I’m unlikely to take umbrage. I take it what has gone on is you’ve used what I’ve said to talk about things you’ve found interesting, and I’ve been trying to draw the line between what I’ve said and what you have talked about. (And also make clearer the line between what I said and what I meant.)
Given the right qualifications I doubt that there should be any conflict. I haven’t spelt out exactly what I meant, and maybe you haven’t spelt out exactly where you intend to go beyond what I said. In any case, the argument given about Cantor’s proof was going beyond what I meant. (I had in mind simply; if you want a syntactic object for each real, and there are uncountably many reals, a countable language won’t have enough syntactic objects. That much should be uncontroversial.)
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(There is still the interesting observation of Cantor’s theorem; it’s just that that one line’s particular perspective on it doesn’t pan out, for subtle, underappreciated reasons. Everything else you said was, of course, correct.)
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The former (this intricate but fallacious argument), I don’t think, was among the things I said. So if everything else I said was correct, everything I said was correct.
In the above, I’m having some trouble following where you’re expressing the common view and where you’re expressing your own view. Can you do me the favor of laying out starkly what the common view is, what the problem is with it, and what your view is?
Yes, fair enough. (Extending to this the same charity as for everything else: on the appropriate interpretation, etc., etc. :))
Sure.
The common view is that one can use Cantor’s theorem to entail a corollary, “There is a real which I cannot express, because the phrases of my language are countable but the reals are, by Cantor’s theorem, uncountable”.
Or, just as well but more directly, the same assertion replacing “real” with “subset of the phrases of my language”.
This is the end of the common view with which I take objection to.
To express the problem, we must first recall how Cantor’s theorem works (that is, the argument for it). Cantor’s theorem works by explicitly defining, from any countable enumeration of reals E, some real DiagonalReal(E) not in that enumeration. Or, more generally, by explicitly defining from any mapping I from some set L to its powerset, some element Diagonal(I) of that powerset not in the range of I.
[Very specifically, Diagonal(I) is “The set of those x which are not contained in I(x)”. Since each x is in Diagonal(I) just in case x is not in I(x), no I(x) can match up with Diagonal(I). Thus, Diagonal(I) cannot be in the range of I.
DiagonalReal works the same way, with some messiness about coding the result as a decimal string.]
In short, Cantor’s theorem works by an explicitly definable process of diagonalization.
[This is the end of the review of Cantor’s argument]
Now, the problem: If we go back to the common assertion above, and inline the argument for Cantor’s theorem wherever it is invoked, it reads like so: “There is a real [specifically, DiagonalReal(the enumeration of the reals via the the interpretation of the phrases of my language)] which I cannot express…”
Or, on the more direct version, “There is a subset of the phrases of my language [specifically, Diagonal(the interpretation of the phrases of my language); i.e., the set of those phrases of my language which do not express a set in which they are contained] which I cannot express…”.
In both cases, once the actual argument for Cantor’s theorem is inlined, the assertion is made “There is a specific object x [constructed in precisely such and such a fashion] which I cannot express.” But in making this assertion, one has also made very clear reference to that specific object x. One has expressed x, in order to give the argument that one cannot express x. If our reasoning framework allows us to go here, we reach contradiction and paradox. [Essentially, the same paradox as in the more direct “The opposite of whatever this phrase expresses”]
(And there’s no use saying “Listen, I just wanted to use the theorem established by Cantor’s argument. I didn’t want to inline the argument.” The only warrant one has to use the theorem established by Cantor is because of the argument given by Cantor, and if it should be ok to invoke this argument indirectly, it should be just as ok to spell it out directly)
[This is the end of the problem with the common view]
The standard formal resolution is to note that “DiagonalReal(the enumeration of the reals via the the interpretation of the phrases of my language)” and “Diagonal(the interpretation of the phrases of my language); i.e., the set of those phrases of my language which do not express a set in which they are contained” are both defined with reference to the function mapping phrases of my language to their denotations.
If my language is such that I am unable to actually refer to this function, the paradoxical argument will be halted. And this is uncontroversially what is accepted to happen with many formal languages of interest; the language L does not have the means to refer to the function mapping phrases of L to their denotations.
Thus, the paradox is halted; one is unable to conclude or even express the proposition “There is an object which is not the denotation of any phrase in the language L” from within the language L. One can say “For any fixed function of some countable collection, there is some real(/subset of that collection) not in the range of that function”, and one can say “The phrases of my language comprise a countable collection”, and one can even say “For any fixed function of the phrases of my language, there is some real(/subset of those phrases) not in the range of that function.”
