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.

Nested numerical structures

Typical view of various number sets, accurate and useful for most purposes, but not necessarily all purposes

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 $\mathbf {C}$. In symbols this would be $\mathbf {N} \subseteq \mathbf {Z} \subseteq \mathbf {Q} \subseteq \mathbf {R} \subseteq \mathbf {C}$. At the right is a picture used currently in our number article, showing these relationships.

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 are identical to the finite ordinal numbers.

Then the integers are represented as ordered pairs of natural numbers, for example $(n,m)$, intended to be understood as $n-m$, up to the equivalence relation $(n_{0},m_{0})\sim _{\mathbf {Z} }(n_{1},m_{1})\iff n_{0}+m_{1}=n_{1}+m_{0}$. (The universe of the structure $\mathbf {Z}$ is then the quotient of $\mathbf {N} \times \mathbf {N}$ by $\sim _{\mathbf {Z} }$.)

Note that this already breaks the subset relationship from the first paragraph. For example, the natural number 0 is the finite ordinal number 0, which is literally simply the empty set. However, the integer zero is an infinite set, namely $\{(0,0),(1,1),(2,2),\ldots \}$, and indeed all integers are infinite sets (for example the integer $-1$ is $\{(0,1),(1,2),(2,3),\ldots \}$). Therefore the natural number 0 is not an integer, so $\mathbf {N} \not \subseteq \mathbf {Z}$.

The universes of the structures

I don't want to spend a lot of time on this, but it seems worth putting down the construction of the remaining structures:

• Rationals ($\mathbf {Q}$): A rational number is represented by a pair of integers $(n,m)$, with $m$ nonzero, understood as ${\frac {n}{m}}$, up to the equivalence relation $(n_{0},m_{0})\sim _{\mathbf {Q} }(n_{1},m_{1})\iff n_{0}m_{1}=n_{1}m_{0}$.
• Reals ($\mathbf {R}$): A real number is represented by a Cauchy sequence of rational numbers, up to the equivalence relation $(q_{n}|n\in \mathbf {N} )\sim _{\mathbf {R} }(r_{n}|n\in \mathbf {N} )\iff |q_{n}-r_{n}|\to 0~{\textrm {as}}~n\to \infty$.
• Complex numbers ($\mathbf {C}$): A complex number is an ordered pair $(x,y)$ of real numbers, understood as $x+iy$.

With these choices, taken literally, none of the subset relations hold: $\mathbf {N} \not \subseteq \mathbf {Z} \not \subseteq \mathbf {Q} \not \subseteq \mathbf {R} \not \subseteq \mathbf {C}$.