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How do we know when we are allowed to use transfinite induction in a proof?

Edit: considering the replies, I should say the following:

Consider an infinite sum of fractions.

By induction we can show that for any finite step of this sum we get another fraction. However the infinite sum (step at $\infty$) might be irrational.

So we cannot use induction till ordinal w / cardinal $\aleph_0$ / infinity.

Similarly I ask when we are allowed (or not allowed such as in the example above) to use transfinite induction.

I appreciate the answers and they are not 'wrong' but I don't think they address this, hence the edit. My apologies for not being clear enough.

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    Um... if you refer to the use of Zorn's Lemma, you can use it if you assume the axiom of choice. If you refer to "when can I use it so that it works?", well, no one can tell you precisely when the axiom of choice can prove something or not... do you have a more precise question in mind?2012-09-14
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    Your edit makes even less sense.2012-09-16
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    Concerning your edit, it seems you are misunderstanding the definition of a series. Adding an infinite number of rationals is not well defined; what you are doing is finding the limit of a summation process. And of course, a limit of rational numbers can be irrational - otherwise the rationals would form a complete metric space.2012-09-16
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    This is why you are led to think that induction does not work in this case. All induction allows you to do here is to show that at the $n$-th stage (for $n$ *finite*) you have a rational number. Nothing else.2012-09-16

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