I'm just thinking aloud on this one. So let's imagine a hypothetical function S_r that's the real analogue of S_n (the successor function over the naturals). Now let's say a, b in R are defined as follows: b=S_r(a); b is the real successor of a. Now of course if such a function existed then b-a must equal zero. If it didn't, then we can always find a c in R that's between a and b which would then make c a new candidate for a's successor. But if a-b is equal to zero, then a must necessarily equal b and so b can't be its successor. Such an argument seems to show that consecutive reals cannot exist, or at least can't be expressed using real numbers themselves. On the other hand, if we are working with the reals under the assumption that consecutive reals do not exist, then there is no intuitive way to refer to the "next" real number. Certainly there's no way to define it using reals themselves as above. But if one says that there actually and certainly is no next real number then how are we moving along the field at all? How do you move between one and two if there's no conceptual way of referring to the real number that comes immediately after the number 1 in the set of reals. If b-a must equal zero then you can add all the zeroes you want in the world to a but it won't change a thing unless you give infinity some mystical power to turn those zeroes into real values. I'm somewhat familiar with Cantor's diagonal argument now and I understand that that is precisely the whole point: that there exists no bijection between the reals and the naturals, and so no consecutive real by definition. But while I submit that such a thing can't be formalized in anyway using current language, doesn't the above reasoning at least justify the existence of consecutive real numbers in some as of yet to-be-defined form?
A couple of points: You can choose an ordering on the real numbers which is different from the one we normally give it such that you can in fact have a well defined successor function for each real number. This ordering is probably terrible and doesn't correspond to what we think of when we think of real numbers being larger or smaller than each other. That said, the problem of defining successive reals is independent of the ability to "move along" the real number line. What does that mean mathematically? Most people think of it as "I give you a position for each point in time". Time typically takes any real number as a value, so to say "where am I at the next point in time" is a meaningless question to begin with. More generally you will be hard pressed to give a good definition of moving along the real number line such that an actual contradiction can exist.
Yeah that occurred to me actually, just wasn't completely sure as I have just a touch-and-go familiarity with it. I know that this really sinks back some ways to that basic question which has been solved, for all practical purposes, by calculus. So this is really more of a, I suppose, metaphysical question: are consecutive reals a justified, if not expressible, idea?
The phenomenon that you are investigating is more related to the fact that the normal order on the reals is what is called a dense linear order. It's the dense part that makes it seems that there is no "successor-like function". The rationals are also densely ordered, and so there is no function on the rationals which is successor-like and respects the normal order. So cardinality (and thus Cantor's diagonal argument) is really not the issue. It turns out that, assuming the Axiom of Choice/Well-Ordering Principle, there is a successor-like function for every set. This isn't so hard believe to if you believe that every set has a cardinality in the sense that it can be put into 1-1 correspondence with a cardinal number.
That depends of course on what you mean by "moving along" the real line. I'm referring to the intuitive idea encompassed in the "intermediate value theorem" that if you are moving from one to two, you must "move along" all the reals between 1 and 2 to get to 2. You can't be jumping around seemingly random reals using some arbitrary mapping with the naturals. And the problem I have with using time is that it is a circular argument, much like defining finite vs infinite (i.e. we might define something as having a finite length if it takes a finite amount of time to measure it). So I could just as well ask how does one move through the field of time between the 1st and 2nd second if there is no conceptual framework validating the move to the next real after the 1st second.
I think I've seen it for the rationals, but I actually wasn't aware that there was such a discreet mapping for the reals. Even so, I guess I'm just defining S_r differently to mean what is encompassed in the intermediate value theorem; namely that to move from 1 to 2, you must move through all the reals between them starting at 1. Or if I have the correct terminology, there ought to be a real successor function (S_r) that is totally ordered to allow one to "move along" the field of reals.
There is no mapping from the naturals that covers all of the reals. Cantor's proof still applies. What is provable (given the Axiom of choice) is that there exists a "well ordering" of the reals. However, the proof of this is not constructive. That is to say that it is not possible to explicitly exhibit such an ordering. An "order" (or sometimes "partial order") is a binary relation, "<" between pairs of elements that is transitive, irreflexive and anti-symmetric. Transitive means that if a < b and b < c it follows that a < c. Irreflexive means that is it never the case that a < a. Anti-symmetric means that if a < b it follows that it is not the case that b < a. A "total order" is an order that also satisfies trichotomy. That is, given any pair of elements, either a < b, b < a or a = b. A "well ordering" is a total order that also has the property that any non-empty set has a smallest member. The natural numbers are well-ordered by their usual ordering. The rational numbers are not well-ordered by their usual ordering. For instance, the set of all rationals has no smallest member. But if you map each rational to a natural number then the rational numbers are well-ordered by the order that is induced by that mapping. The real numbers cannot be mapped (one to one) to the integers. So that trick does not work for them. That terminology is not apt. The difference between what "ought to be" and what "is" is often large. Note that if you are going to invoke some hypothetical well-ordering of the reals then the intermediate value theorem and the prerequisite notion of continuity become largely irrelevant -- those are tied to the standard ordering. Given a well ordering... For every point x "between" [in the sense of the well ordering] 1 and 2 there would be a "next point" [the smallest real that is greater than x and less than or equal to 2]. That's guaranteed by what it means to have a well order. But that doesn't mean that you could get from 1 to 2 a step at a time, even with infinitely many steps. It could be that some numbers "between" 1 and 2 are not successors of any other number.
