Simple mathematical problem (?)

[xouper]

Suggestologist: There are no real decimal number representations between .999... and 1;

Correct.

Suggestologist: ... but this does mean that the distance between them is 0 and that they are the same number;

That's exactly what it means.

Suggestologist: ... the distance between them is iota and they are different numbers. Otherwise .99999...99998 would also equal one;

LW: That one is not a real number.

LuxFerum: 0,8+sum n=2..inf, 18*10^(- n)

Rewriting this so that the summation goes from 0 to infinity,

= 0.8 - 18*10^(-0) - 18*10^(-1) + sum n=0 to inf, 18*10^(-n)

pulling the 18 out of the summation (distributative property),

= -19 + 18 * sum n=0 to inf, 1*10^(-n)

using the geometric series theorem, we know that the summation part is 10/9, thus

= -19 + (18*10/9)

= 1

Ergo, 0,8+sum n=2..inf, 18*10^(-n) is not less than 1.

Suggestologist is still wrong.

 

[Suggestologist]

Hey, why not pretend 4 is really 5 and look at the consequences?


An IQ like a bag of roasted peanuts.

Hey, why not pretend that the square root of negative one is a number and look at the consequences? An IQ like a bag of burned marshmellows....

 

[T'ai Chi]

Seriously, what are we discussing again?

If .999... = 1, or if .999... is a real number (ie. a member of the Real number system) ?

 

[Suggestologist]

I take it this is your impression of the standard definition of decimal expansions of real numbers. Yes?

Your impression is not correct. A decimal expansion does not have a last digit, not even one labelled "omega". Every digit is followed by another. The three dots are at the end, not in the middle. (.999. . . is a decimal expansion; .999. . .999 is not.)

You seem to be treating omega as merely a very large finite number. It is not a finite number. It is a limit ordinal, which means it has no immediate predecessor. You appear to think it has one called "omega - 1".

Are you familiar with Cantor's notation or not?

1,2,3,...,1,2,3 represents a set of omega+3 elements. Do you see the dots in the middle? Is Cantor being silly?

If you prefer, we can use a modified version which corresponds to my proposition, as such:

6,7,8,9,10,...,1,2,3,4,5 : In which up to and including the elipses, we have a set of omega MINUS 5 elements, and then we add another 5.

"Cantor's theory of infinite sets provoked a host of protests. [...] Leonard Kronecker, who was also personally antipathetic to Cantor, called him a charlatan. Henri Poincare thought the theory of infinite sets a grave malady and pathological. 'Later generations,' he said in 1908, 'will regard set theory as a disease from which one has recovered.' Many other mathematicians tried to avoid using transfinite numbers even in the 1920s" (Mathematics: The Loss of Certainty, Morris Kline, p.203)

Perhaps someone should have offered Cantor the JREF Million Challenge?

Also:

Regarding 16th and 17th century mathematical thought: "Some of the more advanced thinkers, Bombelli and Stevin, proposed a representation which certainly aided in the ultimate acceptance of the whole real number system. Bombelli supposed that there is a one-to-one correspondence between real numbers and lengths on a line..." (ibid, p.116)

 

[69dodge]

Originally posted by Suggestologist
Pretend that they can be in the middle, and look at the consequences.

That might be an interesting thing to do, certainly. But then we wouldn't be talking about real numbers and their decimal expansions; we'd be talking about some new mathematical construct that we just invented. There's nothing wrong with inventing new mathematical constructs, of course, provided we don't claim that whatever is true of them is necessarily also true of old ones.

Real numbers and their decimal expansions are a well-understood mathematical system. In that system, the statement<blockquote>.999. . . = 1</blockquote> is true. That we can invent a new system, with its own rules and notation, in which a statement that looks the same is false, doesn't change that.

 

[Dorian Gray]

If you have a Liberal Arts degree:

"Alter" and "change" are synonyms, but when applied to your pants, they mean different things - that's because of the practical application of the terms to your pants (i.e., 'changing' your pants and 'altering' your pants are different things).

