The other thing avoiding that particular morass of common terminology does is it helps keep straight that there are many different coherent notions of size/measurement/comparison/etc., and there isn’t just one handed down from God appropriate for all purposes; it is not wrong to want to consider, say, “{0, 1, 2, 3, …}” as larger than “{0, 2, 4, 6, …}” in some sense, just as it is not wrong to consider the measure of a long line segment as larger than the measure of a short line segment, in some sense. It’s just that the relevant senses are different ones, in those cases, than those given via cardinality. But that’s ok; cardinality is not the only idea in our conceptual toolbox; sometimes, it’s what we care about, and sometimes, it isn’t. However, the amount of unduly monopolostic attention it has been given as a tool for exploration of ideas concerning infinitude tends to obscure this, making it misleadingly appear the only game in town for such purposes.
You’re right to be confused by my terminology–that’s part of why I used scare quotes, after all, to signal that I knew I wasn’t using the standard terminology, because I don’t know it.
By “infinity of convergence” I meant infinity as the measure of the limit of a series which diverges. Someone upthread said such a series “converges on infinity,” and that’s where I picked up “convervence.”
As for what you’ve said about infinity being no kind of number at all, have you seen the post upthread that explains that in set theory, infinity is a kind of number, namely, a kind of number greater than any counting number?
But, as explained in that same post, it turns out that the measure of the limit of a diverging series, which is typically called “infinity,” is not the same thing as a number greater than any counting number, which is also typically called “infinity.” Same name, but it turns out, two different concepts. The second concept is the concept of a kind of number.
This was actually the only thing I really stepped into this thread to talk about.
There is an extension of standard base p arithmetic to a system called “p-adics”, where one is allowed to have infinitely many non-zero digits to the left of the decimal point as well. (“p” is usually assumed to be prime, for various reasons, but let that not concern us now; we can still do this with base 10 just fine). Thus, in the system of 10-adics, one might consider a quantity denoted …999999 .
Note: In this system, there is no great reason for calling this “infinity”. Pleonast played along, but why should we? No, no, no. There’s a much better name for it, though, as we’ll see in a second.
Now, what happens when you add 1 to this? Well, the way 10-adics work is that you follow all the usual add-and-carry rules for calculating sums. Thus, …99999 + 1 = well, the last digit is 0, and then we carry a 1 over to the tens digit, which becomes 0, and then we carry a 1 over to the hundreds digit, which becomes 0, and then… well, they all end up 0, in the end. So, …9999 + 1 = …00000 = 0. We might as well think of …999 as a denotation of negative one.
By the same reasoning, …9998 acts as a denotation of negative two and so on. And, similarly, …1111 acts as a denotation of -1/9 (since adding it to itself nine times and then adding one yields zero).
In fact, quite a lot of negative and non-integral numbers can be represented this way without having to resort to negation signs or decimal points. It’s quite an interesting system.
(It does have one unfortunate property: it’s possible to multiply non-zero numbers together and get zero as an answer in this system, which means, among other things, that it has no good notion of division. But this problem goes away if one uses a prime base instead of 10)
So, interesting stuff. There’s nice ways to think about …9999, where it corresponds to a very natural concept. The only thing is, that concept isn’t “infinity”; it’s “negative one”.
This is right of course. I was giving expression to the fact that the way I was raised, in set theory, transfinites are treated as being of a kind with finite numbers. I didn’t mean to make any strong metaphysical claims about what is and isn’t “really” a number–I just took myself to be making clarifications about what is (or what I’ve always taken to be) standard terminology. When Exapno says transfinites are not numbers, I expect his audience would become confused when they see mathematicians talking about them as though they were numbers. I was trying to head off that confusion for those folks–probably doing more harm than good on that front I admit.
And so is the first; say, a number in the extended reals, or the projective reals, or whatever is most appropriate to the particular application. The same license you are invoking for the second concept can be invoked for the first. In the words of Wittgenstein:
ETA: Ah, I didn’t see your previous post when I made this one. Still, I like the Wittgenstein quote, so I’ll leave it up.
I guess for any physical application of the concept of a limit I can imagine, finite limits are best modeled as having an order–but the “infinite” limit of diverging series is best thought of as not being ordered together with the others. A million is to the right of a thousand, but infinity is just, basically, an anomaly or a useless result (for any phyiscal application of the concept of a limit I can imagine).
For other applications of the concept of a limit, I imagine you can model limits however you like. Ordered, unordered. With infinity “way over to the right” or standing outside the order imposed on the other limits. With several infinities or none. Etc. Just depends on what you want to do with your model. (Right?)
Yep, you can model limits (or anything; this is the general nature of math) however you like, depending on what you want to do with your model. Exactly.
That having been said, I’m surprised you don’t feel that for physical applications, there is a natural, useful interpretation of this ordering extending to (certain) divergent limits. If a series diverges in the particular way of eventually growing and staying larger than than any particular finite quantity (whatever the relevant notion of these is), then you quite possibly will want to consider the result “positive infinity, larger than all the finite quantities”. Similarly, you may want to consider a distinct possible limit of “negative infinity, smaller than all the finite quantities (including the negative ones)”. There’s nothing wrong with considering an ordering here; that’s what the (affinely) extended reals do.
Physical applications? Sure, why not? For example, it is natural to define the average rate of change of a quantity Q(t) over the time period from a to b as (Q(b) - Q(a))/(b - a), and then to define the instantaneous rate of change at a single instant c as the limit when a and b both approach c. [The usual limit-based definition of the derivative (well, technically, slightly different, but the technicalities are irrelevant)]. Of course, fancier words for instantaneous rate of change include “speed” or, particularly, if the quantity Q is some kind of distance, “velocity”; a physically relevant quantity, no? But, of course, we could consider such hypothetical cases as where, say, Q(t) is proportional to the cube root of t. In this case, what would the instantaneous speed be when t = 0? Well, this limit goes off to positive infinity. And it certainly makes sense to consider this as faster than any finite speed; if the quantity Q grows like this, then, for any quantity growing from the same starting value at a constant finite speed, Q outstrips it in the suitably short-term. There are any number of reasons one might be interested in this kind of ordering. The particular example using cube roots isn’t important; the point is that one could easily make sense of rates of change including the infinite and want to consider the natural ordering upon them, even in a physical context.
Now, granted, it may be the case that for a particular kind of quantity Q, physical laws prevent such a hypothetical situation; but it’s not conceptually incoherent! It’s certainly at least conceptually possible for this to be a relevant physical employment of the idea of the ordered arithmetic with positive infinity (or negative infinity, similarly, or various such things). And so on for all mathematics; I cannot even fathom what it would mean for an abstract idea to be a priori impossible to use in modelling some physical situation.
(But all of the above is only concerned with using positive/negative infinity to stand for the limits of series which diverge in very particular ways; you would say {0, 1, 2, 3, 4, 5, …} goes off to positive infinity, but not {0, 1, 0, 1, 0, 1}. Perhaps it is consideration of the latter sort of thing which led you to consider these as only kind of anomalous “error code” results for which linear ordering is not sensible?)
The reason I left math was that I found that I was far more comfortable with and far better at words. I totally appreciate the distinction between the technical usage of certain terms and their bastardization by the general public and I sympathize with the technicians who find that useful distinctions get so slurred by sloppy translations into the vernacular that all meaning is left behind.
I need to straddle this gap professionally all the time and it makes me hyperconscious of the problem. That’s why I think that too many of the threads here don’t serve their OPs as well as they could.
You (plural) are unquestionably correct to draw upon the nuances of terminology to dispute the very use of the term number. Yet I’m certain that anyone who would ask the kind of question asked by the OP is not going to be able to appreciate the discussion of the last several posts, or many before them. There is plenty of solid popular science usage for number and infinity that makes the distinction between them in ways that present the basic concepts without needing to introduce what in some ways is almost theology.
I see this in multiple threads here on science. I see it on Wikipedia, where it is now almost impossible for any non-specialist to use any article on a technical subject.
Any professional writer has to continually ask: who is the audience, what level of understanding does the audience have, what concepts and vocabulary need to be defined, how deeply a subject can be studied. That’s Basic Explaining 101 and it’s an art like any other, and has no one right answer.
But when you get beyond me, I’m 100% positive that you’ve gotten well beyond an OP. Discussing the deep stuff among peers is more fun and in many ways much easier than Basic Explaining. In some ways I’m lucky that I have people to say to me “I don’t get it. Start over.” (Not that I feel lucky when I have to face that blank screen again. :smack: )
I think you’ve hit the point in this thread where someone needs to say “I don’t get it. Start over.” Not that you can’t continue what is a fascinating discussion in its own right. Just remember the rest of the audience at times.
(This might be better off split into its own thread. But since ATMB has been taken over by the the anti-mod forces, I wouldn’t know where to put it. 
)
I don’t think it is bigger than the reals if we’re sticking to continuous things. A continuous function is defined by its values on the rationals, which are countable. The cardinality of a real vector space with countable dimensionality is the same as the cardinality of the reals, right?
