If you're having math problems, I feel bad for you, son...

[tridim]

I got 98.9999... problems but .9999... *is* one."

Vi Hart's New Video: 9.999... reasons that .999... = 1

&feature=youtu.be

[longship]

1. R&K for the first math post I've seen on DU

Not that I see them all, but as a former math teacher, I can't not R&K this.

[wake.up.america]

43. Question for math teacher - Please. At the end of the year I have 100,000 Pesos...

Every month for 12 months I make 12 percent increase. How much did i start out with?

With formula please.

[longship]

44. Well, it doesn't come out even.

The formula to calculate it is:

100,000
----------=25,667.5092945
1.12^12

To the significance of my calculator.

If it was dollars and cents, rounding to the nearest penny:
25,667.51 x 1.12^12 = 100,000.00

If I round after each compounding period, you get a different answer and I am not about to do that one before AM coffee, especially for an Inet post. Just too damn many calculator key presses to solve it, or even to write a little program.

[napoleon_in_rags]

2. I disagree. Is PI a rational number?

I mean, can't we write PI as 314159.../100000... A rational number who's denominator and numerator are INFINITELY long integers?

No, we can't. PI is not a rational number, because integers with infinitely long representations aren't integers. If they were they would not be countable, thus not integers. So rationals that are infinitely long, also uncountable, are a not a proper part of the set Q.

That brings us to the next question: is .999... really a decimal number? No. By the same logic that excludes pi from being a rational number, and the numerator of the "rational pi" above from being an integer, a decimal number with an infinitely long representation is not a number. (because I consider the decimal numbers countable.) If you believe the decimal number are uncountable, fine. But now subtract the set of countable numbers from the uncountable numbers. Which is 1 in? Which is .999... in? How can a number be both countable and uncountable?

You can resolve this by considering .999 repeating to be a computable number... Its a simple algorithm which prints 9's to infinity, taking finite code. So in this sense its countable. But within the context of the computable numbers, is it equal to the program which prints "1" and halts? No.

And my final example: If I gave you two choices: 1) I will give you 1 million dollars today, or 2) The amount of money I give you will converge to 1 million as time approaches infinity, which would you choose?

Good choice, and that's why you agree with me, not the OP.

PEace.

[TalkingDog]

3. Woot! for critical thinking and logic!

[Jim__]

4. A set of numbers is countable if it has the same cardinality as some subset of the natural numbers.

Whether or not a set of numbers is countable has nothing to do with the how members of that set may be represented. For instance, 1/3 is a rational number, and its infinite decimal expansion, 0.33333... is a rational number, even though its exact decimal expansion is infinitely long.

... PI is not a rational number, because integers with infinitely long representations aren't integers. ...

Pi is not a rational number because there do not exist 2 integers, say n and m, such that pi can be written as n/m. Pi is not an algebraic number because there is no non-zero, one variable polynomial with integer (or rational) coefficients that has pi as a root.

... If you believe the decimal number are uncountable, fine. But now subtract the set of countable numbers from the uncountable numbers. Which is 1 in? Which is .999... in? How can a number be both countable and uncountable?

Your statement doesn't actually make sense. It's sets that are countable or uncountable. If you have a set of numbers that contains pi, then pi counts as one member of that set. The set of real numbers are uncountable because they cannot be put into one-to-one correspondence with any subset of the natural numbers. Since 1 and .999... are the same number, any set that contains one of them contains the other; the only difference is representation.

[napoleon_in_rags]

5. "there do not exist 2 integers, say n and m, such that pi can be written as n/m"

Last edited Sun Mar 25, 2012, 05:59 PM - Edit history (1)

What do you mean? I just gave them! 314159.../100000... Two infinitely long integers that when divided equal pi. But you don't like. Why? I can give a way of counting that will include 314159... (it would count all the finite algorithms that count infinitely long integers) so prove its countable. Yet you know this is wrong, you know Z plus all integers of infinite length would probably have the same cardinality as R. And I don't blame you for not liking that.

Ignore my comments on .999 being part of any set (it can be argued both ways) and look at the notion of countability itself. You right, its about creating a 1-to-1 map between the naturals and some set of numbers. But I say look deeper at that idea of a map. I say you need to be able to specify, in finite time, each number of the new set for each natural. Because if a number can't be specified in finite time, its really questionable as to whether its really been specified at all.

You may be thinking about the number e for instance. It is defined through an infinite series, with the dreaded '...' at the end. But what does the "..." mean? It means "you see the pattern." and because of that, you can write it in sigma notation as an infinite sum, using finite information. You can specify it precisely in finite time, same with pi. However if I told you had a number like e, except that the denominator of each part of the infinite sum was TOTALLY RANDOM, it could not be expressed with finite information, because by definition of randomness, there is no simple pattern which can be reduced down to a finite algorithm.

So that's my point: For x to BE a countable number it has to have a finite specification in the family of numbers it belongs. So 314159... is not a an integer, 314159.../100000... is not a rational and 3.14159... is not a decimal. The "..." is a social agreement, saying "you see the pattern" but neither of these numbers really says for sure what the pattern is. The form of e here in sigma notation IS precisely specified with finite information:
http://mathworld.wolfram.com/e.html
with no "..." at the end. And you can produce the same kind of precise, finite definition for pi, and you can produce it for 0.9999 but as soon as you do that, you are giving more information than you had with the "..." hand wavy notation.

The issue isn't about whether 0.999... is equal to 1, the issue is that 0.999... was never a precisely specified number in the first place, anymore than 0.134250978... (you know the rest) is. So the question is, what finite form are you going to reduce it to for me? If you assumed 1/3 = 0.333... and say therefore 0.999... = 3*1/3, of course its equal to 1. If you are defining it by being infinitesimally off from 1, than its infinitesimally off from 1.

[Jim__]

7. "... you know Z plus all integers of infinite length would probably have the same cardinality as R."

... I can give a way of counting that will include 314159... (it would count all the finite algorithms that count infinitely long integers) so prove its countable. Yet you know this is wrong, you know Z plus all integers of infinite length would probably have the same cardinality as R. ...

That statement is nonsense. It shows a lack of understanding of elementary set theory.

[Hawkowl]

8. +!

[napoleon_in_rags]

9. I will make it simpler for you.

My point is that integers of infinite length aren't allowed, and we should look at why. For instance, in the video, she says that for any given integer n, there is a larger integer n+1. But lets suppose instead, my integer is defined as

m = 1+1+1+... (to infinity)

can you add 1 to this infinitely long integer and make it larger? Its value is already infinite. Can you get a larger number with infinity plus 1? No.

Now consider the real number line. between any two numbers A and B with A < B, there are infinite more real numbers in the interval between them. And furthermore, I can choose a new number C which is 9/10ths of the way between A and B, but still less than B. So for example, if A = 0 and B = 1, I can choose the point C = 0.9 which satisfies that condition. And looking at the new interval between 0.9 and 1, I can choose a new point, 0.99 which is 9/10ths of the way between those. And yet another point between those, 0.999. Generally, I realize, that I can just stick another '9' on the end of the decimal to attain the decimal representation of the next number which satisfies that condition starting at 0 and 1. But what about the number 0.999... (repeating) can I stick another '9' on the end of it to find the next point? No. Just like I can't add 1 or 1+1+1... repeating to get a larger number. So does the common sense idea of there always being an interval between two non-equal points really make sense for this kind of number? No.

The point is that the rules change when you get to this numbers with an infinite expression, like 1+1+1... or 0.999... Specifically, they are ill defined. For a person who believes 1/3 = 0.333... of course .999 repeating equals 1. But why didn't you just say 3*1/3 in the first place?

[one_true_leroy]

10. "God created the integers"

And when "God" created the integers, she said they were of finite length. There are no integers of infinite length, and there is no "look[ing] at why," as you implore. The very careful definitions of integers, rationals, etc., cannot be muddled ( as you suggest) without crumbling the edifice upon which math is built.

<You've made an argument for 1/3.> {sorry- incorrect attribution} It is NOT an integer, but a ratio of integers, and its repeating nature places it in the rational numbers.

