User:Trovatore/Arithmetic and exponentiation in various numerical structures

Many times, usually in the context of discussions about the expression $0^{0}$ , I have wanted to express certain ideas that I think are relevant, but it would be too long-winded inline, and also I didn't want to spend the time. So I'm hoping that pointers to this essay may prove useful.

Opening remarks

It is often claimed that the natural numbers $\mathbf {N}$ form a subset of the integers $\mathbf {Z}$ , which in turn form a subset of the rational numbers $\mathbf {Q}$ , which in their turn form a subset of the real numbers $\mathbf {R}$ , and the reals a subset of the complex numbers $\mathbf {C}$ .^{[note 1]}

For most purposes this is true.

However, it's worth noting that in any of the most common implementations of mathematics in set theory, it is not literally true. In a typical implementation, the natural numbers^{[note 2]} are identical to the finite ordinal numbers.

Notes

^ Many prefer blackboard bold for the names of these sets. In most cases I prefer to save blackboard bold for blackboards. Since this is my essay I'm going to use ordinary boldface.
^ Including zero as a natural number.

[1] Many prefer blackboard bold for the names of these sets. In most cases I prefer to save blackboard bold for blackboards. Since this is my essay I'm going to use ordinary boldface.

[2] Including zero as a natural number.

[note 1]

[note 2]