How do you define the natural numbers for different base systems?

Question

When we define the natural numbers through set theory does it matter how the numbers are represented? For example, does the definition apply to a base 9 or base 16 system just like it would to a base 10 system? Is the radix independent of the set theoretic definition of the natural numbers or should we somehow work that into the definition. I’m confused on what the natural numbers actually are like their ultimate structure versus how we represent them on paper.

Answer 1

No, it does not matter at all.

Natural numbers are not defined by their representation in base $10$, or in any base whatsoever. This comes a long way after numbers have been defined.

One common way of defining them is based upon set theory. Set theory provides the empty set and the possibility to make sets that include other sets; so a definition of integers is:
$0$ is the empty set $\{\}=\emptyset$
$1$ is the set $\{\emptyset\}$
$2$ is the set $\{\emptyset, \{\emptyset\}\}$
and more generally, the sucessor function $S$ is defined as
$S(n)=\{\emptyset,n\}$
Then the set of integers is defined as the smallest set that includes $0$ and its successors - which actually requires some axiom of infinity to make that work.

(I can remember when I was young, and my father explained this to me, and I thought "Oh, these mathematicians are just mad men" :-)).

Actually, this set-theoretic definition of natural numbers, being a recursive construction, explains why recurrence demonstration is so often used for properties on $\mathbb{N}$.

Defining integers by their representation in base $10$ would be more troublesome. We would certainly rather choose base $2$, but that would still be complicated. Think how you would define the successor of a series of $0$'s and $1$'s...

Similarly, real numbers are not defined by their decimal expansion, notably because you get the additional complication that the representation is not unique, i.e. $0.999999... = 1.00000...$.