If you are to continue a MATHEMATICAL argument, you must accept the mathematical definition that there exist NO integers of infinite length. Now, there is no limit to the length of an integer, but this is not the same as admitting infinite length.

In other words, if you pick a number, and it is an integer, it WILL be finite in length. If it is not finite, it may be cyclic (rational) or not (irrational), but not an integer. You can dig around and find a proof that every ratio of ANY pair of integers will have a terminating or cyclic decimal representation.

If you are unwilling to accept these definitions, you are not making a MATHEMATICAL argument, but merely a semantic one. Math is the elimination of semantics in pursuit of truths.

If you are sincere, read Howard Eves Foundations and Fundamental Concepts of Mathematics. Best introduction to higher math I've ever read. If you insist on being ornery for ornery's sake, read something by Bukowski.

Peace and waffles,
leroy
On edit: it wasn't you that mentioned 1/3, and that post argued correctly about the decimal expansion.
Double edit: be more careful in math arguments. It is a VERY unambiguous field, especially at this level.

[napoleon_in_rags]

11. This is a teachable moment.

First things first.

If you are sincere, read Howard Eves Foundations and Fundamental Concepts of Mathematics.

I'm always looking for new truth and sincerely appreciate your attempts to share it with me, its going on me Amazon wish list now.

That said, let's start at the end.

It is a VERY unambiguous field, especially at this level.

There's nothing that can create more confusion than when people believe meanings are not ambiguous when they actually are. How much conflict is rooted in failures to communicate? Math is, after all, a language.

Let's keep it simple. I'm a working man, a simple man. And I have my own down home definition of the decimal numbers, a vernacular if you will, which I assure you most common folks like me share. Let's say you have a number n =
3.xxx....
where the x's are unknown, and the ... means repeating. What do I know about this number? Well, in the common down home sense of decimal numbers, I can tell you that n is greater than or equal to 3, but less than 4. (equal if all the x's are zeros) So the first digit reveals a certain interval: 3, as well as every number up to but not including 4 is in that interval. If next we find out n = 3.1xxx.... that specifies an interval to us that includes 3.1, and all the numbers between 3.1 and 3.2, but not including 3.2.

So based on this common interpretation of the decimal numbers, what somebody like me hears when one of these pointy hat people comes down from on high and tells us that 0.999... = 1, what they are in effect saying is that that there is exists a number in the interval that is defined as being less than 1, which is equal to 1. And that is just plain false on its surface. Its like saying there is a cat that is a dog, they clearly have a different definition of the word "cat" than us, which includes chihuahuas or something.

So what is this different definition of the decimal numbers where 0.999 can equal 1? Thinking about it and turning it over in my head, I see that not only must 0.999... = 1, but 1.999... must equal 2, and 3.1999... must equal 3.2, etc. In fact every decimal number with a finite amount of digits must have an alternative form with its last digit decremented, being trailed by infinite 9's. Okay. So its a system where any number representable with finite digits has two possible representations, where in our common down-home system, each number only has 1 representation. Okay. its seemingly consistent, and interesting. We'll call it the elite decimal system, where ours is called the common decimal system.

But something strikes me as odd, in having our common down home decimal system dismissed in this way. Specifically, its the idea that even if its discarded, I can create a new one: Consider two numbers, say - e and pi. I can define an interval which includes all real numbers which are equal to or greater than e but less than pi. I can write it ((e, pi). Then, because the real numbers are closed under division, there will be a midpoint c halfway between, defining a new interval ((c, pi). I can refer to e as ((e, pi).0 and this new midpoint as ((e, pi).1 Then I can define a new midpoint for that new upper half, and designate it as ((e, pi).11 and a new midpoint in that even newer interval becomes ((e, pi).111 etc. (binary numbering, basically) Eventually, I see I can keep this process going, Zeno's paradox style, until I get ((e, pi).111.... (repeating). Now we got us a funny number, by definition less than pi, but infinitely close to pi. And what my intuition tells me, (and my intuition is pretty durn good) is that all the arguments for 0.999... = 1 apply equally to the idea that this new number ((e, pi).111... = pi, when by definition is does not... And furthermore these arguments can be extended, and reduced to the argument that any interval on the real line defined as containing all numbers less than c also contain c, and that, is what we call in these parts, a problem.

So, sorry to be Bukowskiesque, but I see a real cultural problem. When 99% of people see the decimal system the way I do, where the decimal number 3.x is less than 4, but an elite pointy hat crowd comes down with a new definition of the decimals where it may also equal 4 without explaining their assumptions, the concept of math as a cultural phenomenon to enlighten us all is lost. We have to at least recognize that there ARE differences between the 99% concept of the decimals and the 1% concept. (and thus between .99... and 1, such as it is) And use that as a springboard to spread the joy of mathematics, making sure its not just locked up for a few.

PEace

[one_true_leroy]

13. A few points...

You're flirting around with the definition if a limit, but making incorrect arguments. Zeno's paradox is no longer a paradox. The idea of a limit (which is central to analysis, the theoretical basis for calculus and the study of functions) is that if you can get as close as you want to a value- and then still get closer- then the sequence you are building actually IS that number. The number you constructed IS pi, just as .99999... IS 1. I remember ripping my hair out about Taylor series and infinite series. Then came Fourier series, a beast conceived in Hell (or at least in thermodynamics). If you want a book to focus on infinity, look for The Pleasures of Pi,e by YEO Adrian. It is very germane to this discussion.

Infinity is a strange beast. The Greeks, because of Zeno, were very suspicious of infinity ( and irrational numbers) and it was not until the 19th century that the field of real numbers and infinity were well enough elucidated to build a rigorous foundation for calculus.

I hear you when you talk of simple and down home understanding, which works 99% of the time for 90% of the people. However, it's precisely in those 1% moments in those 10% of minds that math reveals its beauty and wonder.

Math is a house of cards, and has been since Euclid. At the foundation are very careful, VERY precise definitions. These truly eliminate ambiguity, but must be accepted exactly as stated. The next layer up are the axioms, the 'rules' obeyed by the objects defined. Then from these two all else in that field of math must be built. Every theorem, every 'truth', must reduce to these axioms and definitions. To 'tweak' these definitions for a down home interpretation is to kick over the foundation. While your down home definition may be more satisfying in the short term, we pointy heads (and remember that everyone begins math with a simple approach) will reject those because the pointy head approach has and continues to yield beautiful results, while the simple approach collapses under the rigor needed to approach and understand that beauty.

Now, that's not saying the simpler definitions prevent you from appreciating that beauty, but they are not going to be sufficient to build that beauty.

Get the Eves book. It's of medium rigor, and your writing suggests that you'd be able to follow and hopefully it can convince you better than I can about the need for pointy heads to make math something more than a bunch of operations. Also, do some research on the history of geometry from Euclid to Hilbert ( which Eves covers in his first three chapters). In that history, you should come to appreciate the liberation of math from ambiguity.

[napoleon_in_rags]

15. Yes, I've always had something of a flirtation with limits...

Something of a dance, a desire to cross... Whoops, lost context. Its the Bukowski in me.

I was delighted to learn about limits in calculus in college, they answered some deep questions I had since childhood. But I disagree fundamentally that any argument I've made is incorrect. The limit a thing approaches is not thing thing approached. The classical examples of limits show a number where, at a certain value it is divided by zero, and is thus undefined at that point, but through limits, a value can be approximated through smaller and smaller intervals. But ultimately, we can't throw out the fundamental fact, the number is STILL undefined when divided by zero. The limit is not equal to the value at that point. Everything that comes after the axioms must obey the axioms.

then the sequence you are building actually IS that number

That's where I disagree, because I believe math has a logical basis. If we defined an interval which includes all numbers less than but not equal to C, than C better damn well not be in it, or our system is logically inconsistent. In fact I remember from my college days something called "proof by contradiction" where if we could show a hypothesis leads to a logical contradiction with that starting hypothesis, that hypothesis must be wrong.

