[.333…][.333…] = .09 + 9x .00222… + 9x [.0111…]^2
Multiply through by 3
[.333…] = .27 + 27 x .002222… + 27 x [.0111…]^2
Multiply through by 3 again
1 = .81 + 81 x .00222… + 81 x [.0111…]^2
I have a conceptual model of a system that makes it easier to understand the concept of infinitesimal and its properties of “finiteness”, the conceptual model is what is guiding me through these intuitive proof attempts.
You need some sort of algorithm for multiplying two infinitely long decimal fractions together, because you can’t start at the right like you do with finitely long fractions. One way is to set up a sequence, where you multiply truncated fractions, and lengthen by one digit at a time:
As you keep going, you find that the sequence 012345679 gets repeated indefinitely in the decimal fraction. If you multiply that by 81, you get 999999999, so multiplying .00123456790… by 81 gives you .0999…
These “calculations” you have made are all erroneous. If you are trying to demonstrate that you know of an algorithm for multiplying these two infinitely precise sequences together you are going to have to do it by showing an initial first step of applying the axioms that is not erroneous …else you have not demonstrated anything and your initial “equation” which was
[.0111…]^2 = 0.000123456790123456790123456790…
stands as nothing more than an assumption such as the assumption 4 = 5 (which can be
invalidated via the application of the axioms of the set)
Nowhere in this thread has anybody demonstrated an algorithm either conceptually or rigorously for multiplying two infinite sequences together.
Yeah, I was not exactly being coherent. I was trying to say that if you have a system, and you find that it contains an unprovable proposition, that does not imply that your system is self describing. Sorry about that.
Well Indistinguishable is the real mathematician here. I just have dim memories of studying this stuff as an undergraduate.
Since it looks like this thread won’t die, let me point out that the set of real numbers has a precise definition. Dedekind cuts were mentioned above, but it’s easy to show that 0.999… = 1 using the construction of R as the completion of Q. By definition, 0.999… corresponds to the sequence (9/10, 99/100, 999/1000, …) and 1 corresponds to the sequence (1, 1, 1, …). The difference (1 - 9/10, 1 - 99/100, 999/1000, …) = (1/10, 1/100, 1/1000, …) tends to 0, so 0.999… and 1 are equal by definition— not “infinitely close,” or some idea of ‘infinitely close by near enough that it’s easier just to call them the same’, but equal. While I’m on the subject: The real numbers do not contain any infinitesimals, the idea of an “infinite number” is not a meaningful term, and Goedel’s incompleteness theorem has absolutely nothing to do with any of this.
Why is this thread 23 freaking pages long? This is a Question, not a Debate, and it’s been answered repeatedly in this thread and in many, many other places on the Internet. I don’t know what anyone arguing against the result 0.999… = 1 could possibly want at this point.
In this case, I don’t think a general algorithm is required. The sequence [0.111…] is simple enough that starting with limited examples such as Giles gave, it’s easy enough to see that that squaring it gives the result indicated. Or perhaps more precisely, that it couldn’t be anything else.
As a former math major who didn’t continue, I know just enough to be dangerous.
However, I do understand the value of and need for absolute precision in definitions. And when you say “Nowhere in this thread has anybody demonstrated an algorithm either conceptually or rigorously for multiplying two infinite sequences together.” you seem not to understand that in all your “proofs” you have been making blithe assumptions about the multiplication of infinite sequences yourself. You do so regularly, and also square infinite sequences and divide one into another. How do you explain that?
Giles roughly described such an algorithm in the post immediately previous to yours.
We could also specify such an algorithm as follows: Recall, from post #36, that the standard interpretation of an infinite decimal sequence is as representing a value >= than its rounding down and <= its rounding up at each digit (you don’t have to use the notation this way, but it is the standard). So, for example, “.0111…” would denote “A value in the interval [.0, .1] and also in the interval [.01, .02] and also in the interval [.011, .012] and also…”.
If we then suppose multiplication has the usual properties with respect to order, we can readily and mechanically extract information about the product of such values. For example, since .0111… is in the interval [.01111, .01112], we will have that [.0111…]^2 is in the interval [.01111^2, .01112^2] = [.0001234321, .0001236544]. Since the two endpoints of this interval agree on the initial digits “.000123”, we will say that so must any point within that interval, and thus so does [.0111…]^2. In this way, we can continue to extract further digits of [.0111…]^2, using more precise rounding intervals capturing .0111… (and can do similarly for any other basic arithmetic combination of such decimal sequences).
You could, if you liked, suppose 81 x [.0111…]^2 was infinitesimally smaller than .01, in much the same way you could suppose .999… was infinitesimally smaller than 1, along various lines previously discussed throughout this thread.
But that’s not how everyone else standardly interprets the notation of infinite decimal sequences. It has been explained throughout the thread how this notation is standardly interpreted, and on that standard interpretation, “.0111…” is defined so as to mean exactly the same thing as “1/90”, making “81 x [.0111…]^2” exactly equal to “.01”.
Why don’t you tell us why you think “81 x [.0111…]^2” should denote a value less than .01?
Alongside everything else, by the way, the real numbers are not subject to Gödel’s incompleteness theorem, as their usual axiomatization is not sufficiently expressive. That is, the system of real numbers is both consistent and complete. They’re also decidable, i.e. there’s a procedure such that for every statement expressible in the language of the reals, it can determine whether it’s true or not. For 0.999~=1, it says ‘true’.
You can, of course, use different axiomatizations—but then, you’re really just talking about different things than everyone else.
Let’s be more precise: what is algorithmically decidable is the truth or falsehood of every statement expressible in terms of polynomial equations, Boolean operators, and first-order quantifiers over real numbers (“for all real numbers…” and “there exists a real number such that…”). In fact, every such statement follows by standard logical derivations from the simple rules that the reals comprise an ordered field satisfying the intermediate value theorem for polynomials. [I suppose these axioms are what you mean by “the system” in “the system of real numbers is both consistent and complete”?]
But if we were to want to say other things about the reals, we could lose this algorithmic decidability. E.g., if we add cosine to our language, we lose such algorithmic decidability.
Also, it’s a bit silly to say our axiomatization of the real numbers, in itself, tells us “0.999… = 1”; the structure of the real numbers is orthogonal to our conventions for how to interpret notation as mapping into it. We could still take the reals to comprise an ordered field satisfying the intermediate value theorem for polynomials, while choosing to interpret “0.999…” as denoting something other than the real number 1.
Still, you’re right to note that Goedel’s incompleteness theorem is a total red herring here. As you say, we do standardly interpret “0.999…” in a certain way, and anyone who doesn’t is talking about different things than everyone else.
If you like, of course, we could hook up a validity-determining computer program of this sort to furthermore allow among its inputs constants specified via eventually cyclic decimal notation (e.g., “24.1[638 repeating]”) in such a way as that it justified the conventional interpretation of this notation, and thus would happily tell you “0.[999 repeating] = 1”. Perhaps this would convince some people of something (and it certainly puts the lie to the odd assertion that “0.999… = 1” is an unprovable proposition of the sort Goedel’s incompleteness theorem demands! I realize now this is the whole reason you brought it up.).
But others might say this is no more a strong counterargument to their particular views than the much simpler program “PRINT ‘0.[999 repeating] = 1’;”, and, in fairness, they’d have a point. The debate is situated at a different stage.
Yes, we know about Wikipedia. Someone linked to the article on Zeno’s paradoxes, for instance. What information is in the Wikipedia article on 0.999… that isn’t in this thread?