Could you please expand on this? Particularly, how is the notion of continuity tied to standard ordering? And is this a hypothetical or has this actually been shown to be the case?
How do you define continuity? The definitions I am familiar with involve epsilons and deltas and "less than". How do you do "less than" without an ordering? If you change out the ordering, you've changed out the definition of continuity.
Oh nvm, I should've looked up what standard ordering means first. But I still don't understand why invoking a well-ordering on the reals removes standard ordering. This is probably over my head, but I'll do my best to follow.
Sadly, I'm not able to make heads or tails of this just yet (at least not using what definitions I could find for the terminology used), though it has given me the impetus to finally open this book on set theory I got some time back. Having said that, are you essentially saying (in laymen's terms) that since I'm basically quantizing the field, I can't then base that quantized field on the concept of continuity, the definition of which requires that the field not be quantized?
Yes, this is correct. Continuity in particular requires the notion of an interval: take two numbers, and every number in between. This doesn't make sense once you consider a well-ordering of the reals (which says for every real there is a next real). The reason why is that the "next real number" is going to look really weird from our usual perspective, the first real number might be 0, then the "next one" 3, then the "next one" -2.4, then the "next one" 2^{e}. And you lose the idea of real numbers being "really close" to each other - in the ordering I described above, the closest number to 0 is 3; every number is farther away in the ordering so talking about things like limits stops making sense.
To find a formal disproof of the existence of a successor in the standard ordering, see, e.g., the (constructive) argument that between two Rationals there is a Rational; basically between any two Reals there is a Rational, and there is an Irrational number. Or look at the decimal representation of any two Reals, that differ in the n-th decimal place, you can also go farther back than the n-th place to find an in-between Real.
Ok, please stop me if I'm just putting words in your mouths, but when I take this: together with this: can't I interpret this as saying that a well-ordering implies the existence of consecutive reals (in the sense of there being a "next point" after 1), but there is no mapping to the naturals that can take you across a particular interval, or subset of the reals, in a step-by-step manner.
Yes if such a successor function were to exist for the reals, it couldn't be defined in terms of reals as that very quickly leads to the problems you mentioned and in the OP as well.
A next point that need not be anywhere near 1 in our normal sense of "near", yes. A subset of the reals you might map one to one to the naturals. For instance, {1.0, 2.0, 3.0, etc} would be easy to map. A non-trivial interval you can't.
I appreciate you clarifying the terminology for me. Ok, sorry if this is getting repetitive, but as it's been proven that the reals are well-ordered, is it this proof in particular that rules out the possibility of constructing a standard order of the reals? Or is it ruled out by more intuitive notions that a standard ordering of the reals can't be described in any sensible way (i.e. standard-ordered reals in the interval (0,1) have no least element)? I'm asking because I've been playing around with this idea of consecutive reals on-and-off (just realized I posted about them here 2 years ago) and I figured if there was anything to this, then it should be possible to infer their existence with available terminology. Though at this point I get the feeling that nothing short of directly challenging Cantor's arguments on infinity would cut it if one's trying to will such a concept into existence.
A set is not "well-ordered". A set is well-ordered by an order. Or an order is a well-ordering of a set. The reals are not well-ordered. However, it is provable that there exists a well-ordering of the reals. Given a model of the reals (e.g. as equivalence classes of Cauchy sequences, as Dedekind cuts or as canonical infinite decimals), the standard order is perfectly constructible. No well-ordering of the reals is constructible. That just demonstrates that the standard ordering is not a well-ordering.
I'm gonna stop embarrassing myself now in trying to use these terminologies -_- Just one last question, though I'm not sure if it needs to be in a separate thread/forum. Isn't the basis of Cantor's ideas on infinities, namely the existence of completed infinities, essentially a philosophical position rather than a provable theorem? The whole one-to-one correspondence idea makes sense and all, but it seems to me that such a one-to-one mapping only gives information on cardinality if one assumes the existence of these completed infinities.