".999999......." and "1" are "mathematical synonyms", but there is not really a practical application for this fact, I think. This is what makes them exactly equal - just different names for the same thing.

 

[69dodge]

Originally posted by Suggestologist
Are you familiar with Cantor's notation or not?

I assume you are using the same notation as MathWorld's web page on ordinal numbers? That page says, for example,<blockquote>In order of increasing size, the ordinal numbers are 0, 1, 2, ..., omega, omega + 1, omega + 2, ..., omega + omega, omega + omega + 1, .... .</blockquote>And the table on that page defines "omega + 1" as the set {0, 1, 2, ..., omega}. Is this basically what you are talking about?

Notice that there is no mention of numbers like omega - 1 or omega - 2, etc.

1,2,3,...,1,2,3 represents a set of omega+3 elements. Do you see the dots in the middle? Is Cantor being silly?

The notation is not silly. However, it is somewhat unclear, and I believe you are misinterpreting it. The dots are not in the middle, exactly. They are attached to the "1, 2, 3" at the beginning, but not to the "1, 2, 3" at the end. Think of<blockquote>1, 2, 3, ..., 1, 2, 3</blockquote>as meaning<blockquote>(1, 2, 3, ...)(1, 2, 3),</blockquote> instead of meaning<blockquote>(1, 2, 3, ...)(..., 1, 2, 3).</blockquote>In other words,<blockquote>1, 2, 3, ..., 1, 2, 3</blockquote>might be the same as<blockquote>1, 2, 3, 4, ..., 1, 2, 3,</blockquote>depending on what the dots stand for, but it definitely is not the same as<blockquote>1, 2, 3, ..., 0, 1, 2, 3.</blockquote>In "1, 2, 3, ..., 1, 2, 3", infinitely many numbers precede the second 1, but no number directly precedes it; while in "1, 2, 3, ..., 0, 1, 2, 3", there is a number that directly precedes it, namely, 0.

In "1, 2, 3, ..., 1, 2, 3", there are infinitely many numbers (namely, the initial 1, 2, 3, ...) that are each a finite distance from the beginning, but only three numbers (namely, the final 1, 2, 3) that are a finite distance from the end; this is also true for "1, 2, 3, 4, ..., 1, 2, 3" because that's just another way to write the same thing. But in "1, 2, 3, ..., 0, 1, 2, 3", there are four numbers (namely, the final 0, 1, 2, 3) that are a finite distance from the end.

If you prefer, we can use a modified version which corresponds to my proposition, as such:

6,7,8,9,10,...,1,2,3,4,5 : In which up to and including the elipses, we have a set of omega MINUS 5 elements, and then we add another 5.

(Here, as above when talking about "a set of omega+3 elements", you seem to be confusing ordinals and cardinals, trying to describe the size of a set using an ordinal instead of a cardinal. The natural numbers, considered as a set, has cardinality aleph₀; the natural numbers, considered as a well-ordered set, has order type omega. But cardinals and ordinals are not the same sort of thing: the order type of {0, 1, 2, ..., omega} is omega + 1, yet its cardinality is still aleph₀.)

In any case, there is no cardinal number named "aleph₀ - 5"; removing 5 elements from a set of cardinality aleph₀ results in another set of cardinality aleph₀. There is, likewise, no ordinal number named "omega - 5"; removing 5 elements from a well-ordered set of order type omega results in another well-ordered set of order type omega.

I still say you are thinking about transfinite ordinals as if they were finite, when the two are in fact rather different.

 

[LW]

First, I'd like to wish Merry First Christmas of this thread to all readers, and express my wishes that there will not be a second one.

Next, I'd like to apologize everyone by not realizing before what I should have realized way, way, before. I guess that some of you have already noticed it but if somebody has posted it on this thread I have missed it, so I'll twist some iron wire into a model:

But anyway, Suggestologist has repeatedly used the expression "the omegath digit of the decimal expansion of 0.999...'.

I'm now wondering how on earth I haven't noticed earlier that there is no such digit. Or well, I have been feeling uncomfortable about using omega as a common integer but I felt that the problem was in mishandleded arithmetic instead of the more simple answer.

