## Friday, May 15, 2009

### Is 1 = 0.999999... ?

Some years ago a computer science guy who looked eager to pick a fight crossed my way. He asked me if I think 1 equals 0.99999... or doesn't. Notice the three dots supposed to say the nines continue infinitely. This infinite continuation is commonly abbreviated with a bar and looks like this Searching in what was left from my 6 semesters maths I said I think so, upon which the guy looked disappointed. I figured it was the right answer - right at least for my purposes. We then had a pleasant conversation about graphic cards I believe.

Anyway, recently this question crossed my path again. It seems to have some eternal fascination, so let me just offer my take on it. The left side is the limit for n to infinity of  the sum over 9 x 10-m from m=1 to n. The limit of this sum for taking n to infinity is one, according to the most straightforward proof you can think of, namely if you pick any number only an epsilon smaller than 1 there will always be an n0 such that the sum is closer to 1 than that number for all larger n. Thus, both are equal.

You find a furious discussion of the topic here.

This post was created by random association to Mark's post You can't write that number; in fact, you can't write most numbers even though you can write that number. Wishing you all a good start into the weekend!

PS: As was pointed out in the comments, Wikipedia has an excellent site on the matter.

1. 1/3 = 0.333333.....

3*(1/3) = 3* 0.333333333

1 = 0.99999999

2. which merely states that the first statement is identical to the last, not that either is correct (and you forgot the dots...)

3. I find the "1 = 0.9999..." bit annoying because it means every rational number of the form a/(b*2^m*5^n), where m,n are nonnegative integers not both 0, has two decimal representations.

It's a complication which makes Cantor's Diagonalisation argument a bit tricky to get right.

4. Hello Bee,

There are lots of ways to see this result.

I like the observation that all periodic numbers are equal to their period divided by as many nines as the number of digits in the period. For example 0.48484848...=48/99
In this case 0.999...=9/9=1

Or else, one must agree that there exist an infinity of numbers between two real numbers.
Since there is no X so that 1 > X > 0.99999..., then the numbers are equal.

We could go on and on with descriptions of this fact, in rather different ways.
But from a simple depiction to a convincing proof, there is quite a leap.
"The reasons one need in order to be convinced by a reasoning are psychological in nature, in mathematics like anywhere else."
Henri L. Lebesgues (1875 - 1941), "Courses on integration and research of antiderivative functions", p. 328 (traduction from french, sorry if it isn't perfect).

The reasons why it is so difficult to accept for non specialists, are the same as in Zeno's paradoxes.
The concept of infinity is startling, even when only considered as potentially, merely trough the concept of limit.
And because one can live an entire life without coming across such a dilemma, most people still think like the ancient Greeks...

Best,
Johann.

5. Hi Johann,

Yes, the issue is of course here as in many other cases that infinity is startling. It's a concept that seems to be difficult to grasp. There must be an infinite amount of 'proofs' where people treat infinity like a number and come to contradictions. I am not entirely sure how come though. At least we learned how to take limits already in high school. I think a lot of damage can be done by careless statements (as unfortunately many physicists like to use them).

Either way, I've found in several cases that one is likely to find online a correct rather than a wrong answer to questions like this. (That is not to say you wont find discussions about it with many misunderstandings.) So maybe we'll see some improvement on these matters thanks to bloggers like Mark and others who spread mathematical soberness.

Hi Blaise,

A subtlety that admittedly hadn't occurred to me before.

Best,

B.

6. From the algebraic prespective ,the question " 1 = 0.9999999... ? " is very annoying!Let R* be the multiplicative group of the real numbers,now since R* is a group then there exists only one identity e such that e times x = x for all x belongs R*.Clearly in R* ,e = 1 is the only element satisfying e times x = x for all x !Now putting 1 = 0.9999999....will be seen as trivial and inconvenient change of notation! Regards, Serfo

7. I understand the 1=0.9999... concept. And I won't deny that it took a lot of the math to accept that (lots of internal resistance to it).

My question, however, is that is there a different representation then for something like 0.7777...

Or does 9 being the largest single number in the decimal system, give it this special property of having a different representation (i.e. 1 in this case)? That seems a little arbitrary to me, however.

8. And as expected, Wikipedia has a very nice page about the subject :

http://en.wikipedia.org/wiki/0.999

9. 0.999... is no more or less strange than 1.000... except that by convention we don't write the extra zeros and dots.

