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Unformatted text preview: ECE 3250 NUMBERS Fall 2008 Engineers, scientists, and applied mathematicians think about numbers all the time, but usually in a utilitarian way. We manipulate them, calculate with them, make plans based on them, drive cars and fly in airplanes whose design depended on them, and so on. We don’t spend a lot of time thinking about what numbers “actually are” or “where they come from.” Numbers are just kind of “out there.” We all have our own ways of visualizing numbers, and we understand on some level how different sorts of numbers are related. We think of the integers, for example, as forming a subset of the rational numbers, which in turn form a subset of the real numbers. And the real numbers constitute a subset of the complex numbers. Once in a while, it pays to spend some time pondering numbers a bit more deeply than we usually do. That’s what I’ll attempt to do in what follows. 1. Sets, Cardinality, and the Natural Numbers Let’s talk about sets. You have to be careful when you define what “set” means if you want to have a “set theory” that works in the sense that it doesn’t lead to logical contradictions. For example, if you define a set as an arbitrary collection of objects, Russell’s Paradox comes into play. Here’s how Russell’s Paradox goes. Specifying a set entails, in some sense, listing the elements of the set. So you can think of a set simply as a list. Here’s an example of a list: (1) Collegetown Bagels (2) Radiohead (3) One teaspoon of salt (4) Nitroglycerin Here’s another one: (1) Sibley Dome (2) Stella’s (3) This list (4) The Foo Fighters (5) The song “Video Killed the Radio Star” The first list, although it enumerates some rather unrelated things, is not as unusual as the second list. The second list contains itself as a list item. Let’s call a list anomalous if it lists itself as an item. Next, define L R as the list of all non-anomalous lists. Question: is L R anomalous? I.e., does L R list itself as an item? If you have a clean answer to that, please see me so we can flesh out a third-millennium alternative to Aristotelian logic. If L R lists itself as an item, then L R is by definition anomalous — but that’s a contradiction, since L R lists only non-anomalous lists, by construction. On the other hand, if L R does not list itself as an item, then L R is by definition non-anomalous — but that, too, is a contradiction, since L R lists all non-anomalous lists by construction, hence would have to list itself if it were non-anomalous. Behold Russell’s Paradox. What went wrong? Essentially, what happened was that we attempted to define “set” too broadly. It won’t do merely to say that a set is any collection of objects. Only certain collections of objects can qualify as sets if we are to have a set theory immune to Russell-type paradoxes. Different stipulations of exactly what collections to deem sets lead to different approaches to set theory. Many such different approaches are mathematically viable and several are particularly popular with working...
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell.