Understanding Zorn's Lemma and Its Implications in Vector Spaces

[Yuqing]

I was reading the proof that every Vector Space has a basis which invoked Zorn's Lemma. The proof can be found http://mathprelims.wordpress.com/2009/06/10/every-vector-space-has-a-basis/" .

Now I have an issue specifically with the claim that $$U := \bigcup_{S\in C}S$$ is an upper bound for C. Applying the same idea as the proof, this seems to imply that the natural numbers has a maximal element. Let C be a chain of natural numbers and similarly let us define $$A:=\sum_{n\in C}n$$ Then $A\in \mathbb{N}$ and is an upper bound for C. Applying Zorn's lemma then implies that the naturals have a maximal element.

What exactly am I missing here?

 

[Hurkyl]

Let C be a chain of natural numbers and similarly let us define $$A:=\sum_{n\in C}n$$ Then $A\in \mathbb{N}$

No it's not... (if C is infinite, anyways)