The dictionary defines a paradox as a “seemingly” or “apparent” self-contradiction. It’s often used to describe two things that strike us as both true and mutually exclusive, which would make them impossible. And they always are just “seemingly” mutually exclusive — given time, we always solve them.

Let’s consider the Sorites paradox, also known as the “paradox of the heap”. If you have a pile of sand containing 10,000 grains of sand, it fits the definition of a “heap”. If you remove a single grain, isn’t it still a heap? Intuitively, it’s still as much a heap as it was before. How about if you remove another? And another? At some point you’ll be left with a single grain of sand and still referring to it as a heap. You could arbitrarily set a point at which it passes from heap to non-heap, but that doesn’t feel right.

The Sorites paradox has been answered multiple ways, but my favorite solution is fuzzy logic in which the heap becomes less of a heap as grains are taken away, and no true boundary has to be set but rather you could describe it with terms such as “small heap” or “medium heap” or “somewhere in between a small heap and medium heap”. Because we have answers like this, the paradox is no longer truly a paradox but rather one that only seems like a problem with mutually exclusive conclusions.

Another set of paradoxes to consider are Zeno’s paradoxes, such as Achilles and the Tortoise. Achilles chases a tortoise but can never catch it because every time he has gone half the distance the tortoise has still gone further, and once he has traveled half of that new distance between them the tortoise has still traveled further so he can never catch up (even though we know he obviously can in real life). We also have the paradox of the arrow in which an arrow can never hit its target because in any instant of time the arrow is motionless (motion occurs over a change in time), and the elapsed time between the arrow leaving the bow and hitting the target is just a series of “instants of time” in which the arrow is motionless. And yet, arrows obviously do travel and hit targets.

A good answer to Zeno’s paradoxes is the geometric series that calculus improved upon with the convergent series, and we’ll be visiting and expanding upon that in the near future (because it is an alleged method for proving that .999… = 1).

So my paradox (which is strangely not seen as a paradox) is this: the ratio of all integers to even integers is inconsistent. Mathematicians argue that there are just as many integers as there are even integers because you can multiply any integer by 2 to get an even integer. So if 1 x 2 = 2 and 2 x 2 = 4 and 3 x 2 = 6, etc then the set of integers [1, 2, 3…] and the set of even integers [2, 4, 6…] have the same number of numbers in them. But that’s obviously not the case. If you match the numbers among all integers rather than multiply them, you can match every other integer but all of the odd ones have no partner. In this instance, there are twice as many integers as there are even integers (which follows intuitively).

In other words, you can match all integers to even integers on a 1:1 ratio by multiplying them by 2 (2X = Y) or you can match half the integers to even integers on a 2:1 ratio by multiplying them by 1 (X = Y). Either these sets are equal or one of them is larger, but they can’t simultaneously be both — that would give us mutually exclusive conclusions, otherwise called a paradox.

Perhaps this is a known paradox and I haven’t come across it yet, or perhaps it has even been solved. In any case, I think modern mathematicians are too dismissive of it and simply deny that there is more than one way to look at it (they see it only as a 1:1 ratio), and I think it deserves some deeper thought.

Galileo’s paradox comes to mind or is it not the same thing?

I was not aware of others stumbling upon this first. Thanks for the post!

“In any case, I think modern mathematicians are too dismissive of it and simply deny that there is more than one way to look at it (they see it only as a 1:1 ratio), and I think it deserves some deeper thought.” I think it’s because you’re looking at it wrong. The reason you try to match them to natural numbers is so that you can assign a number to the set. For example, in the set of naturals starting from 1, N={1, 2, 3, …}, the first is 1, the second is 2, the 43rd is 43 and so on. It’s about being able to look up N(n) for any natural number n. Likewise, in the set of even naturals, E={2, 4, 6, …}, the first is 2, the second is 4, the 43rd is 86 and so on. You’ve assigned every E(n) to a natural number so you can look up the first number in your set or the 157th or whatever. While yes, I can see your point that there seems to be twice as many natural numbers as evens, because the odds aren’t assigned to a spot in the evens, it’s because we don’t count like that. The point isn’t to assign arbitrary sets to other arbitrary sets for the sake of it, it’s so we can look up items in the set like looking up pages in a book.

I understand how to match sets. The fact that “we don’t count like that” isn’t because we *can’t* count like that but because mathematicians have chosen not to do it that way… but we obviously *can*. Your argument basically boils down to the idea that mathematicians only look at it one way and ignore the other, which is more or less what I said myself. The point here is that I think they ought to resolve this paradox through means other than denial.

Really? When you’re counting how many oranges you’ve got, you count 2, 4, 6 and wind up with double the number you actually have?