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.

  • Comments are not for extended discussion; this conversation has been moved to chat. Please do use comments only to suggest improvements to the question. – Philip Klöcking Jul 28 '18 at 22:23
  • If what you're asking is so complex and subtle that everyone who has looked at it has missed exactly what you're arguing, both in the technical aspects and the philosophical aspects, then you should take the question to mathoverflow and ask there. If you really have the credentials you say, and "have taught mathematical logic at a university level", then your concern can be considered research level. Either way, I guarantee you that this question won't be reopened here, especially in it's current state, so there isn't much use editing it here. – Not_Here Jul 30 '18 at 21:21
  • @Not_Here. I didn't edit to get reopened but because I couldn't delete and felt bad about people still reading this and wasting time. "Credentials" aren't the point. My main interest is in philosophical theology, not set theory. This stuff is a necessary tangent, but I didn't misrepresent what I've done in the area, just brought it up because you were questioning whether I understood very basic things. Yes, I do. I made a mistake being on this site because there were too many things I needed to assume from previous results I'd gotten. They just won't let me erase the mistake. It's frustrating. – Nightspore Aug 1 '18 at 5:36
  • @Nightspore If the question remains closed for a certain amount of time, I can't remember off the top of my head how long that is, it will automatically be deleted. – Not_Here Aug 1 '18 at 6:11

It's possible for A to be finite.

Take the function f(x) = 2/pi * tan^-1(x), an order preserving bijection from the positive reals to the interval.

We see that f(x) < 0.5 if and only if x < 1. So in this case A is just {0}.

  • 1
    Note that the mistake in the question is clearly marked with the word "clearly", as it happens so often :) – Arno Jul 28 '18 at 9:40
  • Thank you. I am more than willing to admit any number of counterexamples. No need to be this sophisticated, just map the reals beginning "0." to (0, .5], etc. However, your objection carries some force if it means that I cannot rely simply on the existence of a possible bijection but must produce the specific function. I don't think I'm obligated to be that specific, but only to assert a possible mapping with the apposite properties. For the present, I can say your counterexample doesn't work as a refutation: you would need to prove that NO such bijection is possible. I'll think about it. – Nightspore Jul 28 '18 at 10:17
  • @Nightspore Ok so my example of mapping P0 to (0,.5] is correct. So what is the next step in your argument? Earlier you said one couldn't map only finitely many packets to the left half of the unit interval, but it's clear that we can. – user4894 Jul 28 '18 at 10:20
  • @Nightspore In your argument you make a choice of order preserving bijection f and then talk about certain sets A and B which (crucially) depend on the choice of f. If your proof that A is infinite, which is without reference to any special properties of f besides the fact that it is an order preserving bijection, was correct, you'd have proven that for any choice of f, A is infinite. As demonstrated, this is not true. – Daniel Mroz Jul 28 '18 at 10:31
  • In fact it's not hard to prove that A cannot be infinite. Pick a real number x, and consider its integer part n. Because A is infinite, it contains some m > n. That is, m is the integer part of a real number y, and f(y) < 0.5. Also, since the integer part of y is strictly greater than the integer part of x, x < y. Finally, because f is order preserving, f(x) < f(y) < 0.5. That is, every real number gets mapped to something in (0, 0.5), so f is not a bijection. Contradiction. – Daniel Mroz Jul 28 '18 at 10:38

It's difficult to discern what precisely you are asking, but I think the following theorem is relevant:

Theorem: If A and B be two nonempty subsets of the natural numbers with the following property:

  • If a ∈ A and b ∈ B, then a < b

then A is finite

I.e. if everything in A comes before everything in B, then A has to be a finite set.

Of course, < refers to the usual ordering on the natural numbers. You can pick different orderings for which this theorem doesn't hold.

For example, define a new ordering "{" where x { y means that one of the following is true:

  • x and y are both even and x < y
  • x and y are both oddand x < y
  • x is even and y is odd

For example, 2 { 4, 7 { 31, and 8 { 1 are all true, but 7 { 4 is false.

It turns out that { is an example of a total ordering where all of the even numbers come first, and then all of the odd numbers come afterwards.

I strongly suspect that whatever idea you have in mind about "dividing the natural numbers in half sequentially" is not about the usual ordering on the natural numbers, but some nonstandard ordering derived from the line of thought you are pursuing.

  • Thanks for this. You're right that I am not pursuing what could be called a standard line of inquiry. I am trying to tease apart the precise mode of thought necessary to infinitary reasoning, which entails holding constructivity at bay to the extent possible to investigate the nature of the ideality involved in claims of infinite actuality. This puts me more or less on the edge of, among other things, clarity. Granted. Well it's a dirty job but somebody has to do it. – Nightspore Jul 28 '18 at 15:08

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