Could you explain Perron's paradox to me, please?

Question

This is Perron's paradox:

Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$.

What does it mean? I get from it that a very large number does not exist or $\infty=1$. Am I right? Or maybe the paradox is wrong?

Answer 1

It means that if there is a largest integer, then that integer is $1$. That assumption is what led to a silly result.

The point of this "paradox" is to not assume that something exists; the question of whether it exists or not is important to investigate. It has nothing to do with infinity.

Answer 2

It tells us that we can say anything about a number that does not exist.