When you look at the sequence of nines you can enumerate them using natural numbers: the first number gets the index 1, the second the index 2, and so on up to the infinity. Even though the sequence is infinite, every single index is finite. No matter what nine you choose, only a finite number of nines precede it.

Now, the limit ordinal omega by definition is the smallest number that is greater than any natural number.

So, if all nines can be indexed with a natural number, then all their indexes are strictly smaller than omega, so no nine is "the omegath".

Even though there are countably infinite number of nines, their indices never reach omega.

Also, I believe that Suggestogist's repeated usage of 'omega - n' indicates that he believes that there is a number x different from omega such that x + n = omega.

If this was really the case, then it should be that:

x + (n - 1) + 1 = omega
x + (n - 2) + 1 + 1 = omega
...
x + 1 + ... + 1 = omega (with n ones)
(((x +1) + 1) + ... ) + 1 = omega
y + 1 = omega

Now what's wrong with this? Well, the number y = x + (n-1) is the immediate predecessor of omega. If this doesn't raise any suspicions, then maybe recollecting the definition of a 'limit ordinal' helps a little: An ordinal is a limit ordinal if and only if it does not have a predecessor.

 

[BillHoyt]

Hey, why not pretend that the square root of negative one is a number and look at the consequences? An IQ like a bag of burned marshmellows....

It is a number. It isn't on R.

 

[Suggestologist]

I assume you are using the same notation as MathWorld's web page on ordinal numbers? That page says, for example,<blockquote>In order of increasing size, the ordinal numbers are 0, 1, 2, ..., omega, omega + 1, omega + 2, ..., omega + omega, omega + omega + 1, .... .</blockquote>And the table on that page defines "omega + 1" as the set {0, 1, 2, ..., omega}. Is this basically what you are talking about?

Yep. That's basically what I'm talking about.

Notice that there is no mention of numbers like omega - 1 or omega - 2, etc.The notation is not silly. However, it is somewhat unclear, and I believe you are misinterpreting it. The dots are not in the middle, exactly. They are attached to the "1, 2, 3" at the beginning, but not to the "1, 2, 3" at the end.

They are not mentioned on that page? The way one represents an ordered set of omega-1 elements is: 2,3,4,5,...; the way one represents an ordered set of omega-2 elements is: 3,4,5,6,...; etc.

Think of<blockquote>1, 2, 3, ..., 1, 2, 3</blockquote>as meaning<blockquote>(1, 2, 3, ...)(1, 2, 3),</blockquote> instead of meaning<blockquote>(1, 2, 3, ...)(..., 1, 2, 3).</blockquote>In other words,<blockquote>1, 2, 3, ..., 1, 2, 3</blockquote>might be the same as<blockquote>1, 2, 3, 4, ..., 1, 2, 3,</blockquote>depending on what the dots stand for, but it definitely is not the same as<blockquote>1, 2, 3, ..., 0, 1, 2, 3.</blockquote>In "1, 2, 3, ..., 1, 2, 3", infinitely many numbers precede the second 1, but no number directly precedes it; while in "1, 2, 3, ..., 0, 1, 2, 3", there is a number that directly precedes it, namely, 0.

Infinitely many numbers precede the second 2, and the second 1 directly precedes it. By removing (an finite number of) elements from the beginning of the first infinite ordered set, and adding the same (finite) number of ordered elements to the end; you end up with the same number of ordered elements as omega; and you can uniquely reference them. Although, I personally suffer no cognitive dissonance when referring to an "omegath" digit.

In "1, 2, 3, ..., 1, 2, 3", there are infinitely many numbers (namely, the initial 1, 2, 3, ...) that are each a finite distance from the beginning, but only three numbers (namely, the final 1, 2, 3) that are a finite distance from the end; this is also true for "1, 2, 3, 4, ..., 1, 2, 3" because that's just another way to write the same thing. But in "1, 2, 3, ..., 0, 1, 2, 3", there are four numbers (namely, the final 0, 1, 2, 3) that are a finite distance from the end.(Here, as above when talking about "a set of omega+3 elements", you seem to be confusing ordinals and cardinals, trying to describe the size of a set using an ordinal instead of a cardinal.

A ordinal is the size of an ordered set -- the number of elements in the set. A cardinal number like aleph_naught, does says nothing about the ordering of the set. Ordinals in language are words such as: first, second, third, fourth, last, next-to-last, etc. Natural numbers like 1,2,3,4 etc. serve both the functions of ordinality and cardinality; but when you consider transifine sets, ordinality and cardinality are usefully dissociated.

The natural numbers, considered as a set, has cardinality aleph₀; the natural numbers, considered as a well-ordered set, has order type omega. But cardinals and ordinals are not the same sort of thing: the order type of {0, 1, 2, ..., omega} is omega + 1, yet its cardinality is still aleph₀.)

I agree. I have no need to reference aleph_naught.

In any case, there is no cardinal number named "aleph₀ - 5"; removing 5 elements from a set of cardinality aleph₀ results in another set of cardinality aleph₀. There is, likewise, no ordinal number named "omega - 5"; removing 5 elements from a well-ordered set of order type omega results in another well-ordered set of order type omega.

That's not correct. "omega - 5" is the set: 6,7,8,9,10,...

I still say you are thinking about transfinite ordinals as if they were finite, when the two are in fact rather different.

I am thinking about transfinite ordinals as if they were ordinals, not cardinals.

 

[Cabbage]

They are not mentioned on that page? The way one represents an ordered set of omega-1 elements is: 2,3,4,5,...; the way one represents an ordered set of omega-2 elements is: 3,4,5,6,...; etc.

This is wrong. Both of the sets you mentioned (and, in fact, any tail of the natural numbers beginning with any arbitrarily large integer) have order type omega.

Although, I personally suffer no cognitive dissonance when referring to an "omegath" digit.

Nor do I, and nor do any of the other posters to the thread, I expect. The simple fact is that when you talk about an omegath digit, you're no longer talking about representations of real numbers, which was implicit in the topic of this thread.

By removing (an finite number of) elements from the beginning of the first infinite ordered set, and adding the same (finite) number of ordered elements to the end; you end up with the same number of ordered elements as omega; and you can uniquely reference them.

If by "you end up with the same number of ordered elements as omega" you are referring to cardinality, then you are correct, of course. But the set you get has a different order type than omega--it "represents" a different ordinal.

A ordinal is the size of an ordered set -- the number of elements in the set. A cardinal number like aleph_naught, does says nothing about the ordering of the set. Ordinals in language are words such as: first, second, third, fourth, last, next-to-last, etc. Natural numbers like 1,2,3,4 etc. serve both the functions of ordinality and cardinality; but when you consider transifine sets, ordinality and cardinality are usefully dissociated.

I think I get what you're trying to say here, but I'm not crazy about the wording. I've never thought of an ordinal in describing the "size" of a set in any way--that's really the realm of the cardinals, which is the natural way to think of the "size" of a set. To fully describe an ordinal, not only do you need to describe the cardinality ("size") of the set, you also need to describe the well ordering of that set. Also, in the modern way of thinking, cardinals and ordinals aren't really dissociated, in the sense that a cardinal is a special type of ordinal--the cardinals form a subclass of the ordinals.

That's not correct. "omega - 5" is the set: 6,7,8,9,10,...

Wrong again. As I mentioned before, this set has order type omega.

 

[BillyJoe]

Should it be omega^th? (just wondering)

 

[Suggestologist]

This is wrong. Both of the sets you mentioned (and, in fact, any tail of the natural numbers beginning with any arbitrarily large integer) have order type omega.

Nor do I, and nor do any of the other posters to the thread, I expect. The simple fact is that when you talk about an omegath digit, you're no longer talking about representations of real numbers, which was implicit in the topic of this thread.

If by "you end up with the same number of ordered elements as omega" you are referring to cardinality, then you are correct, of course. But the set you get has a different order type than omega--it "represents" a different ordinal.

I think I get what you're trying to say here, but I'm not crazy about the wording. I've never thought of an ordinal in describing the "size" of a set in any way--that's really the realm of the cardinals, which is the natural way to think of the "size" of a set. To fully describe an ordinal, not only do you need to describe the cardinality ("size") of the set, you also need to describe the well ordering of that set. Also, in the modern way of thinking, cardinals and ordinals aren't really dissociated, in the sense that a cardinal is a special type of ordinal--the cardinals form a subclass of the ordinals.

Wrong again. As I mentioned before, this set has order type omega.

A review of the literature seems to show that you are correct with respect to Cantor's concept of transfinites. However, other systems do allow for numbers such as omega-1 (omega MINUS 1), notably Conway's surreals. Interestingly enough, I found this reference on the Internet Infidel's website regarding the subject: http://www.infidels.org/library/modern/graham_oppy/t_finite.html

 

[Suggestologist]

Should it be omega^th? (just wondering)

Yep, I'm just too lazy to use superscripts. omegath, omega^th, omega^th, whatever...

 

[Cabbage]

A review of the literature seems to show that you are correct with respect to Cantor's concept of transfinites. However, other systems do allow for numbers such as omega-1 (omega MINUS 1), notably Conway's surreals. Interestingly enough, I found this reference on the Internet Infidel's website regarding the subject: http://www.infidels.org/library/modern/graham_oppy/t_finite.html

I'll agree with you on that. Omega - 1 does have an existence in the surreals, but not in the ordinals.

In fact, honestly, in certain aspects I believe we agree more than you may give me credit for. I do understand where you're coming from with most, if not all, of the points you've been making in this thread. My main complaint has been whether or not those points are applicable in this or that context.

I have a general familiarity with surreal and hyperreal numbers, and I can certainly understand the concept of sequences of decimal digits that are omega+1 long, omega+2 long, or of any ordinal length.

My point, however, has been that those concepts aren't applicable when discussing the real numbers (and here, as I have always done throughout this thread, by "real" of course I'm referring to the formal mathematical meaning of "real number". I do not and never have meant to imply that the real numbers have any more reality than imaginary numbers, surreal numbers, hyperreal numbers, or any other algebraic construct. That's just the name they've been given, for better or for worse).

My impression is that you seem to be arguing that restricting our attention to the formal definition of real numbers is, in some way, limiting, and with this I must disagree. Certainly, it is worthwhile to study various extensions of the real numbers, such as the surreals and hyperreals; I think they're quite interesting. At the same time, it's also worthwhile to restrict your attention to the reals, and learn how they operate, by themselves. It's also interesting to study the properties of integers, or rational numbers, or algebraic numbers,... by themselves. There's no need to immediately jump to some "largest" class of numbers and study those alone--it's also interesting to take a smaller piece and see how that behaves on its own.

In other words, I don't find it limiting at all to restrict our attention, at times, to real numbers--quite the oppposite. Every piece of it has a place. The natural numbers are the way we count things--that's what makes them interesting (one of the things, anyway). The rational numbers are the closure of the integers under division--that's what makes them interesting. The real numbers are the only complete ordered field--that's what makes them interesting. The complex numbers are the algebraic closure of the reals--that's what makes them interesting. The surreals have infinites and infinitesimals--that's what makes them interesting. They all have a place.

Do we have an agreement here?

Anyway, I hope that clears up all of this "talking past" each other that seems to be going on in this thread, which, as far as I can tell, it all seems to boil down to. And I do want to apologize for any insulting comments I have made along the way.

Merry Christmas to everyone out there reading, and I have a small amount of hope (perhaps naively ) that this might put an end to it.

 

[phildonnia]

...
Merry Christmas to everyone out there reading, and I have a small amount of hope (perhaps naively ) that this might put an end to it.

So we all agree; it equals 1.

 

[Cabbage]

So we all agree; it equals 1.

I don't know if we have an agreement or not; I'm just trying to put everything in its proper context and wait and see what Suggestologist has to say. We haven't had an agreement on the original topic so far, but I'm hoping my last post may put everything in its proper place. I guess we have to wait and see....

