Most of my justifications for claiming that 道 is U follow from the many comments that discuss features that 道 has or lacks, which I read in an attempt to understand 道 as a set. These are a few of my observations in that attempt:
- 道 is infinite and the forefather of everything.
"道沖而用之或不盈。I've deemed this "hollow" and "inexhausible" container (沖) talk as one about the 道 as something such that it does not merely inject to the natural number line, that there are, crudely, infinitely many things which are of 道. If 道 really is U, then it would be a "source of everything" or an "ancestor to everything," at least in the sense that everything about which we could possibly have a discourse would be a subset of or element of U.
"Tao is a hollow vessel, And its use is inexhaustible!
Fathomless! Like the fountain head of all things."-- Daodejing, 4:1~4:2 (trans. Lin Yutang)
- 道 is abstract, and its empirically known subsets are not 道.
"道可道，非常道。"Chan's translation simplifies a matter very well. When they discuss 道 as a verb (as in "可道"), most translators often translate it to some sort of (often presumed) illocutionary act -- referring, addressing, describing, speaking, telling. This is less than mundane activity for Laozi, since it's practically a litmus test for determining if something is 道, or rather 常道 ("the eternal Tao"). If 道 is U, then Laozi outlining a basic fact about U, that none of its subsets except for those that equal U are U.
"The tao that can be described is not the eternal Tao."-- Daodejing 1:1 (trans. Chan)
My interpretation further presumes that Laozi's "describing" (道, v.) refers to "describing from our five senses." In a pre-logical world, the abstraction of U would always appear as exist "before" the existence of everything else and of anything that we could see, or smell, or taste... The Daodejing offers that kind of treatment. 道, the U, is the source of everything; but more, it doesn't have any of the features of the things that we can access with our five senses.
- 道 is the union of all possible contraries (whereas the intersection of them would be the empty set, ∅, or "nothing").
"反者道之動；弱者道之用。Legge's flowery translation does favorably portray the set-theoretic reading of these lines. The "contraries" (反) are the "action" or "proceedings" (動) of 道, since failing to account for complements of subsets of U would not generate U. People sometimes allude this first sentence to Taijitu, that of complementary forces working together (and seeing the 道 as U would make that move easier) and determining the nature of everything in nature, and for lay primitive theory, that is probably plenty adequate. However, what convinces me more that they're discussing an abstraction prior to set theory is the line that follows it: "Everything comes from something" (that is, everything that we can know is a subset of 道, of U, but also from subsets of 道 or U). "Something comes from nothing," though, interprets more openly under the presumption that 道 is U. There are things that we can derive from the empty set, such as all of the theorems of logic and the definition for natural numbers. However, it's a stretch to believe that Laozi or any pre-Han Chinese thinker had any such thoughts. Instead, thinking prior to a formal means of understanding subsets and nested sets, we may claim that our denotations for things don't "come from" anywhere. They're just presumptions, just in the way that U is a presumption. We need to assume U for discourse on any topic, just as ancient Chinese thinkers need to assume 道 for their discourse on references to make sense. Our noises and scribbles have to correspond to referents, and the collection of those references, in a pre-logical China, would be some part of 道.
"The movement of the Tao By contraries proceeds; And weakness marks the course Of Tao's mighty deeds."-- Daodejing, 40:1 (trans. James Legge)"The things of this world come from something; something comes from nothing."-- Daodejing, 40:2 (trans. Red Pine)
- There are narrowed 道, such as a "Way of the heavens" (天之道) and a "Way of mankind" (人之道). In logic, we can stipulate a narrower U (certain sets of numbers, for instance), but it is still defined as a union of all of the sets and elements that would still fit the definition.
"天之道，不爭而善勝，The ability to make stipulations like this allows Chinese thinkers to discuss other philosophical areas, especially ethics and politics. However, these metaphysical areas and ethical areas connect together in a very special way for Daoists, especially when it comes to pursuits of 天之道 and 非道, that is, those unchanging and overarching principles the guide our mortal world (and that is, pursuit of truth).
"Heaven's Way (Tao) is good at conquest without strife:
Rewarding (vice and virtue) without words,
Making its appearance without call,
Achieving results without obvious design."-- Daodejing, 73:3 (trans. Lin Yutang)
The defense of such a link is not foreign to us. Spinoza, Kant, and Wittgenstein derive their ethical stances from highly (logically) formal considerations of "the world"; and it would not be out of place for the philosophers of ancient China to appeal to an impartial, all-pervasive force (namely, 道) which would inform their ethos, as well. This is the approach that I will take when I approach the Ethical/Political Side, the sense that one could make of Laozi's ethics from a set-theoretic reading of the text.