Set signatures and finite syntax strings

A set signature $L=(C,F,P,a)$ has pairwise disjoint sets of constants, function symbols and relation symbols, with $a:F\cup P\to \omega \setminus \{0\}$. Its alphabet ${\mathrm{\Sigma }}_L$ is a tagged disjoint union of these sets, variables $v_i$ for $i\in \omega$, constructor tags $\mathsf{v}\mathsf{a}\mathsf{r},\mathsf{c}\mathsf{o}\mathsf{n}\mathsf{s}\mathsf{t},\mathsf{f}\mathsf{u}\mathsf{n},\mathsf{r}\mathsf{e}\mathsf{l},\mathsf{e}\mathsf{q},\mathsf{n}\mathsf{e}\mathsf{g},\mathsf{a}\mathsf{n}\mathsf{d},\mathsf{e}\mathsf{x}\mathsf{i}\mathsf{s}\mathsf{t}\mathsf{s}$, and punctuation $(,),,$. Equality is a logical symbol.

A word is a function $w:n\to {\mathrm{\Sigma }}_L$ for a natural number $n$, its length. The set $W_L$ of words is obtained by Separation inside $\mathcal{P}(\omega \times {\mathrm{\Sigma }}_L)$. The empty word has domain $0$. Concatenation of $u:m\to {\mathrm{\Sigma }}_L$ and $w:n\to {\mathrm{\Sigma }}_L$ is the function on $m+n$ whose value at $i<m$ is $u(i)$ and whose value at $m+j$ is $w(j)$ for $j<n$. Signatures may be empty or uncountable; finite words here are set codes, not necessarily natural-number codes.

Conventions and prerequisites: The natural numbers $\mathbb{N}$ (von Neumann).