In mathematics, the quaternions are a number system that extends the complex numbers." Naturals are a subset of integers.

I don't know the exact answer, but yes I think there's nothing special that distinguish naturals and accidentals from a "listening" point of view.

The difference is, in some sense, arising just from how we represent and manipulate subsets of the naturals (as pertaining to the set of multiples of each prime, as well as their merger into the totality of composites): either as streams of increasing naturals [efficient], or as streams of "In"s and "Out"s [less efficient].

But the question is, what's natural about the Naturals, and whats accidental about the Accidentals?

Naturals are not closed under subtraction or negation, and therefore you should never negate or subtract a length.

So if you have a normal datatype encoding of natural numbers data Nat = Zero | Succ Nat then you can convert Church numerals to that data type very trivially C(n)(Succ, Zero) To maybe see the pattern a little more clearly, let's consider something very similar to Naturals---linked lists!