General conventions adopted throughout the course of this work are given below.
\(N_n\) will represent the set of natural numbers starting at 1 and ending at n,
\[N_n = \{ 1, 2, ... , n \}\]
The cardinality of a set \(A\) will be denoted \(\lvert A \rvert\)
∎ will be used to denote the ending of all examples and proofs.
The terms “set” and “class” are used interchangeably.
The Capital Gothic letters \(\mathfrak{F}, \mathfrak{G}\) are reserved for functions. Functions will be written \(\mathfrak{F}: \mathbb{D} \to \mathbb{R}\), where \(\mathbb{D}\) is the domain of the function and \(\mathbb{R}\) is the co-domain of the function.
It is incorrect to treat the empty quotation marks “” as the English “Empty Character”. Empty Characters do not exist in Language. In other words, \(\varepsilon\) does not get assigned a value under an interpretation (model), unless \(\varepsilon\) itself is being treated as a symbol. In such cases, \(\hat{\varepsilon}\) will be used to refer to the physical inscription of the Character assigned to represent the Empty Character.
The exact meaning of these symbols should be attended with care. \(\mathfrak{a}, \mathfrak{b}, \mathfrak{c}, ...\) represent literal Characters of the Alphabet and thus are all unique, each one representing a different linguistic element. When Character symbols are used with subscripts, \(\mathfrak{a}_1, \mathfrak{a}_2, \mathfrak{a}_3, ...\), they are being referenced in their capacity to be ordered within a String. With this notation, it is not necessarily implied \(\mathfrak{a}_1\) and \(\mathfrak{a}_2\) are unequal Character-wise, but that they are differentiated only by their relative order in a String.
Likewise, when Character Variables are used with subscripts, it is meant to refer to the capacity of a Character Variable to be indeterminate at a determinate position within a String.
Moreover, the range of a Character Variable is understood to be the Alphabet \(\Sigma\) from which it is being drawn.