The concept of infinity has plagued a great many proofs, both formal and informal. I think that there are two foundational problems at play in most people’s thinking about infinity that causes issues.
The first problem people have with infinity is that they treat it as if it were a single value. Because infinity is bigger than all possible natural numbers, people assume that it is bigger than any number, and therefore there is nothing beyond infinity. Therefore, people have the concept that if I have two infinities, then I still have the same number. They believe that 2 * infinity = infinity. However, using that logic can quickly lead to contradictions.
This problem is exacerbated by much mathematical notation. People often will use ellipses to indicate that something goes on “for infinity”, whether a series notation or just a continuation of a decimal expansion. Therefore, they feel that the original set, and a set with just one more or one less added, are the same set. Therefore, you get “proofs” like the following “proof” for why 0.9999… = 1:
x = 0.999…
10x = 9.999…
10x – x = 9.999… – 0.999….
9x = 9
x = 1
The problem here is the assumption that, after multiplying by 10, 0.999…. has the same number of digits as 9.999… It is true that they both have an infinite number of digits, but they are not necessarily the same number of infinite digits.
Infinity is not a single number, but rather a class of numbers—like odd or even. Just like even and odd numbers, we can make some rules about their usage: odd + odd = even; even + even = even; even + odd = odd. However, we can’t simply substitute odd and even for the values they represent. For instance, consider the following obviously false proof:
x = 3; y = 2
x = odd; y = even
odd * even = even
3 * 2 = 6
y = even = 2
6 = even
Therefore: 2 = 6
The obvious problem with this proof is that it is considering a whole class of numbers to be equal to each other, rather than merely a member of a class. This is what we tend to do with infinite values—even though they are a class of numbers, we treat them all as if they were the same value.
Infinity is a Class of Numbers
Once we understand this, the problem with the original proof becomes clear. Let’s be more specific about how many digits that 0.999… has. We will use the Greek symbol ω to represent whatever specific infinity of digits that this number has after the decimal point. If we multiply the number by 10, then the value becomes 9.999… . However, since one of the digits was shifted left across the decimal, that means that, on the right of the decimal, there is only ω – 1 digits! This is still an infinite number, but it is a different infinite number.
The second problem with infinite numbers is that there are actually different, incompatible conceptions of them. The notion I have been describing so far is based on the hyperreal number system. An alternate system of infinities would be Cantor’s cardinal numbers. In Cantor’s system, infinities are grouped based on whether a one-to-one correspondence could be made between two sets. To see how this differs from the previous conception, imagine that we compare the set of all integers to the set of all even integers. On the hyperreal system, the set of all integers would be twice the size of the set of all integers. On Cantor’s cardinal system, since you can define a formula that maps a value from one set to the other (and vice-versa), they are both considered the same “size.” There are different sizes of infinities in Cantor’s system, but the differences are not as fine-grained as in the hyperreal system.
There are other conceptions of infinity as well, but these are the two approaches that I have seen have the most utility. The hyperreal system tends to work best with questions involving manipulating formulas, while Cantor’s cardinals tend to work best with philosophical questions about fundamental incongruences of different ideas (computer science’s “halting problem” can be proven using Cantor’s approach).
Infinity is a Valid Concept
Many view these issues of problems with infinity as reasons to exclude them from mathematics. However, history and logic show us that this is a bad idea. First of all, infinity certainly belongs in mathematics conceptually. If infinity was not a concept, then we could, for instance, name the largest integer value. Since there is no largest integer, it means that infinity is a valid concept, even if we aren’t always sure how to treat it.
However, this isn’t the first time we have added numbers which gave mathematicians fits. We are all very comfortable with having the zero in mathematics, but there was a time when it was controversial for many of the same reasons. Prior to the introduction of zero, every mathematical operation had an inverse operation. The inverse of addition was subtraction, and the inverse of multiplication was division. However, the introduction of zero meant that not every multiplication had an inverse operation. This caused quite a bit of consternation at the time. However, eventually, mathematicians developed the appropriate rules and conceptions for fully including zero into mathematics. This is a historical process that takes time. Imaginary numbers are a more recent example. There was initial hostility, and it took some time to resolve all of the problems, but they were eventually resolved.
My own thoughts are that, first of all, thoughts about the infinite have always been ever-present in mathematics (though sometimes implicit instead of explicit) and can’t be removed. Second, I think that mathematicians will eventually narrow down which conceptions of infinity are useful within mathematics, perhaps even to a single conception, though I have difficulty seeing how the hyperreal numbers and Cantor’s cardinals can be unified. Finally, mathematicians will develop rules to help people know which operations are valid in infinite contexts (or even in which infinite contexts), and how to know which type of infinite value you should be using in which kind of problem.
In all, the best way to avoid being tripped up by apparent proofs that include infinities is to be sure that the person giving the proof is clear about the type of infinity that they are using and the limitations, applicability, and rules of manipulation for that type of infinity. If not, then the proof is just as likely to be trading on the ambiguity of the term “infinity” as it is to be showing something true in mathematics.
See also these MM posts: