Mathematics

Is there any number larger than all other numbers? George Cantor proved that that even infinite quantities may be smaller than other infinities. Still, might there be some infinite number that is greater than all other infinite numbers?

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Infinite numbers are not found in nature but rather constructed through mathematical axioms and reasoning. This is somewhat analogous to how we can also construct the natural numbers by starting from 1 and then adding 1 again and again. We start from a set of cardinality aleph-naught, for example the set of all natural numbers. (Cardinality is a measure of how many elements the set contains; and aleph-naught is countable infinity: the cardinality of any set whose members can be mapped one-to-one into the natural numbers.) We then construct a set of higher cardinality, for example the power set of the set with which we started. The power set of any set S is the set of all the subsets of S -- and Cantor showed that the powerset of the set of all natural numbers has higher cardinality than the set of all natural numbers (i.e., that the powerset of any countably infinite set is uncountably infinite). Sets of even higher cardinality can be constructed through replication of a simple principle, much like ever larger natural numbers can be constructed through adding 1 yet one more time. So the mathematician can, for any infinite number/cardinality you may propose as the largest, construct a larger one -- much like you can easily, for any natural number proposed as the largest, construct a larger one by adding some natural number to it.

What Prof Pogge has said represents one perspective on this issue, but it involves assumptions that can be rejected. The central issue is whether you are prepared to speak of "how many sets there are". If so, then let Fred be how many sets there are, that is, the number of things that are sets. It is sufficiently clear that Fred is the largest number.

In standard set theory, by which I mean Zermelo-Frankel set theory (ZF) and its extensions, there is no such thing as the number of things that are sets. There just isn't such a number. But there are other set theories in which there is such a number, and one can, in fact, consistently add to ZF an axiom known as HP which allows us to speak of (cardinal) numbers in a way different from how ZF by itself allows us to speak of them. And then there is a number of all the sets there are, and it is again the biggest number.

How can the question how many sets there are simply fail to have an answer? The idea to which Prof Pogge is giving expression is similar to what is known as "indefinite extensibility", the idea being that, in some sense, the universe of sets is never completed but is always in process. If something like that were true, then one can see why there might not be a number of all the sets there are. But this point of view is not wildly popular. Most people—and especially most people who work on set theory—would suppose that sets and numbers, infinite and otherwise, are no more "constructed" by us than are photons, quarks, or electromagnetic fields. As Frege famously wrote, "...[T]he mathematician cannot create things at will, any more than the geographer can; he too can only discover what is there and give it a name" (Foundations of Arithmetic, section 96).

So I am a little puzzled what it means to say that "infinite numbers are not found in nature". I wouldn't have thought any numbers were found in nature. Birds, trees, rocks, stars, photons, and the like, yes, but not 2 and not 1.765 and not π. (If 2 is in nature, where is it?) Perhaps what is meant is that infinite sets are not found in nature, whereas finite sets are. But then I'd insist that there are no sets to be found in nature, period. The elements of the set {Chicago, Dallas} are to be found in Illinois and Texas, respectively, but the set itself is not anywhere. So perhaps the question is whether infinity is in some sense "witnessed" by nature. That depends, among other things, upon whether space is continuous. If so, then there are infinitely many points of space anywhere you care to look. And, according to the fundamental physics of our day, highly complicated mathematical structures are in the same sense "witnessed" by nature, and these structures are very often "infinite" in some sense or other. But however that question is answered, it isn't at all clear what its significance is supposed to be. If the power set of the power set of the power set of the set of real numbers is not witnessed by nature, what of it?

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