But one cannot refer to the particular function mapping phrases of one’s language to their denotation, and thus one cannot go all the way to “There is some real/(subset of the phrases of my language) which I cannot denote.”
This is the end of the standard resolution of the problem, which I gave without objection above. For the sake of attaching a name, the core of this is formalized as Tarski’s indefinability theorem.
The above standard resolution of the problem with the common view is everything my previous post was meant to convey, and everything you need to know in order to understand my basic pet peeve. But I will in my next post, because you asked about my own views, go somewhat further in expressing the idiosyncrasies of my personal perspective.
The standard resolution is quite right to note that “I cannot refer to the object constructed in precisely this fashion” is silly, but is silly itself if applied to ordinary language to produce the conclusion that there is no means of referring to the function sending ordinary expressions to their denotation. This is uncontroversially what happens with formal languages of a particular sort (not all, just those of a particular sort), and it’s not erroneous to look at ordinary language in that same fashion (as always, you can interpret terms such as “denotation” and so on in such a way that this is indeed how ordinary language acts), if you would like to do so. I just think it’s a misguided, unhelpful way of looking at it.
One might gain some perspective by considering the analogue in the computable world: here, the Cantorian argument shows up as the fact that there is no program P such that both A) P always outputs either Yes or No after receiving two inputs in a row and B) for any program of just one input which always outputs either Yes or No, there is some first input to feed to P which causes it to specialize to the same input-output behavior. [Because, if P satisfies A), then the diagonalizing program “On any input x, output the opposite of P(x, x)” serves as counterexample to B)].
And Tarski’s indefinability theorem shows up as the fact that one cannot write a one-input program J such that A) J outputs a value on all inputs, and B) Whenever the program P outputs a value, running J with input P produces the same output value [Because, if J satisfies A), then the program “Run J on myself, wait for it to output a value, and then output a different value” serves as counterexample to B)].
That last fact expresses the sense in which the mapping from computer programs to their denotations is not itself expressible by a computer program. Yet, at the same time, there uncontroversially is a sense in which the mapping from computer programs to their denotations is expressible by a computer program: the sense given by the program “Take in a program as input, and just run it; do whatever that program does”. This satisfies B) above, but not A).
But this seems the most appropriate way to understand the notion of the interpretation function for the language of computable expression. The problem with A) is that it enforces that the outputs be computably negateable (Yes to No, No to Yes), without accounting for the nature of general computer expressions as occasionally running forever unpredictably (and this cannot be computably negated; there is no computable means to send nonterminating programs to terminating values and vice versa). And this same perspective sneaks in to the usual presentation of results such as the inability of a language to refer to its own meaning function.
…And I’d like to expound on my foundational views much further into crankery some time (and then defend them). But guilt informs me that, for the time being, I should probably spend more time writing my thesis and less time writing message board posts.
The same holds true for the Zeno time series, 1/2, 3/4, 7/8, … ,(infinity-1)/(infinity) So if you can accepts Zeno, you can accept 0.999… =1
That is the reason I asked it. Figuring out the difference between 0 and 0.000…1, is precisely the same problem as figuring out the the time that the final 0 was written. The fact is that this second to last number was also written at exactly 1 hour, there was no time between it and the final 1, just like there is no difference between 0.999… and 1.
Huh. Now that is interesting. I hadn’t seen the derivation for the Fibonacci sequence or other recurrence relations before.
I find it amusing that the exercise basically amounts to “Of course the sum diverges to infinity. But supposing it didn’t, what would it sum to?”. And then coming up with a perfectly consistent answer.
I also find it strangely compelling (being a fan of 2’s complement math on computers) that the sums of 1+2+4+8+… or 9+90+900+9000+… are -1.
(There is a minor typo there: the rational function should read (a + (b - a)x)/(1 - x - x^2), though luckily I did write the correct value at x = 1 of -b)
Another really fun math topic is “finite difference calculus.”
(Also iterative solutions, or successive approximations, like the Newton method for finding square roots. I discovered that one on my own in high school, and my math teacher showed me how to extend it to give asymptotic approximations to solutions for all sorts of ugly equations. FUN!)
Yes, fair enough. (Extending to this the same charity as for everything else: on the appropriate interpretation, etc., etc. :))
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If there are countably many blah, and uncountably many bleh, you can’t have a different blah for each unique bleh. That comparison is the only one I was making. If you want a syntactic object for each of some collection, and you have less syntactic objects than things in the collection, you can’t have a unique syntactic object for each unique thing in the collection. (I just don’t see why the comment about charity is needed, at least not now.)
This all started with a point about the fact that you have different numbers of syntactic strings depending on how you build them up. Infinite decimal expansions give you a lot of them. All of my points were in that simple context. I didn’t want to make any points about language beyond that context. When you said that you could only use finite strings to represent infinite strings, that was a relevant fact about syntactic objects, and that is the context to read my comment.
The “charity” thing is a joke (hence the smiley). I was noting how uncharitable I had been towards interpreting your post, in comparison to how charitable I had been towards, say, erik150x’s.
As I said, the choice of notation “(infinity-1)/(infinity)” is probably poor on my part as most accept infinity-1 to be infinity. What I meant by that sloppy expression is simply something infinitesimally smaller than 1. The imponderables of infinity go both ways. I could ask you was the last number before infinity? Can you answer that? Of course not? I could ask how much must you subtract from infinity to get back to a real number? Or any number for that matter? Exactly what conditions occur just before you reach infinity… I mean come on man, that is beyond our understanding if the question even has any meaning. But you want me to tell you what the last number an infinite sequence is? And because I don’t know you conclude it’s equal to the limit of the sequence? Are you serious? If could at all be proven it would be probably the ultimate proof of all time.
You asked me what’s the second to last number in the sequence? Okay, what about the 2nd to last? Or 3rd to last, or (10^1000000000) last number just to pick one ridiculously large number (in comparison to most we usually speak of), but yet even I could ask what’s the 10^(number of particles in the known universe), would this still be 1? By your logic it must be?
How about infinity numbers before the last? Would that get us back to reals? if so what would that real be? .9? .99? .999999999? .55378008? 0?
I really had to consider for a bit if you were being facetious or sarcastic because I don’t see the logic?
You would seem to like to say we cannot get infinitely close to something? I would say .999… is infinitely close to 1, but not equal. You would like to say it’s equal. Okay so how would you describe a number infinitely close or as close as you can possibly get to one? You would probably say there is no such number? You would say you can always get closer, if there is some difference between 1 and that number than you can always say take half of it? Right? A little like Zeno’s paradox? But I guess you would say no not forever, at infinity the difference just disappears? Where does it go? Why? You have some secret proof your holding out from the the world of mathematics?
I guess you must have missed my proof where I showed the difference was .000…1?
I briefly state it again, and you can argue it if you like.
Take a line segment which starts at point A and ends at point B
Let’s say the line is of length 1 in whatever units you like
We mark the line on top and bottom as follows:
At point A Top = 1.
At point B Bottom = 0.
Now we mark the halfway point between A and B
Top = 0
Bottom = 9
We continue on that way at each successive half way point taking the remaining half to right on to infinity as such:
1.__________________ 0 _________0_____0__0
_1/2_______3/4____7/8…[We’ll call this unknown# since it is unknown without using Limits.]
0. 9 _________9_____9__9
The “unknown” number is not important here. The important thing is the 1-1-1 mapping which is infinite and must be possible to move from point A to point B in continuous space.
So now we have the supposedly impossible to construct numbers below with their last decimal place. Note that it not actually necessary to be able to physically construct this any more than Cantor had to physically construct his diagonal method.
1.000…0
0.999…9
We do ordinary subtraction (which we don’t actually have to do as the answer is clear by induction)
.000…1 or 1/infinity
If you want to argue that by definition there can be no last decimal place, okay, extended it out as far as you like the the two numbers are not equal in the most ordinary sense of ordinary arithmetic. And I do not consider Limits “ordinary”.
To reach 1 you need to make take one more tiny little step .000…1 or 1/infinity.
To say that either series = 1 you would have to take the Limit of the series. And as we all know the Limit says in plain English if the function or series approaches a number with arbitrary precision, we will get rid of these pesky infinitesimals and do away with our ordinary rules of math which break down and just call it that number.
There is no proof for limits it simply provided as an assumption. And counter to what many people seem to indicate on here the rules of math are not made arbitrarily. It is not like the rules of chess vs checkers (which one could argue are not quite arbitrary either, but I understand the implied comparison). We make rules of math to reflect some truth in the matter of how the universe works. One may take limits as some universal assumed truth, if they wish. It’s perfectly legitimate. However there is no proof provided that they should be accepted. I mean they are extremely useful. A wonderful tool if you ask me. I mean when I am calculating the continuously compounding interest on a bank loan, I am perfectly happy if one uses limits. But if your asking me if .999… = 1, I am not so happy if you simply take the limit which assumes them equal to begin with by the nature of it’s definition.
What does it matter? It matters to me. The truth matters to me. .999… could be equal to 1 and maybe someday someone will prove it. But I have seen none, and I believe I have somewhat informally proved it that it does not equal one. You can ask me what does .000…1 mean? How is it useful? Well, to me it means an infinitely small number and it’s at least useful for describing the difference between .999… and 1.
You can talk about the definition of real numbers all you want, I don’t care. If it makes it convenient for people to define real numbers with conditions that make .999… = 1 (by use of simply asserting it or defining them using limits) so be it, but its a matter of convenience not of some actual necessary truth.
When some asks how is .999… = 1? I am certain for the vast majority of situations, they are NOT asking how did we come up with a number system where we start out with the assumption they are equal and we just define repeating decimals as the limit of the infinite series. They are asking from the very intuitive perspective we all posses that .9 < 1 and .99 < 1 and .9999999999 < 1 how it is that some point it just becomes equal?
You know if you want to say 1/infinity = 0 then that makes sense to some degree. But we don’t because that would lead to all kinds of chaos right? 0 * infinity = 1? Well how about 7 * 0 * infinity = 7? or 1???
So you see we just don’t understand 1/infinity, which is why we say it’s undefined. And it’s why we ignore it through Limits. But that does necessarily mean it doesn’t exist either.
“Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official “Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,” they reappear in the mathematics of the late middle ages and later played an important role in the development of the calculus. Their doubtful logical status led in the nineteenth century to their abandonment and replacement by the limit concept. In recent years, however, the concept of infinitesimal has been refounded on a rigorous basis.” - Continuity and Infinitesimals, First published Wed Jul 27, 2005; substantive revision Mon Jul 20, 2009
"inasmuch as the number of terms in nature is infinite, the infinitesimal exists ipso facto.” - Johann Bernoulli
I guess this is where I think you completely fall apart. I don’t have to “argue” that there is no last decimal place, because, as you say yourself, it’s part of the definiton of the notation.
“extend it out as far as you like” implies an end. There is no end with these numbers. Writing it like this “0.00000…1” is just playing games with the notation. Verbalize what you’re implying here “An infinite string of zeros ending in a 1”, and it falls flat. It’s an infinite, unending, never wavering, string of zeros.
Verbal explanations of the infinite whether large or small will always fall apart. You’re conception of a never ending string of 9s in .999… is what makes a paradox of Zeno’s Dichotomy Paradox I use in my informal proof above. You cannot move from point A to point B without coming to and END of an INFINITE (never ending) series of halfway points. If we were to hold to that notion of never being able to come to an end of some infinite set, then no one could move anywhere, period. This does assume you accept continuous space. As Indistinguishable pointed out you can make the same “paradox” with time. How does one get from one point in time to the next, if one has to complete an infinite number of half way points to get there. So to avoid these paradoxes, 1) you either accept that one can come to an END of an INFINITE sequence or 2) you disavow continuous space and time and assign some granularity at which they cannot be divided any further. In the second case an infinite string of 9s in .999… would not have any real world meaning. It would be a meaningless notion except as to ruminate on some other universe where space and time were continuous.
It’s already been explained (by me and others) that you’re using “end” ambiguously. Reaching the limit of a series is not the same as reaching a final member of the series. These are two different concepts, but you are using one term (“end”) to denote both of them.
I’m sure there are probably actual terms for this sort of concept, but I’m going to expose myself as seriously not a mathematician and ask, aren’t you mixing two “kinds” of infinity? A never-ending string of 0’s is not the same, in my mind, as an infintely precise measurement of half-way points. One is a road that never ends, the other is a ruler that gets forever more precise. Sure, I can travel to the end of an infinitely precise sequence of half-way points, because the concept of a half-way point in itself dictates that the line segment has a beginning and an end. Now, if you’re talking about **marking **those halfway points, you’ll never make it. But you’re confusing the two - marking the halfway points and traveling the total distance of a finite space are different things. To this layman anyways.
Then in your previous post, when you talked about reaching the end of an infinite series, you’re talking about reaching the final member of it–and it has been explained that the infinite series you’re referring to has no final member.
