User:D1ff30m0rf1zm

Denote by $$\Lambda$$ the set of accepted IRC nickname characters. Whenever an input is given, a function is specified:

$$\nu: \mathcal{P}(\Lambda) \longrightarrow \mathcal{P}(\mathbb{N})$$

As you will see $$\nu$$ can be made surjective on a strict subset. Note, among each $$K \in \mathcal{P}(\Lambda)$$, there are $$|K|!$$ possible permutations of letters. For example, one might choose the $$\{a,b,c\} $$ subset, so that the possible combinations are $$abc, bca, cab, cab, bac, acb$$. Thus there is a filtration

$$\Nu: \mathcal{P}(\Lambda) \longrightarrow \text{Perm}(K) \circ K$$ whenever $$K \in \mathcal{P}(\Lambda)$$, where we define $$\text{Perm}(K) \circ K$$ to act pointwise on the discrete $$K$$.

So this gives us the name we want in order. To assign each letter a place value we take the successor function defined inductively: for $$k \in K$$, define

$$s(k) = \min\{K \sim \{k\}\}$$.

As $$K$$ is discrete we note that $$\textbf{rng}s = K \sim \{\min(K)\}$$, where $$\min(K) \in K$$.

Thus we have place value. The point of taking $$\nu$$ into $$\mathcal{P}(\mathbb{N})$$ is so that we can put $$\textbf{rng}s|_{K}$$ into a one-to-one correspondence with our selected numbers.

Note that in particular $$\textbf{rng}\nu \subset \mathfrak{I}$$, the set of subsets of $$\mathbb{N}$$ of form

$$I_{n} = \{1, \ldots, n\}$$.

If IRC is to have a character limit then for some $$\pi$$, $$|\nu(\mathcal{P}(\Lambda))| \leq \pi$$.

Now we create our one-to-one correspondence like this, if $$|K| = n$$:

$$\rho(\min(K)) = 1 \in I_{n}$$, $$\rho(s(K \sim \min(K))) = 2 \in I_{n}$$, $$\rho(s(K \sim s(K \sim \min (K)))) = 3 \in I_{n}$$, ...

Note, we don't have to take $$\mathfrak{I}$$ to consist of only finite subsets if we assign the value $$0$$ to all numbers outside of $$|K|$$ in $$\mathbb{N}$$.