We are taught that infinity is the largest possible number, but does that mean if something is infinitely large something else cannot be larger? I have pondered this occasionally for many years since it first occurred to me during grade school. Perhaps infinity should be allowed to be raised to powers and expressed exponentially.
Take a one dimensional line as an example. Although it has no width, it contains an infinite number of points. Conventional thinking would dictate that there is no larger number than the number of possible points on this infinite line. But what if we compare it to a plane? Now that we have added a second dimension that also extends infinitely, there is an infinite number of points above and below every point on the original line. Would it not be accurate to refer to this as infinity squared? We could then add a third dimension and consider a cube that extends infinitely in all three directions. Now, for every point on the plane there is an infinite number of points in front and behind it. Would this not be infinity cubed? As we continue this thought exercise it becomes much more difficult, but it would seem logical to progress through a higher exponent for each additional dimension, each one being infinitely larger than the dimension before it. We could consider naming this concept beyondfinity, to contrast it with infinity.
Discuss
Kyle Tate on January 18, 2008, 12:45 PM
you do not have a proper notion of infinity. It is not a number like 56, and the usual operations cannot be performed on it. You are correct though that there are notions of different sizes of infinity. Consider the set of integers and the set of real numbers, Cantor proved that the set of real numbers must be larger than the set of integers. The notion of infinity on the plane is different then the notion of infinity on the line, wikipedia the Riemann Sphere for more about that.
Benjamin Dozier on January 18, 2008, 1:21 PM
Mathematicians, and in particular set theorists, have been studying problems like this for a very long time. It turns out that there are in fact different types of infinities, but we need a more rigorous way of comparing the sizes of two different sets that both have infinitely many elements. The concept of dimension is not particularly useful here. Rather, we say that the two sets are equal in size (mathematicians would say the two sets have the same cardinality), if there is a one-one correspondence between the elements of the two sets. So for instance, if your first set is the positive integers (1,2,3,…) and your second set is the even positive integers (2,4,6,…), then you can form a one-one correspondence by matching each element in the first set with its double in the second set, and thus the two sets have the same size. This result seems a bit counterintuitive at first, since the first set has all the elements of the second plus a bunch more, and so we would think that the sets have different sizes. Yet this notion that the existence of a one-one correspondence is equivalent to two sets having the same size is actually a very natural generalization of methods that we are familiar with for determining whether two finite sets have the same size. Anyway, if you study things like this a bit more (maybe start with wikipedia’s “Countable Set” entry), you can prove that there are in fact different sizes of infinity. For instance, the real numbers (or the points on a line) have a different size than the integers or the rationals. However, the set of points on the plane turns out to have the same size as the set of points on the line. There is a very rich, beautiful theory of infinite sets, and I would encourage any one who is interested in infinity to take a look at the topic from the rigorous, mathematical perspective.
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