Is the axiom of choice really related to choice?

See Scott Aaronson's answer to How can the Banach–Tarski paradox make sense to mathematical laymen? There he mentions that denial of the axiom of choice also leads to bizarre results. Could there be a weakened version of the axiom of choice...

I'm curious specifically about processes whose outcomes cannot be computed by other means. It is my understanding that axioms should be assumed as true because they are self-evidently true, and this is what distinguishes useful mathematics from that...

I get the idea of set theory however I do not understand the axiom of choice nor why it is significant.

Here's the proof: First, we note that there's a one-to-one function from SSS to 2S2S2^S given by x↦{x}x↦{x}x \mapsto \{x\}. Now we assume that there's a bijection Ï•Ï•\phi between SSS and 2S2S2^S.  Then there is some xâˆ...

Can it be discarded without changing anything we understand about the physical world?  Would the mathematical expression of any current physical theory be unsound?  Would differentiation or integration suddenly have no theoretical basis, for example...

In other words, what interesting statements require the existence of choice functions for uncountably infinite collections of sets?

Consider an organization of your choice and carefully go through marketing research process to solve marketing related problems or opportunity that may be relevant to chosen ...show more

Multiple choice: The objective with pain is to: (A) Identify the location of the primary pain (B) Identify the secondary locationsof pain (C) A & B (D) None of the above Note: I found a sentence in my text that reads, "The ojbective is to identify...

When choice and responsibility are proportionally linked, that means that to whatever extent I have a choice about something, I am responsible for the outcome or consequence of that choice. And to whatever extent I am held responsible for any sort of...