David St John, Ph.D. is a low-ranked Professor of Mathematics at Malcolm X College, one of the City Colleges of Chicago. Their pronouns are they/them, and they are out in their community as a trans person. Their academic interests include Critical Theory, Critical Pedagogy, Chaotic Systems, Particle Physics, Foundations of Mathematics, Radio Astronomy, and Semiotics. Their views do not necessarily (and likely do not, necessarily) represent the views of the institution, its faculty, administration, engineering staff, nor anyone or any entity anywhere.
In January of 2021, Prof. St John invented* a hierarchical number system and named it Budzmac (BZ). Numerals can be generated by using a special symbol as a counter, / (for BZ numerals) or u (for BZ formulas). The smallest numeral then is / or u, and for any numeral n, nu is also a numeral. This induces the normal ordering of natural numbers, where / < // , // < ///, etc.
Prof. St John then introduces an alternative symbol, \ or d. Reduction is done by the replacement rules: and . This is purely cosmetic. The numerals are usable without this reduction, but then they can get significantly longer.
Finally Prof. St John uses a depth counter, call it _ or s (for sub). If m and n are numerals, then m_n is a numeral. The depth of a numeral is defined as the number of occurrences of the _ or s symbol in its BZ formula or BZ numeral. Numerals are by definition finite, so each numeral represents a generalized natural number with finite depth. This induces the desired hierarchical total ordering using the dictionary rules (that is, and and , etc...for all numerals n).
The numerals can be considered as finite lists of natural numbers indexed by the natural numbers. To do this, the sub-numeral of a numeral with depth p and index k, is defined in three cases of the index k. For , the sub-numeral is defined as the numeral with the symbols immediately before the k^th occurrence of the symbol _ and immediately following the (k-1)^st occurrence of the symbol _ . The sub-numeral of a numeral n with index 1 is defined as either the numeral represented by the symbols preceding the first _ symbol if or as n itself if .
The set of all such numerals then is countable, but not order isomorphic to the natural numbers because of the properties of the dictionary ordering rules.
| BZ Formula | BZ Numeral | Arabic |
|---|---|---|
| u | / | 1 |
| uu | // | 2 |
| uuu | /// | 3 |
| ud | /\ | 4 |
| d | \ | 5 |
| du | \/ | 6 |
| duu | \// | 7 |
| duuu | \/// | 8 |
| dud | \/\ | 9 |
| dd | \\ | 10 |
| ddu | \\/ | 11 |
| dduu | \\// | 12 |
| dduuu | \\/// | 13 |
| ddud | \\/\ | 14 |
| ddd | \\\ | 15 |
| dddu | \\\/ | 16 |
| ddduu | \\\// | 17 |
| ddduuu | \\\/// | 18 |
| dddud | \\\/\ | 19 |
| dddd | \\\\ | 20 |
| BZ Formula | BZ Numeral | Generalized Arabic |
|---|---|---|
| usu | /_/ | 1.1 |
| uuususddud | ///_\\/\_// | 3.14.2 |
| ddddddddduusdddu | \\\\\\\\\//_\\\/ | 47.16 |
This is equivalent to the ubiquitous ordering convention for chapter, section, and subsection headings, where 1 (usually reserved for chapter numbering) comes before 1.1, 1.1 comes before 1.1.1, 1.3.7 comes before 1.4.1, etc. We say that n is above n_m in the hierarchy for any numerals m,n. The map that replaces the depth-zero BZ numerals in each 'slot' between the _ symbols (or at the beginning and end of a numeral) with their generalized Arabic equivalent is trivially an order isomorphism.
Although it is not a numeral, the _ symbol with no adjacent / or \ on either side can stand in as a null character in place of a numeral. This can be useful when whitespace cannot be rendered to create a human-readable visual representation of an ordered list, or for other reasons. It allows enforcing a rule, for example, where each line is non-empty, while still allowing some visual organization via spatial layout. The cost of this is that not every line has a unique BZ numeral label, but only the non-blank ones. A line with only _ would then be read as, 'this line intentionally left blank'.
An example: a book outlined in this format might look like:
/ Book Title
/_/ Author information
/_/_/ Publisher information
/_/_// Other metacontent
/_/_/_/ Other other metacontent
_
/_// Chapter 1 Title
_
/_//_/ Chapter 1 Paragraph 1
/_//_// Chapter 1 Paragraph 2
_
/_/// Chapter 2 Title
_
/_///_/ Chapter 2 Paragraph 1
/_///_// Chapter 2 Paragraph 2
/_///_/// Chapter 2 Paragraph 3
_
etc...
Prof. St John designed the use of the / \ _ symbols and the replacement rules for secret reasons related to numbering each non-empty line in a syllabus file with a unique BZ numeral. There are formatting guidelines for such a document with enumerated lines that remain unpublished. While technically it is incorrect to use the _ symbol at the beginning of a line with so-called 'unnumbered' text, equivalent to a comment in a line of code, Prof. St John created this convention specifically so that they can break it and abuse the notation that way on purpose.
*(unlikely for the first time in history, but independently nonetheless to meet the needs of the task at hand)