A query about countability Suppose we are working in an elementary  context where we want to keep background assumptions modest (not take a stand on fancy issues in set theory, say). What should our attitude be to the idea of countability? Countability is defined by a quantification – $X$ are countable if there is a function $f \colon N \to X$ which enumerates them. But quantification over which functions?
Even if you fully buy into a rich set-theoretic background, taken at face value, different set theories will supply different enumerating functions. Thus the so-called constructible reals are uncountable according to the theory ‘ZFC + V = L’ but countable according to the theory ‘ZFC + there exists a Ramsey cardinal’. More needs to be said even by the orthodox who identify functions with sets, if it isn’t to be left somewhat indeterminate what objects are countable.  But suppose we fall short of  endorsing the set-thereotic orthodoxy because we don’t (in the context, anyway) want definitely to buy into the wildly infinitary assumptions of this or that set theory: then that leaves more indeterminacy in what counts as countable. (If we don't settle just which functions we are prepared to countenance, we leave it correspondingly open which enumerating functions we are aiming to quantify over when we say that some given objects are countable.)
Yet mathematicians — at least when writing in fairly elementary contexts — cheerfully talk about the countable as if that’s unproblematic. How come? Is that just carelessness?? Or worse???
Well, elementary talk about the countable tends (doesn’t it?) to feature  in three sorts of context:


*

*There are claims that certain objects are indeed countable, defended by showing that the objects in question are unproblematically counted by producing a nice enumerating function. (Consider, for a familiar simple example, how we show that the positive rationals $m/n$ are countable by actually constructing the ‘zig-zag’ enumerating function for ordered pairs $m, n$, and so counting them.)

*There are claims that certain objects are uncountable, defended by reducing the assumption that they can be counted to absurdity. (Consider, for another familiar simple example, the usual diagonal argument that the infinite binary sequences are uncountable.)

*There are conditional claims of the kind if X are countable, then ..., supported by general arguments that are insensitive to how exactly we delimit (or fail to delimit) the countable.


(The second is a special case of the third, of course, but perhaps worth highlighting separately.) In none of these kinds of case, at any rate, does such indeterminacy as we might be leaving in the extent of the countable become problematic. So if we proceed with due caution – restrict ourselves to these cases — we can continue to talk about the (un)countable safely enough. And in elementary contexts we do exercise such caution. So we are not in trouble after all. 
Or at least, so it seems. Query: is that a fair description of ‘ordinary’ mathematical practice in elementary, non-set-theoretic, areas? If not, what is a better description of what is going on? While if I’m right, can you think of some elementary texts which overtly ’fess up to the need for this element of caution?
 A: I think you are right about most ordinary mathematics only using the notion of countability in the ways described in cases (1)–(3).  I don't know of any ordinary mathematics texts that acknowledge a need for caution in this regard.  However, I think there might not be any need to exercise caution via a restriction to these three cases.  I think that this restriction might be inherent in the notion of "ordinary" mathematics.
It seems like the other main case would be
(4) Conditional claims of the kind "if X is uncountable, then...."
Note that case (2) is a special case of case (3) where the conclusion is $\bot$.  In classical logic case (1) is a special case of case (4) where the conclusion is $\bot$, but it seems like this might fail to be so in intuitionistic logic.
Case (4) does seem more sensitive than the others to the foundational assumptions in the background.  Even in the special case of "if $X$ is uncountable, then $\bot$," that is, "$X$ is not uncountable," the possibly sensitive issue arises of whether we can then exhibit an actual enumeration of $X$.
However, I don't think that case (4) appears much in ordinary mathematics.  To attempt a proof using the hypothesis "$X$ is uncountable," which is a negative statement, it would usually help to have some knowledge about what the cardinality of $X$ is rather than what it isn't. 
We could argue from the hypothesis "there is an injection $\omega_1 \to X$" (whose equivalence with "$X$ is uncountable" is fraught with foundational issues) but this hypothesis would not usually appear in ordinary mathematics.
Usually in ordinary mathematics $X$ would have the cardinality of the continuum (for example, because Borel sets have the perfect set property.)  Then we could argue from the hypothesis "there is an injection $2^\omega \to X$," which seems more useful.  Similarly to case (3), such an argument would probably not be sensitive to the choice of foundational assumptions, especially if we assume that the injection $2^\omega \to X$ is continuous, as it is when given by the perfect set property.
