Parrochia refers to “ZFC” without any mention of what that stands for or what it is. I challenged people on Facebook to make up their own guesses as to what ZFC stood for.

My favorite answer came from either Joe or Bev Angel at Arhaven House Concerts (they’re both academics), who wrote:

“Zero F***ing Chance you’re gonna understand this book anyway if you have to look up ZFC”,

(Philosophy prof) Rob Koons gave the correct definition, of course. Honored to have him responding to my facebook posts!

Turns out ZFC = Zermelo-Frankel set theory with the Axiom of Choice, and is regarded as the foundational set theory of mathematics, although the AoC is “the most controversial axiom in all of mathematics.” The AoC says that given any (finite or infinite) set, one can choose one item from each set; but it doesn’t say how to do so, and so this offends “constructivists.” AoC is equivalent to several other statements, including the Well-Ordered-ness….? thing? that says that it’s always possible to arrange a “least” element in each subset? and Zorn’s Theory which I have no idea.

Funny quote from famous mathematician Jerry Bona (who was on my PhD committee back in the day):

“The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?”

The AoC gives rise to the Banach-Tarski Paradox and other fun stuff, like sets that have no (Lebesque) measure (not zero measure, rather no / undefined measure).

…So today’s rabbit trails involved a whole lot o’ Set Theory, and “formal logic”. There is “first order logic” which I don’t get, or what 2nd-order logic would mean. The bits about infinite set theory ran into problems, and Skolem’s Paradox which I don’t see as a paradox at all. I remember the Lowenheim-Skolem Theorem…

()(from a philosophy class in grad school on “The Infinite”, taught by Cory Juhl way back in the day – Interesting that his web page lists his interests as “Philosophy of Science, Philosophy of Mind and Language”, so maybe I should have him read by book…)*

….and seem to recall that it had some consequences for language. But my attempting to read about this produced another rabbit-hole of definition after defintion, “Model Theory” and such.

I claim that any word can be ruined by mathematicians by adding the word “theorem” after it. Equivalently, take any word and add “theory” after it, and there will be a field of mathematics about that, in which the subject has very little to do with the conventional meaning of word you chose originally (key culprit: Category Theory! grrr).

I even vented on Twitter & Facebook about the rabbit hole of trying to read up on this stuff:

I even estimated that the CN (henceforth denoted by the symbol for Mercury, ☿) can be given by

☿≃⌈y+20(y+1)π/2⌉,{\Large ☿} \simeq \left \lceil{ {y+20\over (y+1)^{\pi/2}} }\right \rceil,

where $y$ is your number of years of exposure to the field.

Against formal logic, there were “common language philosophers,” which included ‘the later Wittgenstein’. And here’s an interesting bit…

W. B. Gallie (C2, 1956) argued that many concepts, such as art, democracy, and freedom, are “essentially contested”; that is, they necessarily mean different things to different people and so cannot be given definitions once and for all, as required by formal philosophy. – https://funkyeds.blogspot.com/2017/02/analytic-philosophy-part-ii.html

Now, on the topic of all this, I started watching this YouTube video and it contains diagrams like those I see in Parrochia’s book:

…which makes sense. I’ve also been learning about the set-theoretic notions of partitions and equivalence classes, which appear in Parrochia. ==A partition is not just a subset, rather it is the set of all subsets such that each subset has at least one element. (I think. Did I get that right?)==