How is it possible that for the set of natural numbers N, no elem of N is the biggest and every elem of N has something bigger?

Question

(N = {1,2,3,4,...})
It is straightforward to prove this. Assume n is elem of N such that n is the maximum of the set. But exists m that is elem of N such that m > n, then n is not maximum. But we defined n to be the maximum. Contradiction! Therefore no element of N is maximum.

But I cannot comprehend the proposition in the title, it seems like a contradiction. Can anyone help me understand this concept?

Answer 1

This is a known true proposition, i just cannot comprehend it. And for every number n there exists a number m bigger than n, so the number m exists, just define it as n+1.

Answer 2

The set of reals associated with addition is a known commutative group, this means it is closed under addition, meaning that the addition of two reals always gives a real. if n is a real, then n+1 is too because the number 1 is real. therefore n+1 exists if n is real.

Answer 3

But this means no number is the biggest, and what if we define something else as being bigger than any real? does that make sense?

Answer 4

Well, it is a question about infinity so... yes haha. basically, im just saying that i cannot comprehend infinity, and I was asking for help.