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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 N, be if we do not allow anything beyond ZF set theory? Surprise me.

Another question would be about the axiom of countable choice? Why do people accept this but not the axiom of choice proper? It is not intuitive (only less counterintuitive). Yes you can choose an element one by one, but this "intuition" leads to an inductive proof of finite choice for arbitrarily large finite n, not countable choice...

Is it accepted more sheerly on a practical basis? Because a theoretical one seems nonexistent to me.

Edit: I realised this is two questions, but perhaps countable choice is at least a connection between the two.

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    I'm not sure if my answer was what you were looking for. But there is really not much difference between ZF and ZFC when it comes to the natural numbers.2012-11-21
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    It is easy to be confused by some literature about the status of the axiom of choice. Almost all mathematicians "accept" the full axiom of choice. Of the small minority of those who do not, only a smaller minority nevertheless "accept" the axiom of countable choice. The main interest in the countable axiom of choice in set theory is that there are come models of ZF where the full axiom of choice cannot hold (because the models satisfy other things that contradict AC); some of those models do satisfy the countable axiom of choice, which makes useful to know what can be proved with it.2012-11-21
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    I never know when to give writing advice, but the first question can be asked in one quick sentence: "What is a comprehensive list of facts about the countable cardinal that depend on the axiom of choice?" Lots of verbiage (and misplaced commas) in there to ask that question. You get more answers to questions if people can quickly figure out the question.2012-11-21
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    @CarlMummert There is a distinction between "accepting" something as an axiom and "accepting" that it is true. The latter is really not a mathematical statement but a belief system.2012-11-21
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    Also, most mathematicians, if they find a proof that uses the Axiom of Choice, will immediately ask, "Is there a proof that doesn't use the Axiom of Choice?" :)2012-11-21
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    We prefer to use Countable Axiom of Choice, then, because we aren't using quite as "big" an assumption. It is still an assumption (aka an axiom.) We might prefer not to use choice at all, but if we must use choice, we might want to know how "small" we can make the assumptions.2012-11-21
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    @Thomas: Who is *we*? I spoke to several prominent set theorists which expressed some questions about the usage of inaccessible cardinals in Wiles' proof of FLT. No number theorists they spoke with (and all of them asked prominent number theorists) cared about this fact. I always found large cardinals to be much more worrisome (in terms of consistency) than the axiom of choice. Why do people care about choice, but they don't care about large cardinals, then?2012-11-22
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    I didn't say that we prefer to use Countable choice compared to large cardinals, I said we prefer countable choice compared to more general AofC. Most of mathematics is interested in finding smaller conditions under which something is true. I'm not sure why inaccessible cardinals come up in this discussion, however.2012-11-22
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    That said, if inaccessible cardinals are known to be independent of ZF and/or ZFC, what this would show is that FLT is at best "undecidable" in ZF/ZFC. That would mean that Wiles has (intuitively at least) proven that nobody will ever write down a counter-example to FLT.2012-11-22
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    @Thomas: No... in order to show that one would have to show that FLT is **equiconsistent** with inaccessible cardinals. Do note that Wiles proved a vastly stronger statement in a much more general context. The reason I brought up inaccessible cardinals is that adding the whole axiom of choice will not introduce new inconsistencies compared to countable choice; whereas inaccessible cardinals make the theory stronger and therefore *may* introduce inconsistencies. If people are fine with inaccessible cardinals, then the axiom of choice should be just peachy. That's my point.2012-12-12

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