Uncertain: Yes, that’s all correct (with the usual assumptions about cardinals being linearly ordered).
Exapno: You are correct that I haven’t kept the OP in mind; I really only came in here to make post #43 originally, which I think should hopefully be understandable to the OP, or, if not, is what should be explained better.
The business about the word “number” and so on is all rather a sidetrack, of course, though hopefully it’s clear that the point I was attempting to express is that there is no technical definition of the word and people should feel free to call anything they like a “number”, whenever they feel a desire to do so.
I see your point there and must admit that the answers quickly left me behind, but then mathematics generally does after a certain point. I don’t consider myself hard-done-by however. I’ll just have to stick to my flaky, intuitive ‘feeling’ of the infinite which generally serves my day-to-day purposes.
Do you have a cite for this? I would be very interested. I’m no expert on set theory, but I do remember that there was a theorem that said if there were surjections back and forth between groups then there existed a bijection, but presumably this only applied in “classical” set theory.
Well now, that’s quite the carte blanche you’re giving us.
… the number I was attempting to express is that there is no technical number of the number and numbers should feel free to call anything they like a “number”, whenever they feel a number to do so.
Reading that made my brain number.
I implore you not to try this. You will get a cosmic arithmetic overflow error which may cause the entire universe to reboot. 
The “it” in question was the set of continuous real-valued functions on the reals and the assertion is incorrect, essentially because two such functions that agree on the rationals are equal which means that the set of functions from the rationals Q to the reals is an upperbound and that is easily seen to have the cardinality of the reals. (Since the cardinality of Q is that of the natural numbers N and the cardinality of the reals is 2^N, the cardinality of the functions from to Q to the reals is (2^N)^N = 2^(NxN) = 2^N.
Now there are many orders of infinite cardinals, more than you could imagine. There are accessible cardinals (essentially all the ones you could by iterating powerset and union), then inaccessible, then measurable cardinals that are preceded by measurably many inaccessibles, then compact cardinals that are unimaginably larger than measurables and so on to utter madness. (The study of “large cardinals” is one of the more obscure byways of pure math.)
But all this somehow misses the intent of the OP. The study of infinities leads to infinite sets, while the OP wants something involving infinite arithmetic. Now if you are careful (you cannot subtract one infinite quantity from another for reasons more or less similar to why you cannot divide by 0 and there are other restrictions. But if you wanted you could use an infinite string of 9s or for that matter a 1 followed by an infinite string of 0s. Whether anything could be gained thereby is a another question. I doubt it.
(This post is not relevant to the OP)
Sure. In fact, let me go further and state that one can construct situations in which A is in correspondence with a subset of B and B is in correspondence with a subset of A but A and B are not in correspondence with each other.
The classical theorem which says this can’t happen is the Cantor-Schroeder-Bernstein theorem. And it’s a good theorem; it just doesn’t always hold outside of classical contexts (here’s a cite, noting, for example, that it fails in the Sierpinski topos, though that’s not very helpful to anyone not already familiar with topos theory).
For a much more easily understood example, I’ll ask only that you take my word for one thing: in a suitable sense, there are (interesting, important, etc.) non-classical contexts in which all functions are continuous. I can explain in more detail what this means later (or in a different thread), but granted that, we can examine a counterexample to CSB as follows:
Let A be the set of points on one line segment and let B be the set of points contained in two disjoint line segments. Clearly, A is in correspondence with a subset of B; conversely, by splitting A into pieces and then removing the middle, we find that a subset of A is in correspondence with B.
Cantor-Schroeder-Bernstein would then tell us that there must be a correspondence between A and B themselves; however, there cannot be any continuous such correspondence. In fact, there cannot even be any continuous surjection from A to B: any surjection would have to jump between the two disjoint parts of B at some point, thus being discontinuous. Accordingly, in a context in which all functions are continuous, CSB can fail.
There are similar counterexamples using contexts in which all functions are computable and so on. What all these contexts amount to, how to construct them, the properties they do and don’t have in common with classical set theory, and so on, I’ll have to save for another thread with more details, though, as noted before, a good place to start if one wants to read up on such ideas is “topos theory”.
(Though, one detail I should mention is that the main difference between these non-classical contexts and more familiar ones is that in these, the “law of the excluded middle” (equivalently, double negation elimination) is not universally validated (nor, for that matter, is the unrestricted axiom of choice); the relevant internal logic is intuitionistic, not Boolean. The CSB theorem, construction of discontinuous or uncomputable functions, and so on all falter in these contexts upon precisely the lack of general availability of excluded middle.)
Applause.