Infinite and Transfinite NumbersDate: 5/28/96 at 13:49:24 From: Richard Adams Subject: What is difference between infinite and transfinite? Can anyone explain to me, in a simple way, what transfinite numbers are and how they're different from infinite numbers? I seem to recall reading that someone (Hilbert?), found a way to do math with infinites and transfinites - how does that work? How can you subtract an infinite number from another infinite number and get any sort of reasonable answer? Thanks in advance! :) --Richard E. Adams Date: 6/14/96 at 01:12:05 From: Dr. Tom Subject: Re: What is difference between infinite and transfinite? Hi Richard, The word "infinite" in mathematics is a little dangerous, since it is used in many ways. For example, I can ask, "What is the limit, as x -> infinity, of (x+1)/(x-1)?" In this case, I can translate the problem into a form that doesn't use the word "infinity" as follows: "Find a limit L such that given any epsilon > 0, there exists an M > 0 so that if x > M, then |(x+1)/(x-1) - L| < epsilon." The concept of infinity is also used to answer questions like, "What is the area of the entire plane?" Clearly, whatever finite number you name, you can show that the "area" of the plane must be bigger than that. But there's another very precise use of infinity, and it has to do with counting. The set {A, B, C} has three elements, and I can prove it by constructing a one-to-one correspondence with a set that's known to have 3 elements. In fact, often the natural numbers are defined as sets: 0 = {} (the empty set) 1 = {0} 2 = {0,1} 3 = {0,1,2} and so on. In general, n+1 = n union {n} The nice thing about this is that I can show that a set has 5 elements if I can find a one-to-one mapping of the elements in the set with the elements in 5 (where we think of 5 as the set {0,1,2,3,4}). So what is the set: {0,1,2,3,4,...} that is, the set containing all the natural numbers. This is usually called "aleph-nought" (written with the Hebrew letter aleph, with a subscript of 0. Since I can't type Hebrew letters on my computer, I'll call it A0. This set is clearly infinite - it can't be matched up in a one-to-one way with any of the finite numbers. So what's A0+1? Well, why not do the same thing: A0+1 = A0 union {A0} In that way, we can get A0+2, ..., and if we take the union of all of those, we get what is called A0*2, et cetera. This process of unioning can be continued, in principal, forever, and the class of "numbers" generated is called the class of (transfinite) ordinal numbers. I could write about this for a long time, but let me make a few observations (some of which are not obvious): A0 and A0+1 are different, but they are the same size. There is a way to match up all the objects in A0 one-to-one with those in A0+1. There are ordinals, larger than A0, that cannot be matched one-to- one with the elements of A0. For example, the set consisting of all the subsets of A0 cannot be matched one-to-one with the elements of A0. Look up Cantor's famous "diagonalization" proof. The first time an ordinal is bigger than all those below it, it's called a "cardinal" number. Cardinal numbers measure the size of sets; ordinals are more closely related to ordering. There are more real numbers, for example, than there are natural numbers. But the question of whether the cardinality of the reals is the first cardinal greater than the cardinal A0 is undecidable - it is not invalid to assume it is, and it is not invalid to assume it is not. Paul Cohen proved this back in the 1960s using a method called "forcing". Just as you know that certain operations do not make sense in the natural numbers (for example, you cannot always divide one by another and stay inside the system of whole numbers), there are operations that do not make sense in the transfinite ordinals. In fact, addition and multiplication there are not commutative, and subtraction and division do not always make sense. -Doctor Tom, The Math Forum |
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