I’m by no means a mathematician so this is a layman’s confusion after watching Youtube videos.
I understand why the (new) real number couldn’t be at any position (i.e. if it were, its [integer index] digit would be different, so it contradicts the assumption). But how does this **prove** that the set of real numbers is larger? This number indeed couldn’t be at any position, but it can **still** be mapped to an integer, because there are infinite integers – neither set can be larger because neither is finite.
Or are just “larger” or there being “more real numbers than integers” wrong terms to describe cardinality?
Pardon my ignorance.
Thank you.
In: Mathematics
What you’re confused about is the concept of a proof by contradiction.
So let’s break down the logic a bit. We want to show that the set of real numbers is strictly larger than the set of integers. By the definition of “larger” for sets, this means we want to prove that there’s no perfect one-to-one pairing between integers and real numbers.
We use proof by contradiction: we assume the opposite of what we want to prove, and then show that this leads to a contradiction. We assume that we have a perfect one-to-one pairing between integers and real numbers, and then show that, in fact, we don’t have such a pairing, because at least one real number is left out even though every integer is taken. In other words, we’ve shown that if such a one-to-one correspondence exists, then it actually doesn’t exist. That contradiction is enough to show that our initial assumption is false.
> This number indeed couldn’t be at any position, but it can still be mapped to an integer, because there are infinite integers – neither set can be larger because neither is finite.
That’s not true. By assumption, every integer is already mapped to a real number. You give me any integer, and I’ll give you the real number that it’s already mapped to, that’s not the new one we’ve found. You can’t just take the “next” integer—they’re ALL taken, including that “next” one, and the one after that, and all 50 billion after that, and so on. Just because there are infinitely many integers doesn’t mean any are still available—in fact, we assume at the outset that they’re all taken!
Your confusion might come from the fact that the argument is presented as writing out a list of these pairs, and clearly we can’t actually write out infinitely many pairs. But that’s just a simplification that’s made to make the argument easier to understand to laypeople—we don’t actually have to write out every single pairing. We just have to know that every pairing exists, and that every single integer is already paired up with a real number, so there are no integers left.
If you’re willing to wade through more abstraction, you might find the formal version of the argument presented on this wikipedia page clearer (this is a generalized statement, but the fact that the cardinality of the reals is larger than the cardinality of the integers is a consequence, since it’s pretty intuitive that we can establish bijection between the reals and the powerset of the integers using decimal expansions): https://en.wikipedia.org/wiki/Cantor%27s_theorem
Latest Answers