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Foundations of Algebra

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Foundations of Algebra Foundations of algebra The well-ordering principle In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded total order. The set S together with the well-order relation is then called a well-ordered set. Every element, except a possible greatest element, has a unique successor (next element). Every subset which has an upper bound has a least upper bound. There may be elements (besides the least element) which have no predecessor. http://en.wikipedia.org/wiki/Well-order The Well-Ordering Principle for the Naturals (N) is straightforward: Every non-empty subset of the natural numbers has a least element [i.e. the natural numbers are a well-ordered set]. This appears to be common sense; N itself has a least element (0 or 1, depending on text), and of course any subset we examine (the set 4, 5, 6, 7, 8, 9, 10 , the set of primes, the set of odd natural numbers, etc.) will of course have a least{ element. There are} a few interesting things about the Well-Ordering Principle, however: The assumption behind it Saying that a set has a “least element” assumes that there’s some sort of meaningful way of ordering the numbers on that set so that one is clearly less than another. For the naturals, the binary relation fulfills that role (and you can follow the Wiki link to a definition of “total order”). ≤ It’s not always clear cut how to order a set in general, however, even a set of numbers. Imagine the set of ordered pairs (1, 3), (3, 1), (1, 2), (2, 2) . How would you order the pairs themselves? Does (1, 3) come before (3, 1)?{ How about (1, 2)? } No, we’re not going to address that here - just something to think about. “Order” doesn’t just happen naturally - it’s a relation that has to be defined on a set. Name confusion You’d think that something as straightforward as the above would be impossible to botch up, but hey, it’s math. If you start searching a little on the term “well-ordering,” you’ll run into the following well-ordering axiom: Every set can be well-ordered. That’s a pretty strong statement. It’s equivalent to something known as the Axiom of Choice,and that’s one of those historical math controversies that’s been flying around for a while. The Axiom of Choice is to set theory as Euclid’s Fifth (Parallel) Postulate is to geometry - it seems obvious, but you can’t prove the thing, and it has to be taken as an axiom. Replacing it with a different axiom gives you a new system that seems counterintuitive, but is internally consistent. Ditching Euclid’s Fifth gets you some interesting non-Euclidean geometries. Ditching the Axiom of Choice gives you some interesting alternative set theory. We are NOT pursuing any non-Axiom-of-Choice algebra! But if you’re into that sort of thing, follow the links for a while :) Principle of Induction The Well-Ordering Principle on the Naturals (going back to the initial statement) is equivalent to the Principle of Induction - assume one, and you can prove the other. The Principle of Induction is the fifth Peano axiom, and will get a section of its own, coming up next. Generalization of the Well-Ordering Principle We won’t leap right into the idea that every set can be well-ordered, but we would like to generalize the Well-Ordering Principle to other sets of numbers, in particular, the integers. We can’t say that every subset of the integers has a least element; for example the set ..., 3, 2, 1, 0, 1, 2, 3, 4 { − − − } is a subset of the integers, and clearly does not have a least element. Instead, we state the General Well-Ordering Principle: Suppose that n0 is an integer. Suppose that S is a non-empty subset of the integers, and that every element of S is greater than or equal to n0.ThenS has a smallest element. That simply says that any set of integers which is bounded below has a least element. So the example ..., 3, 2, 1, 0, 1, 2, 3, 4 { − − − } wouldn’t contradict the principle, since it isn’t bounded below in the first place. The General Well-Ordering Principle seems trivial, and circular to the point of uselessness: doesn’t it basically say that any set of integers with a least element has a least element? Well, not quite - it’s entirely possible that we know that a set of integers has the property that it’s bounded below (say, we know a rough lower bound for the set is that every element is greater than 10), but we don’t know precisely what’s in the set (we’re not sure if 10 is included, or maybe its least− element is really 9, or even +6). Simply knowing that there is a− lower bound, even with only a rough idea of where− it is, is enough to know that the set has some element which is its least element. So, not useless. Also, not trivial - once again, it’s something that needs to be asserted, and can’t be stated about all sets of numbers. Consider a set of real numbers: the open interval ( 1, 1). That set is bounded below (every element is greater than 1), but we can’t assert that it has− a least element - you can get arbitrarily close to 1. In the same− way that a well-ordering principle for the integers must be stated differently than− the well-ordering principle for the naturals, a well-ordering principle for the reals would have to be stated differently than the one for the integers. So don’t assume that any statements of well-ordering hold outside the set they specifically refer to..
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