But whatever. I see you're a person who cares about math, somebody who is trying to share ideas with me at least through books, and I care about that, I appreciate that. There's a lot of people who don't really have a concept of mathematical beauty. But what I resent is the mathematical purist who has confused rigor with an unnecessary ideological restriction of what the possibilities really are. I took you for a pointy hatter, but your vague reference to real analysis hasn't yet born that out, I must hear far more obscure pedantia to confirm my suspicion. Maybe you are, maybe you aren't. But the bottom line is, I don't care much for any of it.

Especially, infinity, especially the so called "real" numbers. Not too long ago, I read a really good write-up on elliptic curve cryptography. The questions, like many in cryptography, involved not questions about infinite numbers which have solutions in the realm of the Gods, but what real, mortal humans could solve on a real, finite computer. What happens when the system is represented in discrete terms. To me that's the real foundations of math: what us human mortals can know. So in that context, its valid to look at it as information, as a language, as a down home system that down home homo sapiens can represent.

If you're like me, you see mathematical beauty as having a spiritual quality, a language which brings you closer to God. I respect and value people with that insight. But within that context, please meditate on this term I heard some spiritual seeker in India make:

The crowning jewel of my wisdom is the knowledge of my own ignorance.

And ask this question: Could it be that the best math includes the mathematician in the equations?

PEace

[Joseph8th]

22. Had to jump in on this thread...

Last edited Thu Apr 5, 2012, 11:53 AM - Edit history (1)

... as a mathematician. Napoleon_in_rags wants to have a spiritual quality, and that's fine, but it's wrong, Napoleon, to resent eggheads for spending the time, hard work and money to tackle the difficult subjects mathematical. If there is one field of study where the spirit of creativity still reigns supremum, it's pure mathematics. But what isn't acceptable... or rather it's asking way too much... is redefining basic math concepts and objects, then expecting everybody to toss everything they've learned and start over from scratch with someone's half-baked idea. If that person's goal is to somehow justify the existence of their deity by redefining mathematical concepts, then, yeah... most of us mathematicians are going to slam our minds closed in your face.

On to the actual math: the limit of a sequence of real numbers, if it exists, is always a real number. That said, the limit of a sequence of numbers, each of which is in a subset A of R, might not be in the subset A, but will always be in R. An example is the limit of 1/k as k --> infty for k a natural number (positive integer not zero). Every term is 1/k in (0,1], an interval that does not include zero. But the lim(1/k) = 0 as k --> infty, since as k grows infinitely large, 1/k grows approaches zero.

The expansion of Euler's number (e) that you mentioned actually extends this idea to the concept of a series, which the sum of an infinite sequence of numbers. For instance, we might sum 1/k^2 for k=1 to infty. If we fix k, then we call one such sum a partial sum of the series, denoted S_k, and we can write it using an ellipsis (rather than sigma notation) using the familiar S_n = 1/1^2 + 1/2^2 + ... + 1/k^2. Then we can look at whether or not S_k < S_k+1 or not, for each k, and use some theorems, to discover whether or not the sequence of S_n's converges in the reals to some some finite number. If so (roughly), then we can look for the limit of the sequence of sums as k --> infty. This turns out to be profoundly powerful, especially when working with power series, which generalize the notion of polynomials.

Well... this is how e was discovered by Bernoulli as he tried to evaluate the limit of the series (1 + 1/n)^n as n --> infty. Then given this series definition of e, we can define the function log as the inverse of the series that defines e, such that e^log(x) = x, x>0. Or logarithm can be defined 1st as an integral (infinitesimal Riemann sums, related concept for increasingly fine partitions of an interval), and then e can be got from that. The point here is that even though e is irrational (and transcendental), it can still have a series expansion (of infinitely-many terms) anyway. Sorry there's no LaTeX at DU, but...

pi = 4 * \sum_{k=0}^{\infty} (-1)^k / (2k + 1)

is the same number as

pi = circumference/diameter, for any circle.

Both of these equations are precise. That is, they express pi completely as transcendental irrationals, the way that 1/3 is a precise rational number, but .3333... is a decimal expansion is a floating point number with variable precision. A computer can't deal with 1/3, and always does math with rationals using FP arithmetic, and so has limited precision (depending on the language and computer, of about 10^30). This is always true of decimal expansions: the ellipsis at the end indicate that the representation is imprecise. If they follow a repeating pattern, convention says repeat the pattern. If they do not follow a repeating pattern, convention says do not repeat the pattern. That said, in pure math we rarely ever use ellipsis notation and in analysis don't truck in decimal expansions, at all. Decimal expansions are the difference between applied and pure math, in many ways. Even in number theory (source of elliptical curve theory) we are rarely ever interested in dealing with numerals. Letters and symbols keep things nice and general.

In short: arithmetic is not mathematics, and vice versa. So when we write 0.999... it's understood that this is not a precise expression of any number. But it's a number we can write precisely by using our heads to expand decimals to infinity to find the limit of the decimal expansion (sim. as we did with partial sums for e and pi above) as the number of digits --> infty. The precise representation of this number is 1. Really the point is that there's a number of different ways to represent numbers in general, and some numbers in particular. Right? Especially on the real line.

A better way (more 'natural') to real numbers is by continued fractions:

pi = [3;7,15,1,292,1,1,1,2,1,3,1,…] = 3 + 1/(7 + 1/(15 + 1/(292 + 1/(...)))). The sequence of integers in the C.F. representation are apparently random, but nobody's proved it yet.

To get really trippy, yet another way to extend the reals to completion is with the field of p-adic numbers (https://en.wikipedia.org/wiki/P-adic_number). In particular, I draw your attention to the "Constructions" section at Wikpedia, which states:

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000… = 0.999… . The definition of a Cauchy sequence relies on the metric chosen, though, so if we choose a different one, we can construct numbers other than the real numbers.

Ain't math cool?

[Jim__]

12. As simply as it can be put, your statement is in direct contradiction to a Zermelo-Fraenkel axiom.

You know, Zermelo-Fraenkel:

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory such as Russell's paradox. Specifically, ZFC does not allow unrestricted comprehension. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

From your last post:

But lets suppose instead, my integer is defined as

m = 1+1+1+... (to infinity)

can you add 1 to this infinitely long integer and make it larger? Its value is already infinite. Can you get a larger number with infinity plus 1? No.

You don't understand the mathematical concept of infinity. Yes, you can always add 1 to an integer. The Zermelo-Fraenkel Axiom of Infinity:

Let S(x) abbreviate x ? {x}, where x is some set. Then there exists a set X such that the empty set ? is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.

? X ( ? ∊ X ∧ ? y ( y ∊ X => S(y) ∊ X))

[one_true_leroy]

14. Awwwww hell.....

You know it's about to get real when someone brings up AoC/ZFC! I had an old school Serbian professor who would walk out when AoC was mentioned. I can still hear his accent: "well, clearly this argument must be over."

[Joseph8th]

23. HELL yeah! I love it...

... all my math profs made poets and artists look square. The passion it inspires! It's quite invigorating.

[napoleon_in_rags]

16. Ah, my friend. You have forgotten your transfinite cardinals!

The integers are countable, and we can define a countable set as a number where each member would be enumerated within a a finite amount of time.

Yes, you can always add 1 to an integer.

But according to the definition of a countable set, 1+1+1...infinity is not an integer! You can't have integers of infinite length, or else pi, the square root of 2, and the rest would be rationals!

The shape of the argument I was making, was comparing 0.999... with integers of infinite length, which are not allowed for good reason.. That was the basis of my argument that 0.999... is an ill defined number.

[Jim__]

17. Guess again.

The only transfinite number that has anything to do with the integers is ? which is the same as aleph-null which is what we've been talking about all along.

[napoleon_in_rags]

18. So you're saying 1+1+1...infinity is an integer?

Because it would need to be for us to be able to get a larger number by adding one.

All I'm saying is that infinity plus one is not larger than infinity. 1+1+1...infinity has the same value for instance as:
9*10^1 + 9*10^2 + 9*10^3....9*10^infinity. (an infinitely long string of 9's) Its all the same infinity.

