Digits and Decimal Numerals

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ and $0$ be as in The Class Omega of Natural Numbers with Zero §omega and The Class Omega of Natural Numbers with Zero §zero, $1$ as in The Set of Natural Numbers and the Number One §one, $S(x)$ the successor of a set $x$, and $+$ and $\cdot$ the addition and the multiplication on $\omega$. The clauses below are metatheoretic: they concern strings of symbols and introduce abbreviations.

¶ The digits are the ten symbols $0,1,2,3,4,5,6,7,8,9$. The digits $0$ and $1$ denote the sets fixed above, and

$$2=S(1),\quad3=S(2),\quad4=S(3),\quad5=S(4),\quad6=S(5),\quad7=S(6),\quad8=S(7),\quad9=S(8).$$

Each digit denotes an element of $\omega$, since $0\in\omega$ and $S(u)\in\omega$ for every $u\in\omega$ by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity §inductive; so does the base $S(9)$.

¶ A decimal numeral is a string $d_{k}d_{k-1}\cdots d_{0}$ of digits written next to each other, where $k\ge0$ is a metatheoretic number. It abbreviates an expression built from the digits, the base, $+$ and $\cdot$, defined by recursion on $k$: for $k=0$ the numeral $d_{0}$ is the digit $d_{0}$ itself, and for $k\ge1$

$$d_{k}d_{k-1}\cdots d_{1}d_{0}\quad\text{abbreviates}\quad\bigl((d_{k}d_{k-1}\cdots d_{1})\cdot S(9)\bigr)+d_{0},$$

where $d_{k}\cdots d_{1}$ is the abbreviation already defined for the shorter string. Thus a numeral is an expression built from defined set symbols, eliminated by the rule of Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations §defined-symbols; by recursion on $k$ it denotes an element of $\omega$, since $+$ and $\cdot$ are binary operations on $\omega$ by Addition on Omega §addition and Multiplication on Omega §multiplication. For example, $156$ abbreviates $\bigl(\bigl((1\cdot S(9))+5\bigr)\cdot S(9)\bigr)+6$. Strings of digits that the metatheory uses as counts, such as indices and numbers of variables, are not decimal numerals.