Philosophy - 750 words essay (please read all the attached files)

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Philosophy 2: Puzzles and Paradoxes / UCI, Winter Quarter 2020 Professor David Woodruff Smith ( [email protected] ). Teach Assistants: Joost Ziff <[email protected]>, Louis Doulas <[email protected]>, Evan Sommers <[email protected]>. January 22, 2020 Professor Notes: Re Cantor vis-à-vis Zeno. The ancient Greek philosopher Zeno (ca 490-430 BCE) famously posed a series of paradoxes. Zeno’s paradox of motion argued that motion is impossible. For: To walk across the room, say, covering 10 meters, I would have to walk first half the distance, then half the remaining distance, then half the next remaining distance, and so on ad infinitum, or “to infinity”. The same form of argument applies to space itself, holding that space is impossible: For, a finite distance, say, of 10 meters, would have to add together an infinite sequence of partial distances, i.e. 5 meters, 2.5 meters, 1.25 meters, and so on ad infinitum. And the same form of argument applies to time itself: a finite stretch of time, say, of one hour, would have to add together first one-half hour, then one-fourth hour, then one-eighth hour, and so on ad infinitum. On Zeno’s analysis, a finite whole (a stretch of motion/space/time) must be formed by adding together an infinite number of smaller and smaller parts (partial stretches of motion/space/time). Which seems impossible. What to make of “infinity”? Why does such an argument lead to a contradiction (P and not-P), thus a strict paradox? George Cantor (1845-1918) developed a mathematical theory of “transfinite” numbers that arguably solves or resolves Zeno’s ancient paradox. More than 2000 years of mathematical theory led to a direct critique of Zeno-style paradoxes. Cantor developed (with others following) modern mathematical set theory. Intuitively, we think of a set as a collection of various things. We draw a “Venn” diagram as a circle gathering things, where circles representing sets can overlap or not. Formal set theory developed with axioms limiting the definition of sets. (In particular, we must avoid the paradox Bertrand Russell observed: the set M of all sets that are not members of themselves cannot exist. For M is a member of M if and only if M is not a member of M — a self-contradictory proposition. So, Russell observed, not every collection of things can logically count as a set.) As Cantor developed set theory, he applied it to the theory of numbers. Each set is composed of a number of elements gathered together into a set: this measure of the size of the set Cantor called its “cardinality”, a “cardinal” number.

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Two sets with the same number of elements, or cardinality, can be put in a one-to-one correspondence, i.e., so that each element in the one set is correlated with an element in the other set, and vice versa. Cantor used this structure to prove some remarkable results. Cantor proved that the number of points on the “real line” is greater than the number of “natural numbers”. Yet both are “transfinite” numbers! So there is more than one size of “the infinite”. Huh? No wonder the ancients were puzzled! Specifically: Let N be the number of elements in the set of natural numbers: {0, 1, 2, 3, … }. (How many natural numbers are there? N.) N is an infinite number, “countably” infinite — as we might ideally “count” these natural numbers themselves from 0 to 1 to 2 to 3 and so on ad infinitum. Let R be the number of elements in the set of points on the “real line”: {__ 0___________1__ }, say, number of points between the points designated 0 and 1. (Assuming a measure of distance on the continuous line, each point between 0 to 1 is designated by a number such as 0.12578, e.g. measured in meters). (How many points are there on the real line between 0 and 1? R.) By an ingenious argument (now called “diagonalization”), Cantor proved that: R > N, that is, the number of points on the real line is greater than the number of natural numbers. So R is an infinite number, but “uncountably” infinite — as we cannot “count” the points by running through the natural numbers. So, Cantor showed, there are different levels of “transfinite” numbers measuring sizes of “infinitely” large sets: namely, N and, a larger “infinite” number, R. This result is puzzling, seemingly paradoxical, counter-intuitive. Yet proven! That is, within the assumptions of modern mathematical set theory and thus number theory. It is now commonly thought that Cantor’s number theory allows us to solve Zeno’s paradox of motion, or rather to refute Zeno’s argument that motion is impossible. For Zeno seems to have thought that an “infinite” number of partial distances (in physical nature) is impossible to be realized, yet we now know that a continuous line segment (in analytic geometry) is composed of an infinite number of points, indeed, an “uncountably infinite” number of points.