So when I do
9*10^-1 + 9*10^-2 +9*10^-3...9*10^-infinity (an infinitely long string of 9's with a decimal point at the beginning) its equally poorly defined.

In the video, she points out that there is not number which is immediately less than any real number, because any number you choose which is less than c, must have infinite numbers between it and c due to the nature of the reals. Therefore, because .999... is a chosen number, it must be equal to c, or there must be a number between 0.999... and 1, which there clearly isn't. But what I argued above is that for any interval which is defined as containing all numbers less c, we can produce a .999... -like number which is clearly within the interval but by the same arguments must be equal to c, creating a contradiction. (and if you don't see a problem with that, let me tell you about my friend Charlie... he is TALL. So tall, we took him to a party with a lot of people, and everybody there was shorter than Charlie. Including Charlie himself! Now that's tall. )

But the real problem I see is that there is this curse of Babel with mathematics. For laughs, check out this, topic #4:
http://www.cracked.com/blog/6-innocent-sounding-topics-that-are-guaranteed-flame-wars/
Why is that? Its because we don't agree on what things mean. A lot of these things less defined than we like. My two cents is that we should admit human limitations, we should admit that at the end of the day, any number is going to be approximated to a human readable form. In that context, whatever calculations you do with with .999... are going to yield a result exactly the same as if you used 1, after being rounded down. rounding .999... to any degree of finite precision will give you 1. However, using the floor function instead of round will give you what I'm talking about: The largest number that is less than 1 within any arbitrary degree of precision you wish. That's useful too.

Curse of babel dispelled.

[Jim__]

19. The Axiom of Infinity says that 1 + 1 + 1 + 1 ... is an integer.

Do you understand what that axiom is saying? Do you see how you build up the integers? If you understand that then you understand that your question about adding 1 to infinity is not relevant to the process of repeatedly adding 1 to an integer.

[napoleon_in_rags]

20. Yeah, it guarantees the size N is infinite, not that any number in N is infinite.

So you can construct the natural numbers with 1, and with the successor function S(x) = x+1, so that calling the successor function recursively, with 1 as an input, any FINITE amount of times produces any natural number. For instance:

5 = S(S(S(S(1))))

But that being done a FINITE amount of times is critical, that's what makes the set countable. If you enumerate the naturals, every natural number will be enumerated in a finite amount of time, even though the set itself is infinite. So the number

2135307957079005346450789 is a natural, because it will be enumerated after 2135307957079005346450789 steps...A large number, but still finite. That's the quality they all have, even though the number of them is infinite. Its an infinite amount of finite sized things.

Now on the other hand, consider counting the decimals. Would could start with the range -1 to 1 with one decimal place of precision, so our first 21 numbers are -1.0, -0.9, -0.8, ... 0.9, 1.0 then our next 201 numbers would be -2 to 2 with two decimals, -2.00, -1.99, -1.98, ... 1.99, 2.00, then -3 to 3 with 3 decimals, and so on, so that any decimal number with a finite integer part and finite decimal part would be counted within a finite amount of time. So when does pi, or 0.333... get counted under this system? Never. They both have infinite decimal places, so our counting system never enumerates them in finite time. They are uncountable in this system, which is to say that don't have a finite definition. They go on forever.

However if we express 0.333... as a 1/3, than it is countable, as a rational. And if we express pi as a computer program that enumerates all its digits, its countable, by mapping natural numbers to to the binary of representations of all such computer programs. So generally when we say a number is countable, we mean its countable in some system of numeric representation, which is the same as saying it's position on the number line or complex plane can be specified within a finite amount of time. It can be described precisely with a finite amount of information.

So what about these other numbers which are truly uncountable? Not countable in any system? Well, thinking of them in decimal form, they would have a decimal part that goes on infinitely, but with no pattern at all like 0.333.. or that can be described with a computer program, like pi. They would be totally random numbers, requiring infinite information to express them. Because they have no finite definition, I could never specify the location of such a number on the number line to another person, because doing so would take an infinite amount of time.

So the specific value of any particular uncountable number must remain ill defined in the language of math.

Once you see that, you can see that when you take a number like 1/3 from the system of numeric representation where it is well defined and try to express it in a system of representation where it requires an infinite definition like the decimals, and thus becomes uncountable in that system its also ill defined within that system. Just like when you try to express pi as a rational (or even a decimal) its poorly defined. Its like e, you can express it as an infinite sum of rationals, but that doesn't make it a rational, just like an infinite sum of integers isn't an integer. Those infinite sums are just a recipe for good approximations within a more limited system of numeric representation, just like 0.333... is a recipe for good approximation within a system too limited to express the actual number 1/3. The actual number is out there, transcendent of our little system, with a finite definition only in a higher system of representation. Without discussing that finite definition, we have no basis for argument. The number we are arguing over is ill defined.

[Jim__]

21. See post #4.

[Joseph8th]

24. Nicely said... and...

... one might add, though, that transcendental irrationals are only ill defined when expressed as decimal expansions. Pi = C/d is completely precise expression of the abstract number Pi, but because it's irrational, no numerical expression is ever precise.

There's only a few irrational numbers out of an infinitude that have their own names. e & pi & phi are about it. The rest are things like the roots of primes and the like. Each is precise, though, in operation, since they are defined via operation.

That is, we should more properly write pi := C/d where the ":=" indicates definition of a map pi: +R --> R, and C and d are empirically measured quantities, so that's a separate issue of precision. Then, for instance, sqrt(pi) or 2*pi - 3 are both also precise irrational numbers, since we can get pi = (sqrt(pi))^2 precisely by operations.

Of course, we can always approximate pi by inscribing a circle of radius r inside a square, then throwing darts at random, and computing the ratio of darts in the circle only to the number in the square total, which turns out to be pi/r. Every way you cut it, though, the continued fraction or series expansion expression of an irrational number will be the most both precise and useful definitions, since they do not rely on empirical measurement, but were derived via other maths.

[napoleon_in_rags]

26. Now there's some interesting stuff.

I didn't know about the darts... Yeah, I agree with all you're saying. My thing is that a number has to have a finite definition somewhere, and those systems of representation where a number has a precise and finite definition is where it belongs. So 1/3 belongs in the rationals, but not what I call the decimals, which exclude 0.333...

Here's what you say that really interests me though:

C and d are empirically measured quantities, so that's a separate issue of precision.

and

since they do not rely on empirical measurement, but were derived via other maths

Its striking me how odd it is that these two things are conceptually separate. We both share the sense the pi is a well defined number, definable through an infinite series. But we both know that empirical measurement is limited, and pi can never be measured beyond a certain number. So pi has this divine transcendent existence, but where is that? Where does it exist? Far more bothersome to me is the idea that the exclusively uncountable numbers (can be thought of as decimals with infinite random decimals, so they can't be specified in finite time in any system.) are also thought of as real, even though none of them can ever be represented completely in this world. So belief in them is faith based... No let me rephrase that, we can prove their existence plural. But rather the belief in any one of them can never be proven by construction, in fact I can't even think of a logical way to prove that a SINGLE given such number exists that feels right to me. They always exist in collections.

Crazy as it sounds, I'm starting to feel like the real numbers don't make much sense.

I'm thinking back. Why do we have them? We live in a universe where if we have 4 apples are in a basket and you add 3, then you have 7 in the basket, so 3+4 = 7. If we lived in a universe where the same operation yielded 8 apples, it would be a different universe, where 3+4 = 8. So our math comes from our universe. The argument for the reals, as it would be made in Newton's time, was that if a particle of light were moving across space, it must be moving in a smooth continuous line not skipping to the next discrete point, but crossing all points, including those that would be specified by uncountable reals. But we now know that doesn't happen, the light particle doesn't even have a location until a measurement is made and collapses it down to a point. And because the precision of that measurement is limited, its necessarily discrete and finite. But before that, it was just probability wave. (Sayeth quantum mechanics which I make no claim to fully understand at all.) But the point is, the universe doesn't even seem to give a damn about the uncountable numbers at its fundamental level, why should we?

I do know quantum mechanics is intimately related with information theory, and I have a childishly simple idea of what information is that has served me well: if you have a probability space of what might be, information is a function that collapses it down to a smaller space of what is, or more generally makes a new space with a smaller information entropy than the first space. So if that's how QM says the foundations of the universe are, probability spaces undifferentiated until measured, I wonder if you couldn't construct math from foundations based on the same principles. You could just say the uncountable numbers only exist as undifferentiated probability space until somebody actually provides a precise, finite definition of the number they are talking about. It would be so nice to get rid of the two infinities and just have one. Such a math would include the mathematician as observer, acting to collapse the spaces down through pieces of information. Pi and 0.999... might exist more as theories, which predict the finite outcomes of measurements rather than being thought of as "existing" with infinite definitions...

Anyway, I'm just sort of drinking my coffee and dreaming out loud. Sorry to write an essay to you, but you got me thinking...

[Joseph8th]

27. Wellll....

... "So pi has this divine transcendent existence, but where is that? Where does it exist?"

That's not at all what I'm saying even sort of. That's why the distinction between empirical and logical is necessary, and is reflected in math depts everywhere: applied vs pure math. But there's no such thing as the "decimal numbers", as you put it. That's one way to represent a number. There are ways to represent every number, but the most convenient is set theory via countable sets with algebraic extensions.

What I was saying is that we can construct a circle, and measure it's circumference and diameter and so get an approximation of pi, or throw darts and get an approximation, and both of these is empirical (so it exists 'in nature' not as a real thing, but as a property of relationships). That's one way to go about it. But we can consider a pure circle, as well (not drawn imperfectly or susceptible to imperfect measurement) and by that alone mathematically (logically) deduce pi. It is exactly precise, because it is never evaluated.

To get to pi itself, just assume it. Then any bijective function you plug pi into will treat it as a unique point on the real number line, somewhere sandwiched between (always) an infinite number of other irrationals and an infinite number of rationals. If you decided to play Zeno's game, you can play it forever with any point in the reals, including the point pi, by just assuming as much (since one point is the same as any other).

So there's not just nothing special about pi, there's infinitely nothing special about pi. Like I said in another post, there's only a couple named irrational numbers: pi, e, and phi -- and at least two of these have ugly spiritual mumbo-jumbo attached to them by believer types, going back to Pythagoras. The only reason they have names is that they're important to geometers (and from there to physicists, who use lots of geometry) because they appear in nature. So do other relationships that have no names, but are maybe rational numbers. Is C god-magic? What about these two numbers, which I just named for the first time, each of which is equally irrational and transcendent:

Plurp = pi + phi + e
Blurp = pi * phi * e

They have inverses! Plurp - Plurp = 0 and Blurp/Blurp = 1! That's just a few. We could permute these three elements in linear combinations with various operations and bijective funtions to get to any other point in the real number line. So pi's not special. Just named.

If the reals don't make sense, that's probably a good sign. They don't make sense, because as you mentioned we like smooth curves. The deal is Euclid (the ancient Greek geometry) didn't define point, line or plane. These are called undefined objects. We get to just assume them, we name them and label them, then we manipulate them in various ways, these things we think of as numbers.

Not just for science. Lot's of times just for the sheer beauty of it. Number Theory is called the "Queen of Mathematics", and there is no Nobel Prize for us. Fame and glory, baby. Fame and glory.

-- edit --
I take it back. The exponential map (e) is special b/c it's fundamental to both trig and calculus and a host of other areas. For instance your most basic differential equation:

y' = -y

To get the general solution y(t) do

dy/y = -1dt

then integrate both sides wrt to time t

log y = -t + c

We want y(t) so apply exp function to both sides to get

y(t) = e^{-t}.

The solutions y(t) --> 0 exponentially fast as t --> infinity.

[Joseph8th]

28. But then pi's special in its relationship...

... with e, too!

Euler's Identity (I know, he had lots of things named after him ... fame and glory, baby) tell us if we're working with complex numbers (x,y) in RxR, with handy notation x + i*y = z is a complex number in C (the field of complex #s), where i = sqrt(-1) is the imaginary number, that we can derive beautiful results from the utterly beguiling equation:

e^{i*pi} + 1 = 0

Wikipedia (started by mathematicians) puts it nicely:

Euler's identity is considered by many to be remarkable for its mathematical beauty. These three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

The number 0, the additive identity.
The number 1, the multiplicative identity.
The number ?, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics (? = 3.14159265...)
The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828...). Both ? and e are transcendental numbers.
The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.

[tama]

30. Euler's identity

Feynmann said that quantum physics is about understanding Euler's identity, or something in that direction.

In magic math the elements of the identity pi, e(psilon), n and i(maginary) spell 'pieni', which is Finnish for 'small'. .

There is also very deep and mysterious relation between p(rime) and i(maginary), as the non-trivial zeros of Riemanns zeta-function show.

[napoleon_in_rags]

32. Just answer me one question Joseph8th.

Forget about pi for now, forget about everything. Lets just talk about the natural numbers. They can be defined as a set that contains 1, and contains n+1 for every n in N. So they are defined inductively.

Now I think I argued briefly about whether there was an infinitely long number in the natural numbers with somebody above. I argued that there was not. My proof (in my head) looked like this:

a) 1 is a finite natural number.
b) 1 added on to any finite natural number is also finite.

Why? By contradiction. Suppose it is not the case that 1 added on to any finite number was finite. Than it must be the case that there is some finite natural number which when one is added to it, becomes infinite. By which I mean there is some number with a finite amount of digits that attains an infinite number of digits when one is added to it. Absurd.

Therefore the set of natural numbers doesn't include a natural number with infinite length.

Now I will argue the opposite is true, and an infinite number is in the naturals. A one-to-one mapping between any set of natural numbers and a single natural number can be defined as follows:

To convert a set of natural numbers to a single natural number, take the sum of 2^n for each number n in the set. So {1,2,3} = 2^1+2^2+2^3 = 14.

To invert that and construct the set for any natural number m, get the first number s1 by taking floor(log2(m)), then subtract s1 from m to get m2, take floor(log2(m2)) to get the next number, repeat until down to 1. So for 14, floor(log2(14)) = 3. 2^3 = 8. 14-8 =6. floor(log2(6)) = 2 2^2 = 4. 6-4 = 2. floor(log2(2)) = 1. Done. {1,2,3} reconstructed from a natural number by a recursive function.

That defines a 1-to-1 mapping between the naturals and all the sets of the naturals. So by induction, for every n+1 starting at 1, the number representing the set of all numbers up to and including n+1 (call it the set number) can be gotten by adding 2^(n+1) to the set number that corresponded to n. So clearly, for every set of natural numbers, there is a set number that is also a natural number. So because the set of ALL natural numbers is itself a set of natural numbers, there must be a set number for that too, also contained in n. Clearly, the length of this number is infinite, so there must be an infinitely long number in n.

Which of these two arguments is right, the first stating there is no infinite number in N, or the second stating that there is?

Just curious what you think, I bumped into this in my thoughts and its tripping me out.

[tama]

33. You're on an interesting track

Consider also the fundamental theorem of arithmetics: "any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers." http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

Matti Pitkänen has developed the notion of infinite primes:
http://tgd.wippiespace.com/public_html/pdfpool/infpc.pdf

Second chapter of the paper begins with:

2 Infinite primes, integers, and rationals

The definition of the infinite integers and rationals is a straightforward procedure and structurally similar to a repeated second quantization of a super-symmetric quantum field theory but including also the number theoretic counterparts of bound states.

2.1 The first level of hierarchy

In the following the concept of infinite prime is developed gradually by stepwise procedure rather than giving directly the basic definitions. The hope is that the development of the concept in the same manner as it actually occurred would make it easier to understand it.

Step 1
One could try to define infinite primes P by starting from the basic idea in the proof of Euclid for the existence of infinite number of primes. Take the product of all finite primes and add 1 to get a new prime:
P =1+X ,

[napoleon_in_rags]

34. You're awesome Tama.

I hope you know that. I've been really busy lately, but that and linked to document has been occupying my mind since I read it, like a week ago. I chose my set number as I did because it has the same amount of information as the entire set. I've been looking a lot at information theory, how it could possibly be a foundation for math. Really sort of rolling my own math, and having a lot of fun with it.

One thing about the math I'm making is every mathematical concept is tied to the amount information needed to express it. Infinite objects, like e and numbers that go on forever, exist as measurable potentials, with every measurement made on them as finite, though you make more and more accurate measurements involving more and more information: So the third measure of pi might be 3.14, the 4th 3.141, the 5th 3.1415, etc.

So I got to thinking, my system can't have an infinite number except as a potential field, so what would the measurements on an infinite prime be? Well, if you write the set numbers I described above in binary, for all the natural numbers up to n (with n increasing at each step) you get, and the naturals including zero you get:
11 =2 = {0,1}
111 = 7 = {0,1,2}
1111 = 15 = {0,1,2,3}
and so on.
So what might measurements on a certain infinite prime look like? Well, they'd be the best approximation of the number within the information constraints, and so that all the information enumerated at a point was part of the larger definition. (like with pi example above) They would have a quality of there being a prime, greater than all the finite numbers so far.

So anyway there are these primes called the Mersenne primes, which when written in binary are a long string of 1's, just like those set numbers. If you interpret these as set numbers, and calculate the sets, you get the set of all natural numbers up to but not including some prime, which plays the role of the "infinite prime" for that finite set of numbers. So if there are infinite Mersenne primes, each long set of binary one's that corresponds to the next Mersenne prime could be seen as a measurement on that infinite prime.

That thought crossed my head, and I liked it. Its like the "artists interpretation" in some old school book with painted pictures of what dinosaurs might have looked like:

You saw it when you were young, back in the days when Boards of Canada were beings streamed into our subconscious minds by an alien satellite: Its not what the author was talking about, but its the best approximation I can fit into my limited information space at this time, and with my next measure I will be closer still. We are always getting closer.

PEace

[tama]

35. Mersenne primes

Last edited Sun Apr 15, 2012, 06:30 AM - Edit history (1)

are extremely essential in Matti's theory. His latest blogging has a very good feel to it: http://matpitka.blogspot.com/

PS: I share the feeling of being a state of "finite measurement resolution" containing holographic information from larger whole. Very finite and small, especially when trying to understand Matti's theory and physics and nature. But it's also nice to accept some level of holographic comprehension, how ever limited, as a foundation for rudimentary self-confidence.

[napoleon_in_rags]

36. God is Alive, Magic is Afoot.

Name of a good poem. But what interests me is a system that can express the magic of life, by expressing the limitations of our own knowledge. I want to formalize the the idea of the Matrix (movie), the idea we shouldn't assume the existence of anything beyond our perceptions, we don't assume the existence of anything we haven't measured or observed. Sort of a neo-constructivism, in math philosophy terms.

What happens when we do that, I am discovering, is you get a "quantum-ish" system, where the act of a measurement, or an observation, or assumption - whatever term you use - changes the system observed in some way. In my system, which is based on a stripped down probability theory, a measurement collapses a probability density function, which existed as pure potential before that event. So the argument in the video in the OP, that there exists an unlimited amount of points between any two points on the real line, isn't true in my system, all that exists is the potential to specify some point in between, an event which becomes increasingly improbable as the gap between the numbers gets smaller. Its us who builds the numbers.

So as you can see, I'm thinking about seeing us as part of the system, as co-creators of the system. Because I think that's how reality really operates. If you go watching birds, you will observe them watching you. We are part of this universe, and that should be reflected in our symbolic reasoning systems.

I will read that blog eagerly. He is in a good space. I have this idea of scaffolding, the idea of using the old math as a scaffold to define a new math. But eventually the scaffold has to be taken down, and any system has to hold based on its own axioms. So in the end you can't define this thing in terms of Hilbert spaces, you have to end up defining Hilbert spaces in terms of this thing. I for one am not a good enough mathematician to do a lot of this, my focus is on a Duke's of Hazzard simple axiom set based on information theory that normal people could get, and seeing how powerful you can make it. If I had to choose between a system that can give a few geniuses deep insight into the universe vs. one that could give everybody moderate insight, as a small d democrat I'm with the latter.

PEace and thanks for sharing Tama, I love all your posts.

[tama]

37. Category theory

Just found this: http://math.ucr.edu/home/baez/treilles/

Some mathematicians suggest that Category theory (http://en.wikipedia.org/wiki/Category_theory) is more fundamental than Set theory and should be taken as new basis of math. Matti has discussed also Category theory but I believe he prefers his recent idea of Quantum Mathematics to the Category theoretical approach. How ever both Matti and John Baez agree that Set theory is not enough to make sense of this world on scientific level, and the idea of Hilbert space math seems to be in the air and coming through many channels.

I also know and accept my mathematical limitations, my skills don't go much further than developing again in Platonic anamnesis figurate numbers (http://en.wikipedia.org/wiki/Figurate_number) long time again when when obsessing about tetrahedra and bipyramid made of two tetrahedra.

[napoleon_in_rags]

39. I'm just incredibly glad to hear these people seeing the holes in set theory.

There is this psychological type who satisfied with the potential for something to exist rather than actually doing or seeing the thing. In Meyer's Briggs test, its INTP: The theorist. Somehow that mindset permeated math too much with set theory, to the extent the theoretical objects diverged from observable reality, leading to absurdities like the infinite amount of points between .999... and 1, any one of which can never be specified or measured with a finite measuring device.

Those links are really interesting, I really enjoyed that pdf. Its the approachable writing style. Love it.

My thing has been to see if I can draw up working foundations with information theory, which means creating a lean mean version of probability theory from which everything can be derived. On reading that paper, I am pleased: The system I came up with makes all functions invertible, which was discouraging before I read that. What I'm doing that seems unique is really based on enshrining uncertainty at a fundamental level. The observer must be an integral part of the system. I'm also looking at the idea of independent random variables being equivalent in every way to separate dimensions in a space, and going really deeply into that. When you put all that together, you get a math which doesn't try to understand the universe independent of the observer, but rather as totally dependent on the observer: So inquiries into the nature of the universe are also inquiries into the nature of self.

My, that sounds hippy dippy doesn't it? Okay, example: A simplified baby's experience is represented as a two dimensional space, where one dimension is sound information events, one is visual information events. Because the two are independent, a probability density function collapse (aka information event) in sound - "mommy here!" is at first independent from visual events. But the baby's mind is learning, meaning that it is coming to a state where the vast dimensions of independent variables of primal experience are becoming correlated, to produce a space of lesser dimension in his mind. Eventually, the baby correlates the sound event with the visual event of seeing his mother, and maybe even tries to eradicate anxiety of not knowing and keep the dimension low by saying "mommy" himself at some point when she doesn't.

The point is that in a situation where the universe is described by invertible functions, the known and the knower are inseparable, and from one, you can get to the other. I think that's a beautiful thought.

[Dr. Strange]

38. Not quite.

That defines a 1-to-1 mapping between the naturals and all the sets of the naturals. So by induction, for every n+1 starting at 1, the number representing the set of all numbers up to and including n+1 (call it the set number) can be gotten by adding 2^(n+1) to the set number that corresponded to n. So clearly, for every set of natural numbers, there is a set number that is also a natural number. So because the set of ALL natural numbers is itself a set of natural numbers, there must be a set number for that too, also contained in n. Clearly, the length of this number is infinite, so there must be an infinitely long number in n.

You've defined a one-to-one correspondence between the natural numbers and all FINITE subsets of the natural numbers. But if you look at the subsets {2, 3, 5, ...} and {10, 20, 30, ...} (the set of primes and the set of multiples of 10), then they would both correspond to divergent series instead of actual natural numbers.

So, in short, your first argument is right.

However, it is still true that 0.9999... = 1.

And now, for some fun: what is the smallest positive integer that can not be described using fewer than one hundred letters?

[napoleon_in_rags]

40. .999... is not equal to 1.

If there's anything I've become more sure of since participating in this thread, its that fact. You could just as well say:

.999... = 1- (1/infinity), that latter term is also not equal to zero.

To see this, look at the number e defined as a limit:

http://en.wikipedia.org/wiki/E_(mathematical_constant)
And notice that if 1/infinity = zero, than e is equal to 1, which it isn't.
now what would a repeating decimal expansion of 1/infinity be? 0.00... repeating zeros forever, with a 1 on the "end" (which you never get to). subtract that from 1 and you get repeating 9's forever.

The meat of my argument is that a number can't be fully expressed in a number system where it requires infinite representation. The reason I chose those set numbers is because written in binary, a there is a 1 for each number in the set, a zero for each absent in the set. You can go on iterating them forever from the smallest to larger parts, so in this way they are like decimal representations that go on forever as well, just reversed so the decimals are more useful for talking about magnitude. In the big scheme of things, one doesn't make much more sense than the other. Infinite sets are just like infinitely long decimals or infinite integers. They really exist as objects with finite definitions, and that's where they belong, in the context where they can be finitely defined.

[Dr. Strange]

41. The problem is you can't treat infinity like a real number.

That's the whole reason for calculus and analysis: to come up with a rigorous way of dealing with infinity.

You could just as well say:

.999... = 1- (1/infinity), that latter term is also not equal to zero.

I wouldn't be comfortable saying that, because we're putting infinity into the fraction on the right, which is treating it like a real number. That potentially leads to faulty deductions like infinity/infinity = 1, since x/x = 1 for every positive number.
I would however be comfortable writing
lim_(n->infinity) 9/10 + 9/10² + 9/10³ + ... + 9/10ⁿ = lim_(n->infinity) 1 - 1/10ⁿ
which replaces treating infinity like a real number with the idea of limits. It also gives a justification for 0.999... equaling 1.

To see this, look at the number e defined as a limit:

http://en.wikipedia.org/wiki/E_(mathematical_constant)
And notice that if 1/infinity = zero, than e is equal to 1, which it isn't.

True (which is another reason why we can't treat infinity like a number), but notice what's going on in the limit: inside the parentheses, the 1 + 1/n term is approaching 1--however, the exponent is going off to infinity. So, the two parts (the base and the exponent) are battling for control: does the inside go to 1 faster than the exponent goes to infinity?

now what would a repeating decimal expansion of 1/infinity be? 0.00... repeating zeros forever, with a 1 on the "end" (which you never get to). subtract that from 1 and you get repeating 9's forever.

Since we can't treat infinity like a real number, there would be no decimal expansion for 1/infinity.

The meat of my argument is that a number can't be fully expressed in a number system where it requires infinite representation.

It can, but it does require some caution. The construction of the real numbers from the rational numbers is a large part of what mathematical analysis is all about.

The reason I chose those set numbers is because written in binary, a there is a 1 for each number in the set, a zero for each absent in the set. You can go on iterating them forever from the smallest to larger parts, so in this way they are like decimal representations that go on forever as well, just reversed so the decimals are more useful for talking about magnitude. In the big scheme of things, one doesn't make much more sense than the other. Infinite sets are just like infinitely long decimals or infinite integers. They really exist as objects with finite definitions, and that's where they belong, in the context where they can be finitely defined.

Your construction above only works for finite sets, since like I said, the subsets {2, 3, 5, ...} and {10, 20, 30, ...} would not map to any natural number; however, you can adjust your construction in the following way: map a subset {a₁, a₂, a₃, ...} of the natural numbers (where we might as well assume a₁ < a₂ < a₃ < ...) to the number 2^-a₁ + 2^-a₂ + 2^-a₃ + .... This series will converge, and it gives a 1-1 correspondence between the real numbers between 0 and 1 and the set of all subsets of the natural numbers. But it relies on only allowing "infinite" numbers after the decimal point.

[napoleon_in_rags]

42. Agreed, that is the problem, but for both of us.

(Damn, I almost had this post finished, I went to do something else and closed the window. DU needs auto save functionality.)

So basically, yes. .999... is a limit. I write it as
((10^n)-1)/10^n
where ^ is exponent . As each time n increases by 1, another 9 is added to the end. Is it ever equal to 1? No. Roughly, by induction
For numbers positive a and c with a<c, a third number exists b which is 9/10ths of the way between them, which is always less than c.
base case: 9/10 ((10^1)-1)-10^1 < 1.
inductive step: b = (c-a)*(9/10)+a. b < c for all positive a and c with a<c.

Each step of the induction adds another 9 onto the end, and it is never equal to 1. But your saying it gets there at infinity, but it doesn't. The structure of the proof you'd have to make is the same as arguing that there are infinitely long natural numbers. It holds true when the successor function (s(n) = n+1) is called infinitely many times, breaking the inductive step in the proof I made that all naturals are finite above. I don't believe in infinitely long natural numbers, or decimal numbers. I don't think they are proper numbers. But I do think they are both equally as viable.

Your construction above only works for finite sets, since like I said, the subsets {2, 3, 5, ...} and {10, 20, 30, ...} would not map to any natural number; however, you can adjust your construction in the following way: map a subset {a1, a2, a3, ...} of the natural numbers (where we might as well assume a1 < a2 < a3 < ...) to the number 2-a1 + 2-a2 + 2-a3 + .... This series will converge, and it gives a 1-1 correspondence between the real numbers between 0 and 1 and the set of all subsets of the natural numbers. But it relies on only allowing "infinite" numbers after the decimal point.

Oh yeah, that works too. Cool. Also in binary, your numbers are my numbers, but backwards. So the so the set of primes showing the first 3 in mine and yours are:
...101100 = 2^2 + 2^3 + 2^5...
vs.
.001101... = 2^-2 + 2^-3 + 2^-5...

(Where the ellipses in the front is intended to denote an infinitely long natural. )

What I see here though is both of these would require infinite information to denote in full on a computer. They are both just as implausible in this form. What I'm saying is that there is nothing innate about convergence which makes your thing make more sense than my thing, in abstract. Its true that your thing has a decided advantage in terms of magnitudes. For any infinitely long decimal, you can say within a finite amount of steps whether its greater to or lesser than any other (non-equal) one. However, my system makes more sense under other operations. For instance, with the infinite natural

...8761 I can tell you that its remainder divided by 10 is 1, its remainder divided by 1000 is 761, I can tell you its odd, (remainder by 2 is 1) etc. These answers are finite and definitive. Whereas the remainder of pi divided by 10, will be another infinite sequence. If I am willing to get other infinite sequences, I can add, multiply any finite number to any infinitely long number. and in so doing produce a generator that will spill out the results till infinity.

But that's the thing here, what we're really talking about is these generator objects. That's what were doing operations on. For instance with my "set numbers" in binary:
...010101 (even)
+...101010 (odd)
=...111111 (natural numbers)
And the same holds true for your set numbers. We add the generator for all evens with all odds, and get the generator for the naturals . We don't need to expand them to infinity to do operations on them.

This is true because we can see the pattern in the numbers. .01(repeat) is a finite compression of the infinite series .010101... For every well defined number with infinite decimal places, there is such a finite compression, a generator object, specified with finite information which can enumerate its digits forever. A number which cannot be so compressed, consists of an infinite amount of random, uncompressible digits, which can never be specified in any form by human or machine, because to do so would take infinite time. These numbers are uncountably infinite in the consensus thinking, while the numbers specified with finite information are countably infinite. This is really a mess in my opinion: The vast majority of numbers can never be specified, and a particle in space doesn't even at any point in time have a distance from another particle that corresponds to any of these numbers.(Quantum mechanics: At smallest level, particle exists as probability wave until measured and probability wave collapses into 1 of countable measurements) So the vast majority of numbers exist beyond the reach of Man, God, and the Universe. Uh oh.

