Using Zorn's Lemma

Set Theory · Axiom Academy

Walk through the proof that every vector space has a basis using Zorn's Lemma Excellent work! You've completed this example. Here's what we learned: Setup the Poset: We consider the collection of all linearly independent subsets of a vector space, ordered by inclusion. This is the right structure to apply Zorn's Lemma. Verify Chain Condition: The key step is showing that every chain has an upper bound. The union of linearly independent sets in a chain is itself linearly independent because any finite linear combination only uses vectors from finitely many sets in the chain. Apply Zorn's Lemma: Once we verify the chain condition, Zorn's Lemma guarantees a maximal linearly independent set exists. Maximality Implies Spanning: A maximal linearly independent set must span the entire space. If it didn't, we could add another vector and contradict maximality. The Power of AC: This proof fundamentally requires the Axiom of Choice (via Zorn's Lemma). There are models of ZF where vector spaces without bases exist! This technique applies to many existence proofs in algebra: maximal ideals, algebraic closures, and more!

This is the written version of the interactive lesson above. See the full Set Theory course.