I have an integer $N$. I am working with the set $\{1,...N\}$ and it's powerset $\mathcal{P}(\{1,...N\})$.
I want to use define a function, $f$, on the sets $p\in\mathcal{P}(\{1,...N\})$ which represents what seems to me to be maybe the most ``natural way" of ordering the sets. But I am unsure how to define it rigorously and if it already has a name (it feels like it probably does).
As an example, if $N=3$, my function $f$ should map the subsets of $\{1,2,3\}$ to integers as follows:
$$f(\emptyset)=0$$ $$f(\{1\})=1,f(\{2\})=2,f(\{3\})=3,$$ $$f(\{1,2\})=4,f(\{1,3\})=5,f(\{2,3\})=6,$$ $$f(\{1,2,3\})=7$$
Is there a name for an ordering of this sort. i.e. can I just say something like ``let $f$ be the {BLANK} ordering on $\mathcal{P}(\{1,...N\})$" to define the $f$ above or something similar?
It's almost a lexicographic ordering, but not quite. I'm sure if you just said "let $f(A)$ be the index of $A$'s position in the lexicographic ordering on $\mathcal{P}(\{1,\ldots,N\})$," then people would understand what you intend. That said, if you want, here's my attempt at a fully-fleshed out version of what you could do.
For subsets $A,\ B \subseteq \{1, \ldots, N\}$, if $A\neq B$, define $A \prec B$ if $|A|<|B|$, or if $|A|=|B|$ and $\min A < \min B$. Finally, if $|A|=|B|$ and $\min A = \min B$, then say $A \prec B$ if $A\setminus\{\min A\}\prec B\setminus\{\min B\}$. (I've heard this called "top-down" recursion, which is valid if you can prove it eventually terminates. Since our sets are finite, it does, since we remove one element at a time.)
Now define $f:\mathcal{P}(\{1, \ldots, N\})\to \mathbb{N}$ as follows: $f(\emptyset)=0$, and $f(A)=\max\{f(B):B\prec A\}+1$. Since the domain is finite and the range is discrete, this should be valid.