So in my mind, the coup de grace of the girl in the video was based on the argument that "between any two points on the real line there are an infinite amount of other points" so, because no point can be closer to 1 than .999..., it must be 1. To the extent that idea is right, .999... equal 1. but that idea is not right. I would say:
between any two points on the number line both specified with finite information, there exists a point in between them which takes more information to specify. e.g, in binary:
f(0.1, 1.0) = 0.11
f(0.11, 1.0) = 0.111
f(0.111, 1.10) = 0.1111 and so on.
And, because information is based on the reciprocal of probability, the probability of that number being specified approaches zero as the amount of information needed to specify it grows.
And,
for any arbitrarily large limit on the amount of information that can be expressed by a system, there is some number which IS the closest point to some other number. e.g
For ((10^n)-1)/10^n from above, n corresponds to the amount of decimal digits the system can support. If n is finite, then, then there is always a number which is the closest to 1 and also less than it.

The key thing to remember is that those generator objects aren't numbers, they are like functions. The argument they take is the amount of information available to the system, and they provide the best finite approximation for that system. So the generator object for .999 or the set number of all primes or pi is not equal to any single number, though we can treat them like numbers: Like with Eulers identity, we can relate the generator for e to the generators for sin and cos through i, and show that the generators are the same, so the identity works. But the generator for e still takes as an argument for the precision you are going to calculate it to, and that makes it fundamentally different from ordinary numbers. Of course you can cast ordinary numbers to generators, for instance 1 might be:
f1(m) = (10^m)/(10^m)
where as .999...
f9(m) = ((10^n)-1)/(10^n)
Algrebaically, they are different.

[tama]

31. Not much point

fighting against Gödel:

"My thing is that a number has to have a finite definition somewhere, and those systems of representation where a number has a precise and finite definition is where it belongs."

Definition of number is a set of axioms creating a number theory, and Gödel proved that any logical system containing number theory (or rather just primes) does not reduce to a finite set of axioms. And as for primes, the notion is not limited to ordinality or cardinality, but just "one" and "self".

Here's an interview that I liked reading very much: http://www.urbanomic.com/Publications/Collapse-1/PDFs/C1_Matthew_Watkins.pdf

That's where I heard about 'Beurling’s Generalized Primes' first time.

[tama]

6. Transcendentals are strange

AFAIK, being not an algebraic number is other way of saying 'not constructible with compass and ruler (circles and lines). Very few transcendentals are known to be such and proving them are hard problems of math, yet it is known(?) that the points of real line consist mainly of transcendentals. So there is a relation between the notion of infinitesimals and transcendentals to make the real line smooth and continuous.

From a more general view transcendental pi is known to be transcendental because as a mathematical constant it is in some sense the irreducible ratio of compass and ruler, and because it has been proven (at least inside the axioms of standard set theory) that a circle can't be squared.

[Joseph8th]

25. Da! Transcendentals are strange...

... and awesome.

You added a (?) to "known" and the answer is, 'sort of'. There's infinity, and then there's infinity. You've got your aleph_nought infinity of sets that can be mapped bijectively to the natural counting numbers, which includes integers and rationals and the algebraic numbers on the real line. Then you've got your aleph_1 least upper bound of the reals, which includes the uncountable set of transcendental numbers. We think of aleph_1 being infinitely larger than aleph_0, even though aleph_0 is infinitely large. Right? In fact, formally aleph_1 := 2^{aleph_0}, whatever that means. Heheh... So yeah, most of the reals are transcendentals... the larger infinitude, whatever sense that makes.

For which set of q_i's in the rationals does the equation q(pi) = q0 + q1*pi + q2*pi^2 + ... qn*pi^n = 0? The answer is there is no such set. You're right it's some heavy math (abstract algebra), but the proof calls for induction with some long division of polynomials... ie., q(pi) / (q1*pi + q0) searching for irreducible polynomials.

But to get an intuitive sense of 'transcendental', just plug pi in to a polynomial instead of the unknown "x". The result is the same -- "x" isn't a variable, it's an abstract algebraic placeholder called an "unknown" and algebraically, plugging pi in will give you the same result: irreducible polynomials.

Ex., x^2 - 4 = (x + 2)(x - 2) --> x = +-2 are roots in Q. (x + 2) and (x - 2) are irreducible polynomials in Q because x^2 - 2 = (x + sqrt(2))(x - sqrt(2)) and sqrt(2) is not in Q. So sqrt(2) is irrational. But sqrt(2) isn't transcendental (it is algebraic) for this very reason: sqrt(2) is the root of some polynomial with coefficients in Q (x^2 - 2). The same cannot be said for pi.

[tama]

29. Deep shit ;)

Two excerpts from TGD MATHEMATICAL ASPECTS OF CONSCIOUSNESS THEORY (http://tgd.wippiespace.com/public_html/mathconsc/mathconsc.html), about post-Cantorian notions of infinity:

2.9
Comparison with the approach of Cantor
The main difference between the approach of Cantor and the proposed approach is that Cantor uses only the basic arithmetic concepts such as sum and multiplication and th e concept of successordefining ordering of both finite and infinite ordinals. Cantor’s approach is also purely set theoretic. The problems of purely set theoretic approach are related to the question what the statement ’Set is Many allowing to regard itself as One’ really means and to the fact that there is no obvious connection
with physics.
The proposed approach is based on the introduction of the concept of prime as a basic concept whereas partial ordering is based on the use of ratios: using these one can recursively define partial ordering and get precise quantitative information based on finite reals. The ordering is only partial and there is infinite number of ratios of infinite integers giving rise to same real unit which in turn leads to the idea about number theoretic anatomy of real point.
The ’Set is Many allowing to regard itself as One’ is defined as quantum physicist would define it: many particle states become single particle states in the second quantization describing the counterpart for the construction of the set of subsets of a given set. One could also say that integer as such corresponds to set as ’One’ and its decomposition to a product of primes corresponds to the set as ’Many’. The concept of prime, the ultimate ’One’, has as its physical counterpart the concept of
elementary particle understood in very general sense. The new element is the physical interpretation: the sum of two numbers whose ratio is zero correspond to completely physical finite-subsystem-infinite
complement division and the iterated construction of the set of subsets of a set at given level is basically p-adic evolution understood in the most general possible sense and realized as a repeated second quantization. What is attractive is that this repeated second quantization can be regarded also as a model of abstraction process and actually the process of abstraction itself.

(...)

6.2
Leaving the world of finite reals and ending up to the ancient Greece
If strong number theoretic vision is accepted, all physical predictions of quantum TGD would be numbers in finite algebraic extensions of rationals at the first level of hierarchy. Just the numbers which
ancient Greeks were able to construct by the technical means at use! This seems rather paradoxical but conforms also with the hypothesis that the dicrete algebraic intersections of real and p-adic 2-surfaces
provide the fundamental cognitive representations.
The proposed construction for infinite primes gives a precise division of infinite primes to classes: the ratios of primes in given class span a subset of rational numbers. These classes give much more refined classification of infinities than infinite ordinals or alephs. They would correspond to separate phases in the evolution of consciousness identified as a sequence of quantum jumps defining sequence of primes ? p1 ? p2 ...... Infinite primes could mean a transition from space-time level to the level of function spaces. WCW is example of a space which can be parameterized by a space of functions locally.
The minimal assumption is that infinite primes reflect their presence only in the possibility to multiply the coordinates of imbedding space points by real units formed as ratios of infinite integers. The correspondence between polynomials and infinite primes gives hopes of mapping at least the reduced WCW consisting of the the maxima of K ähler function to the anatomy of space-time point. Also WCW spinors and perhaps also the the modes of configuration space spinor fields would allow this kind of map. One can consider also the possibility that infinite integers and rationals give rise to a hierarchy of imbedding spaces such that given level represents infinitesimals from the point of view of higher
levels in hierarchy. Even ’simultaneous’ time evolutions of conscious experiences at different aleph levels with completely different time scales (to put it mildly) are possible since the time values around which the contents of conscious experience are possibly located, are determined by the quantum jump: also multi-snapshots containing snapshots also from different aleph levels are possible. Un-integrated conscious experiences with all values of p could be contained in given quantum jump: this would give rise to a hierarchy of conscious beings: the habitants above given level could be called Gods with full reason: those above us would probably call us just ’epsilons’ if ready to admit that we exist at all
except in non-rigorous formulations of elementary calculus!