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### Why is the axiom of choice controversial? [duplicate]

In other words, what are the arguments for ZF over ZFC, and what philosophical issues have people raised against including it as a standard axiom of set theory?
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### Uses of Axiom of Choice [duplicate]

I am a first-year maths student but I occasionally drift away from our taught material. Some years ago I saw the ZFC axioms for the first time, but now that I am in college, and although the stuff I'...
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### Foundation for analysis without axiom of choice?

Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ...
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### Advantage of accepting the axiom of choice

What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)? It seems that there is no clear reason to prefer over other axioms.. Thanks for help.
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### Axiom of choice - to use or not to use

I was wondering if there are examples of results in mathematics that were first proven using axiom of choice and later someone found a proof of the result without using the axiom of choice.
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### Is there a *simple* example of how the axiom of choice can lead to a counterintuitive result?

This question is quite subjective, but here goes: The axiom of choice notoriously leads to many extremely counterintuitive results, like the Hausdorff, Von Neumann, and (most famously) Banach-Tarski ...
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### What's the thorny issue on: “If all $S\in \ell$ are nonempty, does it follow that $\prod_{S\in \ell} S$ is nonempty? when $\ell$ is infinite?”

I'm reading Paolo Aluffi's ALGEBRA, Chapter 0. Here he proposes that there's a thorny issue: What is this thorny issue?
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### Countable infinity and the axiom of choice

Many of the states of affairs about infinite cardinalities and their size, depend on the axiom of choice. What would a comprehensive list be of the properties of the cardinality of the natural numbers ...