10. Addicted - It is our place value system that gives us two representations for 1.0

There is a closely related problem in finite mathematics well known to computer people. Positive and negative binary numbers can be written in so called one's complement or two's complement form. One of them has a double representation included.

11. It's not a property of the number 9.
It is related to the concept of infinity and therefore of continuity (which it simplifies, again trough the concept of limit) :
If two real numbers (or points on a line for that matter) are exactly consecutives, then they're the same.
For two (differents) of them, an infinity of others lie in between.

It just so happen that we have a name for the exact "successor" of .999... and it's 1. So they're the same.
It's more tricky for the "successor" of .777... but it exists. At least ideally.

Just to prevent misinterpretations, I must add that the concept of "consecutive" or "successor" are not involutary, that is to say that the successor of the successor of a real number isn't the same number (this intuitive representation and its counter-intuitive properties can be backed up by topological concepts).

PS : I struggled with myself to post this answer, because I don't want to participate in transforming your post into an argumentation about this hot subject (I know by experience on some forum that it is)...
By the way I don't know how you can keep up with all the posts and numbers of comments it generates, it must take quite some time.

Best,
Johann

12. A. Gradstudent1:17 AM, May 16, 2009

Another fun (mean?) blog post would be taking all of the hilarious misconceptions taken from forum arguments on this topic and mocking them.

13. The mathematical treatment is just one aspect. More than that the eternal battle of mankind with infinities is a deep philosophical issue.

14. Hi Bee,

I can imagine how such a question which relates to our understanding of the concept of infinity would provoke endless debate. That is as Blaise has eluded it all focuses primarily around whether infinity exists in its entirety or is simply a concept of ultimate potential, as in suggesting direction. This marked for instance the great rift between Cantor and Kronecker. In fact it goes so far that if one accepts Cantor’s definition of an infinite set and the axiom of choice, mathematically thereafter we can get pretty much something for nothing. The question of course that remains is if this extends to the real world or not?

Best,

Phil

15. It's not a property of the number 9. In fact, something similar happens, whichever base you choose. E.g., in base 8,

0.777... = 1

In base 13,

0.CCC... = 1

(where capital letters are used to denote "decimals" with higher value than 9, i.e., A=10; B=11; C=12 and so on).

Some more fun with different bases: 1/7 in base 10 repeats itself, and therefore does not allow two different representations. But in base 7 it is not repeating (in base 7, 1/7 = 0.1), so it does allow another, infinite representation (0.0666...).

It's all due to the base you choose. In a certain sense, 0.999... = 1 because we have 2 hands with 5 digits each (well, most of us do).

16. Hi Bee,

How about this way to frame the argument being that the world of matter exists within the set of .999..., while the realm of energy exists at 1.000..., as it doesn’t have the time ;-)

Best,

Phil

17. The "furious discussion" you link to contains the following, "Notice that I keep putting the word definition in bold face", which is the seed of its own destruction. On all those pages and here there seems to be only one mention of Robinson/non-standard analysis, subsequently ignored. None here yet. Wikipedia, non-standard analysis.

Robinson introduces (consistently) a class of infinitesimals, let's say that delta is one of them. 0.9999... is notation for a limit, as you say; every term in that series is less than 1-delta, hence the limit is less than 1, in Robinson's number system. One might not like non-standard analysis, or think that it adds nothing to our ability to do interesting mathematics, or just prefer not to complicate our definitional system unless we absolutely have to, but it is a matter of definition whether 0.9999...=1 or not.

18. If x = 0.999999....
10 * x = 9.99999.... = 9 + x

or 9 * x = 9

This is a general method to find out what fraction corresponds to a recurring decimal that we were indoctrinated into in high school.

i.e., given 0.abcdef...abcdef...abcdef...

multiply by the appropriate power of 10, etc.

19. I think what nonmathematicians tend to miss is that all this depends on definitions and axioms. We have to say what we mean when we write an infinite decimal expansion.

Bee correctly gives the interpretation of an infinite decimal expansion as an infinite sum, and the rest follows from the axioms of the real field.

If we work in a different system like the hyperreal numbers, then we DO have numbers which differ only by an infinitesimal and which we DO consider to be distinct.

(I just noticed that Peter beat me to the point.)

20. Hi Arun,

Not that I have an opinion either way, yet if .99999... is a potential as in being a direction, rather than a destination, can it be defined simply as equivalent to X ? That is for many X considers that something can be defined, which is what one could say is the essence of being finite. So in this way to insist that .999... is equal to X or 1 would exclude the consideration of the infinite. To be honest I try best to avoid thinking too seriously about infinity, since minds much greater than my own feeble one have paided a heavy price for such consideration. That’s what I most admired about Bertrand Russell is he realized this early enough and switched his focus to philosophy:-)

Best,

Phil

21. Peter, Ivan: You are of course right that it all depends on the definition. It always does, thus my first impulse was to ask well, what do we mean if we write "..." You are right in that with a different definition the above equality might indeed not hold. You could also go and argue that there is in practice no way to ever write down any infinite series, thus the series 'exists' in a merely Platonic sense. (As CIP pointed out above though, the right side has an infinite series of zeros too, so the best you could say realistically is that they agree to some precision.)

22. Hi Phil,

The infinite can be defined, it just isn't a number. But besides this, it is easy to see that 0.999... is not infinite: it is bounded from below by zero and bounded from above by 2, thus you can exclude this case easily. Best,

B.

23. You could also go and argue that there is in practice no way to ever write down any infinite seriesI don't think even the strictest of contructivists would ever argue that, would they?

It is certainly true that you can't write down most infinite series, though, since there are uncountably many infinite series and only countably many finite strings of symbols which describe series (the point of Mark's post). But no one cares about the uncomputable numbers anyway. :^)

I suppose one could argue that the field of real numbers doesn't exist, and only the field of computable real numbers really exists. But the field of computable numbers doesn't have the least upper bound property, which kinda sucks.

24. Grrr... the first paragraph of my previous comment needs to be split in the middle. (The preview looked fine. Weird.)

25. Well, I can with certainty tell you that people exist who believe almost all real numbers don't 'exist' for exactly this reason. ('.' because one can argue about what it means to 'exist' which is a discussion I actually don't want to enter.)

26. Reg the paragraph break: It's a known blogger bug, and I hope it will be fixed soon. All linebreaks after italics, bold face or links will vanish with submission. It's pretty annoying but not your fault.

27. Hi Bee,

“The infinite can be defined, it just isn't a number. But besides this, it is easy to see that 0.999... is not infinite:”

I think that in essence is what I said, yet I will avoid saying anything other than to ask if infinity is not a number, then how can we be confident that it can be logically consistent to have it aid in defining one? Cantor showed that we can envision number sets that while bound still form infinities being uncountable, with proportions that are variable rather than fixed. So if we were to divide an infinity into another we would not necessarily get 1.000... as a result.

The problem as I see it is not with what a series continued to infinity will yield, yet what infinities we consider apply and constitute being responsible for what we call real. It is true that the recursive sequence discussed is considered (and proven) as countable, yet it doesn’t preclude or quantify the density that might exist between each recursion. This is like the old Zeno problem in asking what we need to pass through at each step to reach the next.

So I’m not troubled that within the confines of the definition and rules chosen .999... is the same as one, its simply that this is so loosely defined as being infinite to begin with. I guess what I’m driving at is should calculus be imagined as how we consider the beginning and the end to physical limit or only the outline for which we require something more to fill the gaps. The only thing I’m certain of is the hare does catch the tortoise in the world we live in and yet completely uncertain as to exactly how:-)

Best,

Phil

28. I'm curious what polymathematics thinks of non-standard analysis. With the existence of the Wikipedia 0.999 page, presumably he must know of them now. I'm pragmatic enough that I've never found a reason to use non-standard analysis (except for dabbling with Columbeau functions as a possible way to formulate more rigorously renormalization's desire to manipulate infinite quantities), but the consistent existence of such a system should surely give a True Defender some pause.

Up to a point, non-standard analysis gives a rigorous foundation for many of the informal ideas that non-mathematicians think should be present in a discussion that polymathematics is so eager to deny can be a part of a formal system. Many of the commenters try to say that there are numbers between 0.9999... and 1, polymathematics argues that no there aren't, but non-standard analysis says, I define them thusly, and it is as good as your definitions. Where polymathematics keeps saying no, no, no, he should have been saying, go read about non-standard analysis on Wikipedia or wherever, or Hyperreals, where you will find what you have to do to make your crude intuition work as a formal system. When you've done that, you'll probably agree that it's really easier and more useful to use the standard definitions, or at least we can agree to disagree about our preferences. Personally, I find the dependence of mathematics on how we humans define our formal systems makes me considerably less decided about the status of mathematics.

29. Hi Phil,

Well, it is logically consistent because it is well-defined. I am afraid I don't understand your concern. From the mathematical point of view there is nothing ill-defined about infinity, one just has to be careful to deal with it correctly. A lot of the confusions and misunderstandings arrive from exactly the kind of problems you are mentioning: dividing infinity through infinity, subtracting infinity from infinity, etc, instead of taking the appropriate limits. Best,

B.

30. Hi all,

I have not read the detail of all the comments, I just wish to provide a framework to sharpen a bit our intuition on this matter:
Decimal representation is just a language to speak about real numbers, and as many languages, it has synonymous, 0.9999... and 1 are just two of them that refer to the same object.
Now, an interesting question is: Is there a language to speak about real numbers that does not have synonymous?
The answer is yes, and can be found in the first theorem of the second chapter of Khinchin's "Continued Fractions":
To every real number α, there corresponds a unique continued fraction with value equal to α. This fraction is finite if α is rational and infinite if α is irrational.

Cheers

JP

31. As an outsider I found the topic to be an interesting one from a mathematical perspective. So of course one does their research and are at the mercy of what's out there as to a conclusiveness of design and meaning.

Bee:The infinite can be defined, it just isn't a number. But besides this, it is easy to see that 0.999... is not infinite: it is bounded from below by zero and bounded from above by 2, thus you can exclude this case easily.This quickly dispatches the argument, while I was immersed in my sleep time as to responses to your example, now in relation to the cosmos. So you see, I take this seriously.

The mathematical basis then becomes a struggle for me as to definition. Sure it's easy for all you math types, but lets see if your up to expanding this basis to interpretations to the cosmos? Is it wrong?

"if one traveled in a straight line through the universe perhaps one would eventually revisit one's starting point."

....may mean an objectivity defined expression is a connection with the confines of "Bee's mathematical parameter" so it constitutes a meaning "within only this interpretation."

So you hold to that and see in cosmology there is no other "outside the box" explanation that fits while mathematically holding.

But if one can define a greater "sense of dynamics" then you cannot hold onto that definition any longer? Peter sees a greater sign then in explanation to wiki?

Definitely a learning experience here:)

Best,

32. An Intermediate Polar Binary System. Credit & Copyright: Mark GarlickStephen Hawking’s says:

“Roger Penrose and I worked together on the large scale structure of space and time, including singularities and black holes. We pretty much agree on the classical theory of relativity but disagreements began to emerge when we got into quantum gravity. We now have different approaches to the world, physical and mental. Basically, he is a Platonist believing that’s there’s a unique world of ideas that describes a unique physical reality. I on the other hand, am a positivist who believes that physical theories are just mathematical models we construct, and it is meaningless to ask if they correspond to reality; just whether they predict observations.”
( Chapter Six-The Large, the Small and the Human Mind-Roger Penrose-Cambridge University Press-1997)
Bold added by me for emphasis

This discussion obviously goes far beyond the "mathematical obvious here." While it is distinctly clear as to the explanation of the "mathematical obvious" there is a thought lying just outside of this example, is a whole basis of approach to one's science?:) Maybe Bee's and Peter's views "lie within it?"

Best,

33. http://arxiv.org/abs/0811.0164

34. Yes, one is "equal to" the formal definition of 0.999... because of what "equal means": to have the same value, even if the "means of representation" is different. Well, that's kind of the whole point. Why write "5 = 5", which you already know. Writing " 2 + 3 = 5" actually tells you something: that the sum of two and three equals five. If you believe in infinite series being "real", then 0.999... = Sigma 9/10 + 9/100 + 9/1000 + ... = 1, and therefore the expression is "true" in the same way as "2 + 3 = 5." It just looks oddly put.

I know that some people think a representation should have some proper logic to it, put in the simplest form, etc. OK, maybe, but the statement is still "true". But if you're into surreal numbers and nonstandard analysis (which I barely know or understand), it may be an issue. (Does 0.999.... really equal 1 - epsilon or some weird little infinitesimal?)

tyrannogenius

35. Hi Bee,

I truly didn’t mean to dispute as to argue, yet simply to wonder out loud if you like. I guess one way to express it, is as Cantor has it the uncountables if envisioned on a number line must be either without dimension or not confinable to a line to exist one dimensionally. So it begs the question if all is actually nothing or rather the limit to which it’s considered confined incorrect? Another way to put it is to ask why there is dimension at all, not just simply more than one, to suggest that it’s mandated when mathematics is manifested into something that’s required to be real.

Mathematics and its cousin logic are rife with contradiction and paradox and I see this debate as no different or unusual. For instance to calculate the proportions of a circle or sphere to completion cannot be done without evoking infinity as a requirement and yet they being economical in construction and symmetrical in design are qualities by which they can be defined without calculation and this has been found to be even more fundamental in terms of understanding. So as a bubble or a planet needs not infinity to describe its shape why would a number require it to define its value?

“From the mathematical point of view there is nothing ill-defined about infinity, one just has to be careful to deal with it correctly.”

Would you settle for incomplete rather than ill-defined (no pun intended)? That is when it comes to if there exists magnitudes of infinity between the countable integers and the unfathomable density of the reals has yet to be decided or even proved to be decidable. "God made the integers; all else is the work of man" said Kronecker. If this be true then nature was certainly patient while those who where to create the language to describe and parameterize its action could be formed within such limitation. Yet perhaps this suggests that eternity does actually exceed infinity or that when compared to a true continuum such magnitudes have little if any meaning in measure.

I understand this is purely philosophical and yet I can find no other context in which to express it. None of this of course is to suggest an answer, just simply questions and musings as what could constitute being some of the parameters.

Best,

Phil

36. Phil,

I think by the "uncountables" you meant the irrationals, or perhaps the trancendentals?

You probably need to look into "topological dimension" (there are many definitions of dimension appropriate to diferent settings). For example, with their natural inherited topologies as subsets of R, the rationals have topological dimension 0 and the irrationals 1. Of course R itself has the intuitive value of 1 as its topological dimension (at least with the natural topology...).

The "intuitive" notion of dimension, appropriate for surfaces of spheres, spacetime, etc, comes from their being maifolds and looking locally like R^n, which does work for arbitrary sets.

With infinity, you need to specify what kind? There's the cardinality of sets a la Cantor, used in limits, induction (possibly transfinite), or, perhaps the extended real line where + and - infinity are the very "real" points stuck on each end of the normal real line (with adjustments to the algebra, topology, and order to accomodate them).

37. Bad example - it's well-orderdness (the ordinals) used in induction.

A better example is to take an infinite set (say the integers), and then take its power set (the set of all of its subsets) and show that you can't pair the elements up from each set - so they do not have the same size (or "cardinality"). In fact the power set is bigger.

This process can obviously be repeated giving an infinite hirarchy of infinities.

38. This comment has been removed by the author.

39. This is maybe OT, but in "Extremely loud & incredibly close" J Safran Froer basically writes
1-0.7777... = 0.3333....

Maybe if someone had taught him as a child that 1=0.999... he wouldn't have made such a stupid mistake.

[Yes, I know it depends on the definition. However, there is such a thing as a standard definition of the real numbers].

40. This post is probably dead, but my 2 cents nonetheless:

0.9999... is short hand for an algorithm to yields the series

0.9, 0.99, 0.999,...

It can then be shown that this series converges to 1 in the usual metric topology on the real line.

So the a more complete statement would say:

The series defined by 0.999... converges to 1 on the real line in the topology generated by the standard distance metric.

As a counter example if you use the topology generated by equivalence classes of the Hamel basis over the field of rational numbers for the real line, then the series 0.999... does not converge, let alone to 1.

41. Our posts don't die that quickly...

42. I suppose not, and good thing too!

The other interesting thing is that the topology in the counter example is infinite dimensional, while the usual metric topology is one dimensional. As well the counter example topology is metrizable, and is homoemorphic to the L2 norm topology on the space of all square integrable functions on the field of rationals, the trick being making the appropriate choice of measurable sets on the rational numbers.

43. Yeah, but you need the axiom of choice for your Hamel basis - which just complicates matters...

44. "God created the integers, all else is man's bastardization"

Sounds like someone is an explicit constructionist.

You also need the axiom of choice to define Borel and Lebesgue measures, and without those basic concepts its meaningless to study quantum mechanics, probability theory, most of analysis,...

45. Wikipedia is quite clear on the subject; still 0,9999... does not equal 1.

It's tied up with the concept of infinity. Infinity means *always* a little something will be missing in 0,9999..., no matter how small - the series is unending, or it would not be infinite.
The summing up never arrives at an end - "in the infinite it amounts to 1" has per definitionem no meaning, a simple fact which one tends to conveniently overlook.

The number 1 on the other hand is no series, it has never "departed", so it does not need to arrive anywhere, it just is where it is.

Of course not only "professional" mathematicians will groan at this presentation, but that does not mean they are right. They have taken the concept of infinity and twisted it so it fits with the needs of their (equally twisted) minds, so of course they get the results they desire, and can prove them to their satisfaction, too.

Bee,

"(Infinity is) a concept that seems to be difficult to grasp."

is not quite true - it is *impossible* to grasp, and that's the root of the problem. If you can grasp it, it's finite. *Any* grasping of the infinite is bound to be wrong, and that's that; a fact which guarantees endless discussions, because everyone can prove that the other's opinion is wrong.
Of course one can escape this situation by e.g. "defining" what infinity mathematically is - do you see the absurdity of that?

(... while we're at it, next let's define the indefinable; we already believed more than six impossible things before breakfast ...)

46. So...as the author of the referenced page, here's my take.

Yes, I know about hyperreals and non-standard analysis. In the follow-up posts, I'm very careful to say that I'm discussing the real numbers as they are normally understood.

I do know that .999... equals 1 only in that real number system, but the fact is that most deniers can't argue correctly within that number system, let alone understand the hyperreals. The post and subsequent refutations of counter-arguments are directed at them, not at those in the know about richer number systems.

47. "Of course one can escape this situation by e.g. "defining" what infinity mathematically is - do you see the absurdity of that?"

Defining something and following the rules to reach conclusions!!! NO! NO! NO! - that's just cheating and wrong!!!

Actually, that's mathematics in a nutshell.

48. just wanted to complete a sentence in my previous post:

"... a fact which guarantees endless discussions, because everyone can prove that the other's opinion is wrong"

- and thus (wrongly) infer that one's own opinion must be right ...

49. Hi Ned,

"Bee: (Infinity is) a concept that seems to be difficult to grasp.

Ned: is not quite true - it is *impossible* to grasp, and that's the root of the problem. If you can grasp it, it's finite. *Any* grasping of the infinite is bound to be wrong, and that's that; a fact which guarantees endless discussions, because everyone can prove that the other's opinion is wrong.
Of course one can escape this situation by e.g. "defining" what infinity mathematically is - do you see the absurdity of that?"

I think we are talking past each other. It depends on what you mean with 'grasp'. If you mean with 'grasp' you can picture it in your head, then I would agree that I can't 'grasp' infinity more than I can grasp ten dimensional manifolds or the exterior algebra. What I meant with 'grasp' however is that you can give a well defined meaning to it and learn how to deal with it self-consistently. And unfortunately, many people seem to have difficulties with that 'grasping'.

I don't know what's "absurd" about that. As James said above, that's mathematics which happened to be the topic we were discussing. Best,

B.

50. Bee,

"What I meant with 'grasp' however is that you can give a well defined meaning to it."

Exactly the point - you cannot. If you take infinity and "give a well defined meaning to it", it ceases to be infinity. What you have then is a completely new concept which you just keep on calling infinity, and so delude yourself that it's the same thing as before, in a "well defined mathematical way" at least. You can define it as well as you want, it's still something completely new - not a bad thing in itself, maybe amazingly useful, maybe Fields-Medal worthy - just not infinity anymore.
Call it "Sabine" - that would be more honest, and nice, too (just imagine: "... for mathematically defining the concept of Sabine, basis for the design of the first working wormhole connecting us to another universe").

"and learn how to deal with it self-consistently."

No problem, if one learns some math.

"And unfortunately, many people seem to have difficulties with that 'grasping'."

Maybe they see something you miss - that you have grasped something, but it's not infinity ...

"I don't know what's "absurd" about that."

Trying to define the indefinable is not absurd?

"As James said above, that's mathematics which happened to be the topic we were discussing."

