The thing is that limits of sequences are defined in a way that make you check infinitely many finite conditions:

A sequence (a1, a2, a3, ...) has a limit a if for every e > 0 there exists such k that for every n >= k it holds:

| an - a | < e

The reason I said m < 0, was because if we assume sum(N) = -1/12, then directly from this definition by setting e to e.g. 1/24, there has to exist a k, such that for every n>=k:

| (1 + 2 + 3 + ... + n) - (-1/12)| < 1/24

which means that

(1 + 2 + 3 + ... + n) < -1/12 + 1/24 = -1/24 < 0.

This has to hold for all n>=k, so it should hold for n=k. This implies that for some large enough n

m = (1 + 2 + 3 + ... + n) < 0.

But m is a natural number.

That's it. The sum of all natural numbers looks way too mysterious. Because any infinite sum is defined as the limit of partial sums, we can rewrite it as

sum(N) = lim (1 + 2 + 3 + ... + n) = lim (n (n+1) / 2),

as the sum of the first n natural numbers is

n (n+1) / 2

About incompleteness, If you define your axioms as all the true statements in a certain structure, then you have a consistent and complete theory. The problem is that those axioms will be undecidable (you can't theoretically compute which one is in the set).

I think ZFC has too little axioms to be complete (they're decidable). You can live with incompleteness, you shouldn't be able to live with inconsistency. But from Godel's second incompleteness theorem, we'd only ever be able to prove that ZFC is contradictory. If it's consistent, we'll never be able to prove it.

I have never heard of VGB. It'd be great if you'd give me some links about it and the connection to sum(N) = -1/12.