After searching about the Axiom of Choice for a while, despite starting with a massive headache, it started feeling more intuitive. Basically any uncountably infinite collection of somethings cannot be combined into a Cartesian product of nothing. Choice functions for non empty sets just exist. All vector spaces have bases. Even things like Banach-Tarski which felt super weird at first start to make sense when reading up on more simple concepts like Vitali sets and the hyperwebster. Well-ordering theorem and Zorn’s lemma still feel strange, but I can accept that they can somehow happen even if it is a fact that I have no hope to construct the exact way it is done.
But then comes the Axiom of Determinacy, which is inconsistent with Choice, and which actually felt more intuitive before I started reading about these concepts. If two players enter a game with full information, it should be obvious there is a winning strategy, right? Yeah I know uncountable infinity is weird. Yet this exact axiom gives us two reasons it should be less weird. One is that subsets of R are measurable, that should feel more OK.
But the other is R possibly being a countable union of countable sets. What is the point of cardinalities anymore? If you cannot just take the power set of the power set of the power set… etc and have it make sense, how is it even structured? What seems like a thing that makes the continuum easier makes everything beyond it harder.
What does it even mean for a vector space to not have a basis? Even worse, how can you partition into disjoint subsets and end up with more parts than elements? Unfortunately my math intuition fails completely with this one.
Also how exactly is choice and determinacy at the same time a contradiction? Is it the simplistic “you cannot determine the choice function”, or am I missing some bigger picture here?
Thanks in advance for any answers. I know this is gonna be a hard one, non well structured infinite sets are a very weird concept.
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