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Let's start with the Cambridge Dictionary definitions of the words, indefinite and indefinitely:

indefinite: not exact, not clear, or without clear limits

indefinitely: for a period of time with no fixed end; for an unlimited or unknown amount of time

For instance, if someone says,

"the terms in the sequence repeat indefinitely for every $n≥n_{0}$ for some fixed $n_{0} \in \mathbb{N}$"

what does he mean by that?

Does he mean,

  1. "The terms in the sequence repeat at least more than once, though I'm not sure how many times."
  2. "The terms in the sequence repeat I don't know how many number of times, it could be that they might not repeat at all, i.e $0$ times or once or more than once or infinitely many times."
  3. "The terms in the sequence repeat infinitely many times."

Now based on the definitions above, indefinitely means "unclear or unlimited", so I'd think 2nd statement seems to be the correct choice. However, often times I feel like people tend to mean "infinitely many times" (unlimited times) when they use indefinitely.

I've come across texts (I forget where), which also seem to imply they mean the 3rd statement, that is "infinitely many times" when they use indefinitely.

So really which is it?

Blue
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William
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  • There are indefinite integrals – J. W. Tanner Dec 27 '20 at 10:51
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    Much like in spoken English, it can mean different things in different contexts. "Repeat indefinitely" in Math (and spoken English) means "repeat infinitely many times". – xxxxxxxxx Dec 27 '20 at 13:53
  • @MorganRodgers I agree, but please make your response an official Answer to the question so it no longer appears unanswered – Mark S. Dec 27 '20 at 20:49
  • @MorganRodgers Thank you for your comment but what exactly about this question is not about mathematics here when I ask the usuage of a word in the context of Mathematics? You do know many English words, in Mathematics mean completely different things than what they'd mean normally in an English sentence, hence this question. Because authors don't seem to define what they mean by "indefinitely" and I feel like it's important to know but maybe that's just me. – William Dec 28 '20 at 06:32
  • @MorganRodgers And as you point out, "repeat indefinitely" means "repeat infinitely many times" (3rd statement), which is also what I suspected at times but the reason I ask this question is because then why use such an ambiguous word if you were not going for the 2nd statement (in the question). If you wanted to mean the 3rd statement, would it not be much much clearer if you just said, "repeat infinitely many times" instead? :( – William Dec 28 '20 at 06:53
  • @William I am honestly struggling to find this used somewhere where it seems ambiguous. It is convenient to use natural language; "repeats infinitely many times" is awkward and not really any more precise. – xxxxxxxxx Dec 28 '20 at 09:10
  • @MorganRodgers If you fix the meaning of "indefinitely" as "infinitely many times" then what word would you use, say if you wanted it to mean the 2nd statement? Writing out the whole 2nd statement is one option but using just one word "indefinitely" in this case would seem much more neater and natural since it is suppose to mean the 2nd statement, no? Using "indefinitely" to mean 3rd statement alone, is weird and honestly, waste of good word. – William Dec 28 '20 at 10:00
  • @William Why would you want to say that? It would basically mean "we don't know if it repeats or not, or how many times". You might as well say nothing at all. I would mostly feel the same about the first possible meaning you give. – xxxxxxxxx Dec 28 '20 at 10:14

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