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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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    @GM2001 That's a fair and nice point actually (thanks for pointing it out!), something I have been $w$ondering 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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I've not yet seen, that the "rank of a matrix" has been interpolated to fractional ranks (but I'm surely not very profound with literature...)
(side remark: also your question focuses on whether it has "resisted to attempts" to interpolate ... such attempts may or may not exist, but to know this needed even a bigger radius of insight into literature and non-literature manuscripts...)

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    For arbitrary matrices I basically agree. But for matrices representing *projections*, the rank can be computed from the matrix trace and vice versa--- and there are generalizations of the matrix trace that take values in $[0, \infty)$ (e.g. "traces" on type II von Neumann algebras). And it's not uncommon to think of such things as "continuous" analogues of (or extensions of) the "discrete" dimension function.2011-10-26
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Turing machine is a discrete model.

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    @leslietownes You are right, or no, maybe.2013-05-22
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The Arithmetic derivative.

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    @$\text{}$Mark, @Gottfried: FYI this is a special instance of a [derivation](http://en.wikipedia.org/wiki/Derivation_(abstract_algebra)).2011-10-29
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I don't know if this works, because someone may actually have done something along these lines. I'm by no means an expert on mathematics in general. That said, some of the concepts of number theory may well work. For example, what would the concept of a prime number, composite number, "versatile" number (highly composite number), etc. mean in terms of real numbers (or rational numbers for that matter)? 5 has only 2 positive integer factors (other than 1), but it has an infinity of factors in just the rational numbers. So, if such an extension of the concept of prime number or composite number exists, I don't know how it works at least, and it would take some explaining.

The Fundamental Theorem(s) of Arithmetic don't seem extendable to the reals, since we don't seem to have the notion of a prime number in the reals, in the sense that some numbers exist in the real numbers which have exactly two factors.

I'll add that I know of at least two concepts which can get defined for the integers, but simply can't get extended to the rationals or reals (and never will legitimately).

For any given number n, there exists a least number o, such that o>n in the integers, where ">" indicates the usual ordering relation of "greater than". There does not exist any such number in the rationals or reals. One might say that in the integers, every integer has a distinct-least-upper-bound or distinct-supremum.

For any given number m, there exists a greatest number l, such that m>l in the integers. There does not exist any such number in the rationals or reals. One might say that in the integers, every integer has a distinct-greatest-lower-bound or distinct-infimum.

In other words, given a "direction" which either takes towards larger numbers, or smaller numbers, for any number x, there exists a "next" number "y" and a previous number "v". This does not hold true for the reals or rationals.

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    @Mathemagician1234 : But it doesn't generalize to $\mathbb{R}$ or to fields except in a trivial sense. It's like someone asking you for an interesting topological space, and you mentioning the one-point space (or even worse, the empty set).2011-10-27
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The principle of mathematical induction.

Reduction modulo $n$, as a map of rings. It exists only as a map of additive groups for $\Bbb{Q}$ and $\Bbb{R}$.

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    @Asaf, the Real Induction principle there (e.g. in Pete Clark's notes) is the same as the open set argument and the examples use it in the same way. I don't see how the axioms for poset induction at MO allow a double induction. At the end of the day you get "topological" induction with the same set of examples and essentially the same proofs as existed without induction.2011-10-29