My brief stint on SE has been quite interesting because it forced me to make the premises of my inquiry more explicit. I resisted this initially simply for reasons of economy, but economy proved to be almost universally confusing. As SE will not allow me to delete this, which is my preference, I will leave behind at least a suggestion of the motivation for this question and, as compensation, revise the rest to be much shorter.
So then, consider a possible bijection in general between the real numbers (as a product of the real decimals and the naturals) and the real decimals (or unit line). This is very difficult to do, as the answers to my question below indicate, without recurrence to constructivity. But constructivity tends (and this is why it is used) to cancel the raw force of the underlying idealization, to turn the problem into something else, something presumably more tractable. In this case it is to superimpose something very like the structure of a convergent limit on a perfectly (if purely ideally) uniform sequence.
I can explain this best in terms of an image. Think of the bijection as a string of pearls, each pearl being the infinite aggregate of naturals associated with real decimals in the sequence of reals. That is, the form of a real number is y.xxxx..., where y is some natural and .xxxx... is a real decimal. Thus you have, in sequence, an infinite number of 0's (followed by all the real decimals), then 1's, then 2's, and so on. These aggregates of infinite strings of 0, 1, 2, 3, ... are the "pearls". Why pearls? Because the ideal structure of the sequence is perfectly uniform. In particular, it does not converge (does not, that is to say, depend upon the sequential values involved).
Now ordinarily we understand the sequential division of N to be necessarily into a finite part followed by an infinite part. That is quite correct and expresses one truth about the naturals. But the truth it expresses is an implicit imposition of potentiality or constructivity on the sequence. Yes, you can do that: my interest is in what it means to do it. Which is where this ideal bijection in general between the reals and the real decimals comes in. The chief point is its utter uniformity. Every "pearl", or P-sequence of naturals, is uniformly the same as every other, to infinity, with the trivial exception of its identifying value. (Indeed it is a salutary exercise to imagine the string of pearls without the identifying numbers, merely as otherwise indeterminate aggregates, which serves to emphasize its complete uniformity, i.e. symmetry.)
This means that you can picture the entire sequence being flipped end-for-end symmetrically. In ideal terms there is no rationale for distinguishing the form of the sequence after it is flipped. It is logically (but ideally) perfectly uniform. The difficulty for most people is that we have become so inured to imposing constructivity, say in the form of functions, and therefore potentiality, that it seems quite natural and inevitable. Therefore I get a lot of "yes but don't you see..." comments. Of course I see. What is happening is that the ideal structure of the sequence is being altered into something, in practical terms, more like that of a convergent sequence, even though in ideal terms it is nothing like a convergent sequence. Flip a line divided as 1/2^n end-for-end and the result is anything but symmetrical.
So the whole point of this inquiry is to see what the implications of the underlying ideality of the essential reasoning happen to be. As a perfectly uniform P-sequence mapped onto what is basically a divisional template, you generate a sequential division of the naturals into two infinite sets. Yes, that is paradoxical, and, yes, that is the point. It is the underlying paradox of infinitary reasoning as such, which has been acknowledged as paradoxical from the beginning. There is nothing new, or even particularly controversial, about this. You simply have to be willing to tease apart the underlying ideality from the superimposed constructivity, the reasoning of actualization from the reasoning involving potentiality. The problem has simply been concealed by habituation to systematic equivocation.
Only allow for the ideal separation of the naturals into two sequentially infinite sets, a separation that all construtive/potential intervention (equivocation) is designed specifically to prevent, or at least obscure, and the difficulty is apparent. Can you keep coming up with "answers" to this that essentially involve the imposition of constructive/potential/functional/hypothetical reasoning? Of course. But to me, that isn't an answer, that is the problem. It is the problem with the mode of infinitary reasoning as such, and why the paradoxes still lurk beneath its artifically calmed surface.
So ideally, flipping the P-sequence is perfectly symmetric and, that being so, there is no rationale (ideally, in general, with respect to logical possibility) for distinguishing one P-sequential division of the naturals in this bijection from the other. If not, then if the naturals are infinite, both divisions are infinite, and if both are infinite then one of them is {1, 2, 3, 4, 5, ...), which is identically N. Since both are non-empty this leads, as N is then identically both, to N being a proper subset of itself (not in bijection with a proper subset of itself but identically a proper subset of itself).
Therefore, my question, "Can you divide the naturals in half sequentially?", cannot be addressed without taking into account the perspective from which it is being formulated, and that is the perspective of the ideality underlying infinitary reasoning, without particular regard for the various recursions to constructivity or potentiality or pure hypothesis designed to keep the reasoning afloat in a technical sense, at least provisionally.
Obviously all this takes a lot of explanation to go into, more than I can provide here, so it was a technical error to introduce the question on this forum, but at least I was forced to confront that fact and can make adjustments in my subsequent expositions accordingly. As you will not allow me to delete this entirely, I hope this clarification, such as it is, helps motivate the reasoning. To be honest, even as I read over this I know it probably cannot be clear to anyone coming at it cold. I was assuming an entire discussion of infinitary ideality that simply cannot be included here. It was a mistake, but you won't allow me to correct the mistake properly, so this is the best I can do.
