There is no such thing as “The Limit Theorem” of Calculus. :dubious:
There is a definition of a Limit, which you can accept as some truth as to the non-existence of infinitesimals, or rather as tool when you don’t care about infinitesimals values.
My main point in all of this, if I could make only one, is that we define .999… = 1. All of your proofs in the matter are shenanigans which already have the assertion built in.
I have never asserted that .999… could not possibly be equal to one. Again if your talking about the real number system, which does not allow for infinitesimals to exists (not by proof but by definition), then of course it is equal to one.
I never came on here claiming to have some profound new idea or even beginnings of new number system to replace reals. Though it was enlightening to learn about hyperreals, which in my opinion seem a promising extension to the reals.
Anyways it was enjoyable to debate some of these issues with many of you. I’m actually quite flattered that I have been given so much attention.
No, we don’t define .999… = 1. We define:
(1) the real numbers;
(2) limits of infinite series of real numbers;
(3) decimal notation for real numbers; and
(4) how to write an infinitely repeating decimal fraction using “…”.
From all that we prove that .999… = 1, given all those standard definitions. We do not assume it at all.
Which I will note has been explained clearly and copiously by me and others on this thread.
Erik, did you try coming up with the argument that to traverse a set it must have a final member? I actually thought progress could be made if we went down that path…
I will miss you most of all. Frankly I do not see by what reasonable means of rational or real thought processes one concludes that one may traverse, or iterate if you will, through all the members of a set, be that infinite or not, and yet not have some final member. If the spirit moves me perhaps I will come up with formal proof by contradiction. But it does not at the moment. It’s been a while since I have had to write any proofs, and that one would seem somewhat trivial.
I would like to know by what thought process you justify it, besides “all this stuff” which I would understand if I had a PhD.
It is interesting to note, that perhaps that is the flaw in my proof that .999… != 1 (allowing for infinitesimals of course). If one cannot truly complete a supertask… then my proof fails, and we must also rethink standard resolution to the paradox of motion. Perhaps neither space nor time is truly infinitely divisible, and in that sense the reals would be justified in there exclusion of infinitesimals.
I guess that also raises a point to which many on here seem to believe, which is their right to believe and I cannot prove them wrong, but I do not believe. That is that math is just some arbitrary game of rules we construct to explains things. I do not think so. When I look the fractal images created with complex numbers. I say there is some truth there. The concept of i or sqr(-1) is not just a play thing of math, there is some fundamental construct of nature there, which is REAL. Infinitesimals may or may not exist. I like to believe they do. If they do, real numbers have nothing to say about them. If infinitesimals do exits, there is nothing about the limit of an infinite series that tells me whether 9/10 + 9/100 + 9/1000 + … of an apple actually give me a 100% whole apple. That would assume that such pieces of an apple could exist to begin with, which they couldn’t i don’t think. How would you divide up all the quarks and what not. The question itself could be meaningless in the sense that no such infinite series could in theory even exist in reality, (because infinity and infinitesimals do not exists). Personally I think the matter is yet to be decided, but I am routing for the infinitesimal guy.
This, in a nutshell, is your problem. Math is logic, not belief. Either things are proved or they are not. In this case, there is no room for doubt unless you go outside the system like you are doing and leave logic behind.
Early mathematics was very much tied to physical phenomena. Measuring lengths on the ground, sundials, making notes of the angles of star constellations, etc.
However for more than 100 years, mathematics has diverged from those historical motivations and can be considered as a pure abstract set of rules. (A formalized “game” as some would call it.)
The ideas of non-Euclidean geometry, number theory (primes), etc were formulated purely as abstract ideas before the real applications made practical use of them. Those concepts did not need to “explain” anything about the real world (at first). They didn’t require some touchy feely type of “belief.” Their formalization of rules only needed to be internally consistent and rigorous.
Yes, I had figured this was your standpoint. You raise much much deeper questions than 0.999… = 1. But these are questions that really aren’t part of mathematics but are firmly philosophical. One issue is the mapping from the physical word into mathematics. There is a reason to believe that the real numbers don’t exist at all, not in our universe. I know eminent physicists that don’t believe that imaginary numbers “exist” - but they do think they are great way of representing phase. (Guys that have a list of Phys Rev Let publications. So not flakes.) Plato’s take on this was mentioned in passing early on in this thread. And for many it is about as good a take as any. You can also get into philosophical arguments about whether mathematics itself “exists” or simply is a manifestation of human thought. If someone proves a new theorem, did the “reality” of that theorem always exist, or did it come into existence at the time of the proof? Hint, nobody has an answer to any of these questions, but many have strong opinions.
I mentioned constructivism earlier in the thread. They are an interesting group, not flakes, but people with a strong opinion on what constitutes “real” mathematics as opposed to useful, but not actually true mathematics. You would find a lot in common with them.
A readable introduction to these questions (with a bit of an author’s slant) is Rueben Hersh’s What is Mathematics Really?
In the end these problems are just plain hard. Not 0.999… = 1 is hard, but the underlying questions you raise. It is realising which questions you are raising - which in part comes from realising what your underlying assumptions are, that makes it possible to progress. But there are thousands of years of struggle with these questions, and no firm answer.
I think you found that bit of wisdom in a nutshell, literally.
So the real number system (I assume this is “the system”) you speak of holds some patent on logic? No logic can exist outside that system?
Let’s not even talk about Godel’s Incompleteness Theorem which specifically proves that you cannot prove all truths within a system. In order to prove some things, you actually have to go outside the system!
Also logic is a system of beliefs itself, (of which there are many). So to say that Math is logic, not a belief is somewhat absurd.
When you do, you will find it’s not trivial. You may come up with something that feels trivial to you, but when you offer it up for criticism you’ll discover that your “trivial” proof rests on one or more substantive, non-trivial and not-necessarily-true premises.
The structure of the paradox about traversing endless sets (if there is such a paradox) is this. There are two arguments which each seems correct, but which lead to contradictory conclusions.
One argument goes like this: “Watch this. See me moving? When I move, I’m traversing a set with no final member. That set is the set of all points between (but not including) my starting point and my ending point. And we know that set has no final member because for each member in it, there’s always a bigger one. So it’s clearly possible to traverse a set with no final member, since I’m doing it right now.”
The other argument goes like–well, actually, it’s your job to supply it. I haven’t seen it yet so I can’t recite it here.
But once we have seen that argument, we’ll have, like I said, two apparently good arguments for two conclusions that can’t both be true. It will then be our job to figure out which argument is flawed, and in what way it is flawed.
Until you can present that argument, though, the only argument I need for the view that a set with no final member can be traversed is the one I just gave. It remains unchallenged. There is no argument on offer (yet) to make it seem as though the argument’s conclusion could be questionable despite the apparent airtight soundness of the reasoning.
I have seen you remark on that already, and you know I have a couple professors say that I might align with them. Yet, when I read the description of them… I don’t see my self at all. In fact I sometimes find myself vehemently arguing against Constructivist! I don’t know. Mathematicians may take the opinion I guess that there need be no actual physical reality to their systems, if you do just want to play logic games, (I don’t mean that in a trivial sense at all). In “my” mind and I am sure not the only one who thinks this way, the ultimate goal of math is to describe the fundamental language of nature, of which if we were all knowing there would be some perfect description of. Of course Gödel might have gone and spoiled all that.
Think of all the mathematical sentences you can construct, which state equivalences (or other relations, greater than or equal to, for instance). Take those to be your set of sentences. So sentences like 1=1, 1+1=1, 1=/=1 (that is meant to be a not-equal sign), 1>1, 1>2, 2>1, and so on and so on. Note that nothing is said about these sentences at this point, whether they are good or bad, true or false or whatever. Say you want to construct some formal system that can work with such a set of sentences, so you start with some axioms (call these your system of arithmetic), and from these axioms you might be able to prove interesting things, e.g. that 1+2=2+1. Typically though you wouldn’t want to be able to prove every sentence whatsoever. The basic idea is just that if you can prove every sentence that you can express, you have triviality. (And where triviality is just a technical term.)
Your set is only infinite if the distance is infinitely divisible, which I do not believe you can prove in reality. If you can, I promise you a Nobel Prize
Therefor your argument that you have show it is possible is no argument at all.
Either way you state the argument, assuming space is infinitely divisible seems to lead to contradiction to me.
Infinite Sets have no final member so you cannot complete them any more than you can complete any supertask. And I don’t see how you can argue that a set of infinite midway points that you must have traveled over is not such a situation.
Clearly if space is infinitely divisible you have completed a supertask when you move.
It all leads me to the idea that space is not continuous. I find that most unsettling, and I do believe it leads to other contradictions in itself.
I like constructivism, at least to some degree. This was more or less the basis for my “legalism” metaphor, earlier in this thread. It’s also at the heart of epsilon-delta proofs: you suggest an epsilon, and I can use that to construct a delta. I can never fail to do so, and, when that has been demonstrated, I “win” the “case” – the point is considered proven.
I do know, however, one constructivist extremist, whose views go a bit beyond those of normal constructivists. He says, for instance, that unless a number can actually be constructed, in the physical real world, it isn’t actually a “number.” No one will ever put 10^500 items together in one place, and so 10^500 is “not a number.” Similarly, pi, calculated to more than about fifty decimal places “does not exist,” since there is no real-world application for the value of pi to more accuracy; the “circle” and “diameter” in question simply do not physically exist.
For every “moderate” alternative viewpoint, there appears to be at least one “extremist” interpretation. Isaac Asimov wrote a lovely little essay in which he questioned the “existence” of the number 1/2! What does it mean, really, to have “one half of a piece of chalk?” Looks pretty much like a piece of chalk, doesn’t it? (He was, of course, writing this one with a finger alongside his nose.)
A point, by definition, has dimension zero. This means that between any two points there will always be other points. If this were not so, then every line would have a total length of zero. But since between any two points there will always be other points, it follows that every segment has an infinite number of points.
The only way I can see to construe this as failing to prove that distances are infinitely divisible “in reality” is to insist that there is no such thing as points, lines, lengths, etc “in reality.” Is that something you want to say?
For a sensible discussion to occur in which there is a prospect of progress, you must argue that a set with no final member can’t be traversed–not simply assert it.
There are physicists who theorize that space is digital. I don’t know how this is supposed to work.
Leads to the resolution of lots of problems too. I’ll let a real physicist comment on that. Given the past hundred years of physics, asking that anything in the real world be simple and intuitive is a vain hope at best.
Have at go at Ruben’s book. There is a preview visible on Amazon, so you can get a feel for what it is like.
What you’ve presented so far is logically inconsistent. Logic can exist outside the axioms of standard mathematics, but if it’s not consistent it is poor logic.
