The Long Way Up, Part 1

Just a couple interesting observations about sets, functions, logic, and math I had not realized or considered before.

First off, strings. My physics professors all told me they don’t push. Say we have some set G that contains elements g1, g2, g3, …. A string of elements from G would be some finite sequence of elements of G, e.g. g1 g16 g420 g69. The empty string is considered a string. G* is the set of all finite strings over g: G* = {empty string, g1, g2, …., g420 g69, g3 g14, g6 g6 g6, …}. I work in tech and deal with strings a lot but had not considered a mathematical, set-theoretic definition of strings. It’s rather obvious in hindsight.

Secondly, ordered pairs in set theory. A somewhat arbitrary but effective definition of an ordered pair in set theory equates it to a set of sets: <a, b> = {{a}, {a, b}}. This definition - apparently - preserves the properties of the ordered pair. It’s somewhat arbitrary because it could just as easily be reversed: {{b}, {a, b}}. This definition allows us to define the cartesian product of two sets:
A x B = { <a, b> : a belongs to A and b belongs to B}

in purely set theoretic notation, since the ordered pair can be reduced to a set.

So far it seems like Open Logic Project may be just the sort of background one needs for the CTMU, but it’s 1031 pages and you need to get through at least 300 before anything starts being truly useful. Blargh.