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Do we need axiom of choice for all the theorems studied in basic courses like calculus 1 and linear albgebra? could we build this theory just by using ZF axioms?

Andrés E. Caicedo
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A student
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    Calculus without the Axiom of Choice is like a fish without a bicycle. – Lucian Oct 14 '14 at 07:15
  • In proving for example, that an infinite set of points inside a closed segment has an accumulation point. We take a decreasing sequence of segments, $... \subset I_n \subset ...\subset I_2 \subset I_1 $ and CHOOSE a point in each. Don't we need axiom of choice for that? – A student Oct 14 '14 at 07:25
  • Axiom of choice isnt related to calculus from what I know. It is related to the possibility to count different cardinals alef and for some extravagant theorems as the theorem of Tarski-Banach. Maybe in combinatorics/probability take more uses too, not sure. – Masacroso Oct 14 '14 at 07:31
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    Related threads: one, two, three, four, and there are many many more specific ones. I'll close this as a duplicate of the first, since it's rather general. – Asaf Karagila Oct 14 '14 at 07:33
  • I also found this post interesting http://math.stackexchange.com/questions/126010/continuity-and-the-axiom-of-choice. Basically, you need some form of AC to get the equivalence between continuity and sequential continuity. – PhoemueX Oct 14 '14 at 07:34
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    @Lucian: Not quite. You use the axiom of choice for some basic theorems like the equivalence of continuity at a point (limits vs. $\varepsilon$-$\delta$); A student: No, choosing from intervals is easy, just pick a rational number using the fact the rationals are well-orderable (it is a countable set after all). – Asaf Karagila Oct 14 '14 at 07:45
  • @AsafKaragila: The mere fact that the only ones bringing up that notion are almost exclusively set theorists speaks volumes. When was the last time anyone ever heard such questions being genuinely asked by someone not in that particular field ? The rest of the $($mathematical$)$ world seems to go on just fine without it. – Lucian Oct 14 '14 at 08:29
  • @Lucian: I don't know what you are trying to say. Are you saying the only people who use continuity by sequences are set theorists? That just doesn't sound right. I also don't know how many set theorists care about that either. Your comment, in a whole, is quite insulting as a set theorist. – Asaf Karagila Oct 14 '14 at 08:39
  • @AsafKaragila: Please don't be upset, because that would make me sad also. And, without denying the theoretical $($objective$)$ importance of axioms $($which is made subjectively stronger in those whose inclinations lie towards the more abstract parts of mathematics$)$, I believe the right word describing the relevance of this axiom is “overrated”. This is my sincere opinion; I do not wish to offend anyone. – Lucian Oct 14 '14 at 09:02
  • When we are not dealing with set theory, then, we assume that everyone "knows" which assumptions can be used without a proof and which should be proved. But, the whole point in mathemathic is that you can not relay on anything without formal "objective" proof. This is why, I think, at some point, set theory was needed. This kind of "agreement" which I think Lucian is reffering to, must be satteled, I think. This is why I asked this question. I will have a look in the references. Thank you. – A student Oct 14 '14 at 13:27

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