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Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals?

Does not meet criteria:
Cardinalities of sets
n! / Gamma function
Differentiation / Fractional differentation

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    ...I would say there is always an implicit challenge to analytically continue any function that is supposedly only defined for integer arguments to complex values.2011-10-23
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    Would you mind removing that \*\*AWESOME\*\* nonsense? Seriously distracting.2011-10-23
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    @GM2001: 40 views in 5 hours is not "exponentially". In any case I have removed it. (Everyone: **Please see my comment below before voting to close.**)2011-10-23
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    @GM2001: View count is of course important for the exposure of the question, but more important is to write a *good* question with a reasonable title. Otherwise it's just spam... Even if every person on the planet sees your question, what good is it if no one is willing to answer?2011-10-23
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    The notion of being divisible goes away when you jump from the integers to rationals because you jump to a field and it makes the idea of divisibility trivial.2011-10-23
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    Essentially this question is the complement of [this MO question](http://mathoverflow.net/questions/74252/things-that-should-be-positive-integers-really). I'm not so sure it should be closed, though I will make it CW.2011-10-23
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    @GM2001: I completely agree. It's easier to blame the crowd.2011-10-23
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    "View count" seems like a decent answer.2011-10-23
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    By the way, GM2001, it is a common belief that closed questions get *more* views (as people want to see what is it that was closed) so you should be happy about that, I guess.2011-10-23
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    @GM2001: Exactly how do you think that throwing a snit and insulting people will advance your cause?2011-10-23
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    @IAmBrianDawkins But what if I have only read half of the question?2011-10-24
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    I have to say: The comments of GM2001 and the title thing are immmature and out of place, but I don't think that the question as it is now in its sixth version should be closed, so I am voting to reopen.2011-10-24
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    I have reopened the question. I agree with Phira, I think people were reacting to GM2001's behavior, instead of the question content. I encourage anyone who votes to close again to explain their reasoning in the comments.2011-10-25
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    I don't see how there can be a reasonable answer to this question. Anyone who hedges that a particular discrete concept cannot be extended to the continuous world is asking for trouble, thanks to [Clarke's First Law](http://en.wikipedia.org/wiki/Clarke's_three_laws): *When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.*2011-10-25
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    Too add on Srivatsan's comment, I also don't see how any tag other than [soft-question] would be fitting here. It is certainly not philosophy, nor set theory or number theory. I also fail to see the connection to discrete mathematics other than the title. If anything, it is the opposite - the question requests for things which are no longer discrete.2011-10-25
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    @Asaf What? No. It asks for things which are only discrete.2011-10-25
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    @GM2001: The primes and numbers of the form $\frac{1}{p}$ for a prime $p$ will generate the rational numbers using multiplication.2011-10-25
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    I want to say that the exterior derivative operation defined on differential forms will meet the criteria... could "half of an exterior derivative" be defined? What on earth could it be? What would be, say, the 1/2 exterior derivative of "dx dy" or of "f'(x) dx"?2011-10-25
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    How about prime numbers, is there a similar atomic set for the real numbers for which we can generate all reals through some operation analogous to multiplication?2011-10-26
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    @Sriva And yet you state that a reasonable answer to this question is impossible. You should learn from your own quote.2011-10-26
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    @GM2001 That's a fair and nice point actually (thanks for pointing it out!), something I have been wondering about myself. In fact, to quote my own comment from the chat room a few hours back, "Perhaps someone will actually write a good answer to the question, making the comment bite the dust. It will be like Clarke's First Law coming back to bite it's own tail.".2011-10-26

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