 

[Suggestologist]

I'll agree with you on that. Omega - 1 does have an existence in the surreals, but not in the ordinals.

The surreals extend the transfinite ordinals. Just like negative and irrational numbers extended the reals. You do realize that both negative numbers and irrational numbers met with a lot of "protest" when they were "discovered" and promulgated, and initially they were also thought of as "imaginary" types of numbers, and that eventually people thought of them as part of the real numbers, just like rational numbers; even though the real numbers had "existed" for thousands of years before such "discoveries".

And how long ago was it that limits were "discovered"? Cauchy sequences weren't around before Cauchy. Dedekind cuts weren't around before Dedekind. And yet the real numbers WERE around before all of these "discoveries".

Doesn't it bother people that in the article on the Internet Infidel's website that I referenced earlier, Oppsy states that the reason Cantor didn't allow inverse operations was Cantor's bias against infinitessimals? Let's just pretend that human factors don't enter into the "pure field" of mathematics?

My point, however, has been that those concepts aren't applicable when discussing the real numbers (and here, as I have always done throughout this thread, by "real" of course I'm referring to the formal mathematical meaning of "real number".

Formal mathematical meaning of/since what year? The real numbers imply infinite representations; why aren't infinite representations validly analysed as to their meaning by utilization of infinite concepts of measurement? Is it aleph_naught or aleph_one that represents the cardinality of the real numbers...?

My impression is that you seem to be arguing that restricting our attention to the formal definition of real numbers is, in some way, limiting, and with this I must disagree. Certainly, it is worthwhile to study various extensions of the real numbers, such as the surreals and hyperreals; I think they're quite interesting. At the same time, it's also worthwhile to restrict your attention to the reals, and learn how they operate, by themselves. It's also interesting to study the properties of integers, or rational numbers, or algebraic numbers,... by themselves. There's no need to immediately jump to some "largest" class of numbers and study those alone--it's also interesting to take a smaller piece and see how that behaves on its own.

That's fine, if you want to look at "behavior". But what if I want to look at the underlying "reality" that leads to that behavior? Or perhaps upon closer examination, it doesn't necessarily lead to it, after all?

 

[BillHoyt]

The surreals extend the transfinite ordinals. Just like negative and irrational numbers extended the reals. You do realize that both negative numbers and irrational numbers met with a lot of "protest" when they were "discovered" and promulgated, and initially they were also thought of as "imaginary" types of numbers, and that eventually people thought of them as part of the real numbers, just like rational numbers; even though the real numbers had "existed" for thousands of years before such "discoveries".

So? There was once a cult-like group protecting the existence of alagon. So what does this have to do with underlying reality? Squat.

Doesn't it bother people that in the article on the Internet Infidel's website that I referenced earlier, Oppsy states that the reason Cantor didn't allow inverse operations was Cantor's bias against infinitessimals? Let's just pretend that human factors don't enter into the "pure field" of mathematics?

Humans have biases. Individual humans sometimes make mistakes. Sometimes others follow. Gee, I wonder where a postmodernist whose handle is "suggestologist" might be going with this. I guess I have to dust off my Kreskin's Krystal Ball again to find out.

That's fine, if you want to look at "behavior". But what if I want to look at the underlying "reality" that leads to that behavior? Or perhaps upon closer examination, it doesn't necessarily lead to it, after all?

Oh, please please please don't hold us all in suspense here! Release the pomo dogs, dude.

 

[xouper]

If anyone is interested, I am arguing about this same math topic with a whole 'nother batch of math illiterates over on another forum.

http://forums.crgaming.com/eqbb/viewtopic.php?t=90862

Why do I bother.

I did learn something new though. Apparently Fred Richman, PhD, and Professor of Mathematics at Florida Atlantic University, has written an article skeptical of the notion that 0.999... is equal to 1.

http://www.math.fau.edu/Richman/html/999.htm

I mention that here because I must now revise one of my earlier claims that no mathematician disagrees that 0.999... = 1. Apparently Richman is an exception. Perhaps there are more.