Mathematics, physics, painting, no matter what: if you want arrive at some sort of truth, you have to be honest - if you create a new concept then give it a new name, so there's less chance of confusion.

But mathematicians are human beings, too, who delude themselves and others, start from not thoroughly thought out assumptions, make errors, try to hide them - like all human beings do. The problem arises when they forget that and try to live up to some ideal of a superhuman, pure mathematician who lives only in the highest regions of pure math and therefore can do no wrong - then they find it hard to admit it before themselves and more so before others when they've fucked up, and consequently it becomes difficult for them to correct an error. (Just the same as it's with other human beings, by the way, only of course these aspire to all sorts of other equally impossible ideals.)

51. "What you have then is a completely new concept which you just keep on calling infinity, and so delude yourself that it's the same thing as before, in a "well defined mathematical way" at least."

What exactly was infinity before then? Please define the previous notion. However you define it we can work with it - but you do need to define what you are talking about, be it in maths, philosophy, music, or gardening... otherwise its just hot air

52. Hi,

I was waiting for an ancient Greek to give its opinion... Congrats ned ! You just time-transported your thinking two thousand years ago !
(No offense, I'm just kidding)

Well, the fact is you seem to either know better than anyone what is infinity, and disagree with the common distinction between potential and actual infinite (here there's only a potential infinite concerned in the matter, so if someone tries to "grasp" more than its potentiality, then this someone makes a primitive mistake);
or you've already decided that infinity is an undefinable concept in its potential and actual form (just like any fundamental concept that lies in mathematics' axiomatics or logical rules, like points, lines, sets, etc)...

Accepting a potential infinite is of no difficulty whatsoever (think about Peano's axiomatic), and we've come a long way from Zeno's paradoxes, so if you are unsatisfied with it, it's because you try to grasp it in its whole.
In order to do that, you need to define ordinal numbers, with the concept of set, and it is in the power of mathematics to be able to postulate the existence of an infinite ordinal and build up an entire theory on it.
Maths doesn't care if a concept exists or not, it just tells you what it implies, whether it hurts your "grasping" or not.

If you can't accept that counter-intuitive concepts can extend the realm of science coherently, then I guess all you are left with is counting on your fingers and tell others that the square root of two isn't a number.

Best,
Johann

53. Hi Ned,

“and thus (wrongly) infer that one's own opinion must be right”

When it comes to mathematics what one thinks or what opinion you hold matters for not, as it’s only what you can prove that counts. The only thing open for debate here is the definition of infinity one uses being appropriate (logically) for what one is describing. One definition of infinity is to call it a direction rather than a destination (which seems to be your own contention). The paradox of course is how one can consider a direction without knowing where one is headed, which in turn requires a destination as a reference. So to simply being content with saying that infinity is defined as being without end in itself can be seen as inconsistent.

That is .999... as it’s being considered here is both given a direction and a implied destination which in this instance is 1 . Now with the square root of 2 you can also describe the destination as the place where the proportion found when multiplied by itself will be equal to 2. This destination may be more complex in calculation yet still it exists. The same could be said for PI or any other irrational number, yet it also requires infinity to be used as one of the parameters in defining the quantity of steps taken in the direction to arrive at the destination. So you may be content that one can have a direction without there being a destination yet that doesn’t make it logically consistent to insist there isn’t one.

The .999 in this example describes the steps required to arrive at the destination (route) and infinity describes how many are required to be taken. Infinity is this case is neither the direction nor the destination, yet merely the quantity of steps required to reach it. This may not be a quantity conceivable within everyday experience, yet this is a personal limitation and not one resultant of infinity.

In this context what Cantor proved then is not that some destinations are any further away than others when infinity is evoked, yet rather some types of destinations are more numerous than others, like the irrationals are when compared to the integers. That there is a difference when considering the magnitude of steps in a ordered series and a group whose order cannot be so definitely given. The question then is why? It has been found that the understanding of this lay outside mathematics as now conceived. This is not a failure of mathematics, yet merely an indication that complete understanding lays somewhat beyond its reach. I’ve always liked to think it represents the difference between certainty and possibility with the possible always exceeding the certain.
.
Best,

Phil

54. What it comes down to is this. Yes Ned the game is fixed in a certain way: in that you can make some fairly arbitrary choices of axioms, up to self consistency restraints, a la Godel and Russel.

