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Why not simply use right away? Why is an axiom and what is not?

Some of the obvious statements are made into axioms whereas other obvious statements aren't.

For example, the axiom of choice, arbitrary union axiom, etc. Isn't it so obvious?

Yashas
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  • Why is it obvious? After all, informally speaking (and glossing over what that following statement means formally), the union of ${0}, {0,1}, {0,1,2}, \dots$ is ${0,1,2,\dots}$. That's a qualitatively different set from ${0,1,2,\dots,n}$ (being infinite rather than finite). We need the axiom of union to tell us that this "qualitatively different" doesn't actually make a difference. – Patrick Stevens Aug 15 '16 at 16:50
  • The point is not to be "obvious" if not define some theory properly. You can define, by example, a set theory without the axiom of choice. If you want to understand this way to do mathematics you must read about the develop of formalism due to the "discovery" of non-euclidean geometries. And, moreover, in different theories some axiom can be proved as a theorem and viceversa. – Masacroso Aug 15 '16 at 16:51
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    You might want to read about the Banach–Tarski paradox. If you still think the axiom of choice is obvious afterwards, all the more power to you! – Harald Hanche-Olsen Aug 15 '16 at 16:53
  • What obvious statements are not axioms or derived from axioms? – quid Aug 16 '16 at 15:59

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In general, "obvious" statements sometimes turn out to be false. For example, it was assumed for a long time that any continuous function must be differentiable except at isolated points, until a counterexample was constructed by Weierstrass. There are many other instances of this throughout history.

So if you want to make sure a statement is true, no matter how obvious it seems, you either need to prove it as a result of things you already know, or you need to take it as an axiom. What should be axioms and what should be theorems is a subtle question with no universal right answer, but in particular, the axiom of choice cannot be proven from the ZF axioms (the most commonly-used set theoretical foundation), so if you want to use it, you have to take it as an axiom.

  • Do we really have axioms for everything? Every obvious thing can be either proved or linked to an axiom? – Yashas Aug 15 '16 at 16:54
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    Ah, but it is the curse of mathematics that, as you learn it, what was once obvious is no longer so. Better get used to it … – Harald Hanche-Olsen Aug 15 '16 at 16:55
  • @YashasSamaga "Obvious" is a subjective term, but there are inherent limitations that prevent most useful axiomatic systems from being able to prove every true statement. If that sounds weird, it's because it is. You can read up on Godel's Incompleteness Theorems for more info on that. –  Aug 15 '16 at 16:59
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The Axiom of Choice is a classic example of the danger of "intuition". The quip by Jerry Bona, Professor of Mathematics at the University of Illinois at Chicago, seems appropriate here:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona

The irony is that all three of these principles can be proven mathematically equivalent so if you accept Choice as "obvious" then you must also accept the counter-intuitive well-ordering principle and Zorn's lemma, as well as the Banach–Tarski paradox mention by a commenter.

Gary Kibble
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