Arithmetic

If you depart Naive Set Theory headed east through the Number Theory Range, you will quickly end up in the small town of Arithmetic. The core value these people hold is that sets are reduced to just the number of elements they contain. As such these people don’t care about what’s in the sets, and instead reason about all possible sets with 5 elements (for example).

Consider how much more flexible this approach is, while also thinking about the problems created by not knowing the exact members of the set anymore.

This is almost always the second stop after Naive Set Theory that children make on their pilgrimage to mathematics, though many forget about the underlying set theory as they progress.

Natural Numbers

Arithmetic typically operates over natural numbers (noted $\mathbb{N}$), which are the whole numbers greater than or equal to 0 that you can have a set of that many elements.

Operations

Addition (or the coproduct) $A+B$ is the result of combining A and B such that duplicate elements are retained. Again though, the people of arithmetic don’t care about the underlying sets, and just abbreviate $3+5=8$, knowing that combining any set of 3 elements with any set of 5 elements yields a set of 8 elements.

Subtraction is the inverse of addition, and lets one figure out $A+?=C$, noted as $C-A$. Not all subtraction problems are solvable with natural numbers.

Multiplication (or the product) $A \times B$ is the result of pairing every member of A with a member of B. If you have a set of 3 elements and a set of 5 elements, there are 15 possible pairs. Note the relationship to permutations in statistics

Division is the inverse of multiplication and lets one figure out $A \times ? = C$, noted as $C/A$. Not all division problems are solvable with natural numbers.

Notation