So you can state that the 0.999... does not limit to 1, but to do that you have to chuck out the axiom of choice, and I think trans-infinite induction. If you give up these axioms, but keep the basic axioms of propositional and existential logic you do get a self consistent set of axioms, but one that is much more limited in what can be proven, in fact it is so limited that most of 20-century mathematics and physics cannot be derived.

Axioms are like the metric system, I cannot say that the axioms of logic are absolutely true, I have to take them as definitions. They have shown themselves to be very strong definitions in their ability to allow us to make valid and useful reasoning and predictions about our universe. If you want absolute truth I suggest studying theology.

55. Aaron,

This is off topic, but you earlier said that AOC was needed to define Lesbesgue measure. I'm too lazy to go back to the (lengthy) Caratheodary construction so perhaps you can point me to the place where it is used.
Incidentally, I am not a constructivist - I love the axiom of choice and all of the absurdities that it implies!

56. Not so much the construction using outer measures, but rather computations done with Lebesgue measures. I think it can be found at the end of Rudin:

The Lebesgue measure of the countable union of disjoint Lebesgue measurable sets is the sum of the measures.

57. No, as I suspected, the AOC is NOT needed for the construction of Lesbesgue measure.

It is, however, needed to "construct" an example of an un-measurable set, as first done by Vitali.

58. "The Lebesgue measure of the countable union of disjoint Lebesgue measurable sets is the sum of the measures."

That's just "countable additivity" which any measure has to have - no AOC there either...

59. Hi Bee,

There is one consolation for those that consider that .999... is not equal to 1, which is to say that when examining a calculated result that is say .99999999999999999999999 or for that matter any recurring series less numerous then actual infinity could be (although very unlikely) resultant of a equation or method of calculation not equal to 1.
That is sort of like simply examining a series of numbers and trying to determine if it represents one that is random or not.

For instance if one didn’t know the method for calculating Pi or what proportion it represents it would prove difficult to distinguish any finite series of digits within it as being non random. A random series by the most modern definition is one that can’t be described with information less quantitative then itself. When you evoke infinity into this consideration you find in essence that without all information this then remains definitively undeterminable. That is like much of modern physics mathematics in many instances one can only decide what is probable, rather then what’s completely determinable. In some respect this represents what Godel meant when he implied mathematics as being incomplete rather then it being wrong.

Best,

Phil

60. Hi Bee,

Just as a qualifier and to make more clear what I mentioned above is to say that at some point the sequence .99999999999 demerges, like for example .9999999999999999999991...

Best,

Phil

61. Sorry I didn't state that clearly. It is required in proving that the subsets that satisfy the outer measure addition condition form a proper sigma algebra.

Good ole wikipedia says to refer to Measure Theory by P. Halmos, 1950, section 11, but I'm pretty sure the development of measure theory in the standard text by Rudin covers the theorem.

62. Not convinced.

Although reluctant to return to the formal proof, I found thi son the web http://planetmath.org/?op=getobj&from=objects&id=11280

I am still not convincd that AOC is needed. Please spell it out - don't be afraid of maths ;-)

63. Just read through it and it seems like quite a cute proof - but no AOC in sight.

64. That link covers construction of the outer measure.

It is in fact Caratheodory's Lemma which presents the obstacle. This lemma is referenced in the preceding link.

http://planetmath.org/encyclopedia/CaratheodorysLemma.html

Unfortunately no proof is supplied

65. Among other things in the lemma, showing the set of Caratheodory measurable sets is not empty will require AOC for sets with infinite or greater cardinality.

66. Opps sorry the whole set and the empty set are Caratheodory measurable, so the set of Caratheodory measurable sets is at least trivially not empty.

67. Checked out your link - and no AOC.

There's nothing left but for me to go and re-learn the full construction which I'm not looking forward to. I used to know it off by heart (more years ago than I care to remember). This may take a while...

You could, of course, have helped me out by pointing out exactly where AOC is used (as I asked you to a few
times...)

68. Well, that was an effort - but good for the soul I guess...

No AOC!!!

69. To all of you AOC sensitive people:
http://abstrusegoose.com/133.

70. It's an old joke - but I think it bears repeating...

The axiom of choice is obviously true, the well ordering principle is obviously false, and who knows about Zorn's lemma?

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