Understanding Zorn's Lemma and Its Implications in Vector Spaces

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In summary, the conversation discusses the proof that every vector space has a basis, which invokes Zorn's Lemma. However, there is a discrepancy regarding the claim that U := \bigcup_{S\in C}S is an upper bound for C. This seems to imply that the natural numbers have a maximal element, which raises confusion as it contradicts with Zorn's Lemma. Further discussion involves defining A as the sum of all elements in the chain C and applying Zorn's Lemma, which leads to the conclusion that the naturals do not have a maximal element. The conversation ends with a question about what may be missing in this reasoning.
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Yuqing
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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/" [Broken].

Now I have an issue specifically with the claim that [tex]U := \bigcup_{S\in C}S[/tex] 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 [tex]A:=\sum_{n\in C}n[/tex] Then [itex]A\in \mathbb{N}[/itex] 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?
 
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Yuqing said:
Let C be a chain of natural numbers and similarly let us define [tex]A:=\sum_{n\in C}n[/tex] Then [itex]A\in \mathbb{N}[/itex]
No it's not... (if C is infinite, anyways)
 

1. What is Zorn's Lemma?

Zorn's Lemma is a fundamental theorem in set theory. It states that if every chain in a partially ordered set has an upper bound, then the set contains at least one maximal element.

2. Why is Zorn's Lemma important?

Zorn's Lemma is important because it allows us to prove the existence of maximal elements in certain sets, which is useful in many areas of mathematics such as algebra, topology, and order theory.

3. Who is Zorn and how did he come up with this lemma?

Max Zorn was a German mathematician who first stated Zorn's Lemma in 1935. He used it as a key tool in his proof of the Axiom of Choice.

4. Can you give an example of how Zorn's Lemma is used in mathematics?

One example is in the proof of the Hahn-Banach Theorem in functional analysis. Zorn's Lemma is used to show the existence of a maximal linearly independent set, which is a crucial step in the proof.

5. Is Zorn's Lemma equivalent to the Axiom of Choice?

Yes, Zorn's Lemma is equivalent to the Axiom of Choice, meaning that either one can be used to prove the other. However, Zorn's Lemma is considered to be a weaker version of the Axiom of Choice.

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