Concept of Quantity

Originally Posted by ಠ_ಠ

you cannot shift the last digit of .999..., which is why 10*0.999... - 0.999... = 9, and not 9.0...9 (which I would argue is also = 9)

You actually mean "and not 9.999990.."

But my point is that you cannot shift them at all. I am trying to establish that it makes no sense, because you are implying that you get a digit from place N to place N-1. Which makes no sense if N is infinity and you cannot define "N-1".

If you can please give me a proper definition of what you mean "infinite" digits, otherwise simply we shouldn't use things that are not defined as adding/subtracting from infinity implied by shifting the digits.

Infinity has a definition and we do not have to change the definition given the way we use it here. A number with infinite places has so many places it accounts for all the place values you can fathom (as +infinity is so great it accounts for any +number you can fathom). It is not the upper bound of a quantity.

The upper bound of a quantity is called a ceiling and it's logical inverse is called a floor.

Originally Posted by C_ntua

You actually mean "and not 9.999990.."

But my point is that you cannot shift them at all. I am trying to establish that it makes no sense, because you are implying that you get a digit from place N to place N-1. Which makes no sense if N is infinity and you cannot define "N-1".

Yes, that does make sense. One cannot define the n-1 th digit if n is infinity. But, there's a roundabout way of saying the digit is 9 for all n, which through induction can be said to continue for all amounts of 9s.

Originally Posted by C_ntua

If you can please give me a proper definition of what you mean "infinite" digits, otherwise simply we shouldn't use things that are not defined as adding/subtracting from infinity implied by shifting the digits.

This is hard to do. It could be said that infinite digits means that the number has no exact decimal expansion, and that we successively approximating it, as we do with all expansions of irrational numbers.

The idea of a number being infinitely close is something that shouldn't be possible in the real numbers. One way we can look at it is as a sum of digits to powers of 10. This is the definition of a decimal expansion. Like sqrt(2) = 1 * 10*0 + 4 * 10^-1 + 1 * 10^-2 + ... for example. This same idea carries for all real numbers. So, with this in mind, we can look at the sum that has already been used 10000 times. The only difference is that, this time, there's an explanation of where it came from.

Originally Posted by C_ntua

You actually mean "and not 9.999990.."

yes, well 9.999...0

Originally Posted by C_ntua

But my point is that you cannot shift them at all. I am trying to establish that it makes no sense, because you are implying that you get a digit from place N to place N-1. Which makes no sense if N is infinity and you cannot define "N-1".

place N doesn't become 0 because you cannot shift N + 1 to N, but any place other than infinity can still be defined, place 1 becomes place 0, 2 becomes 1, 3 becomes 2, etc, but N doesn't change, it would remain a 9. Basically this is the same thing that happens when you shift the digits of any normal number, seeing how they can be represented as 1.00000..., 0.10000...., etc and can still be shifted

Originally Posted by C_ntua

If you can please give me a proper definition of what you mean "infinite" digits

all of them

Originally Posted by User Name:

Yes, that does make sense. One cannot define the n-1 th digit if n is infinity. But, there's a roundabout way of saying the digit is 9 for all n, which through induction can be said to continue for all amounts of 9s.

Don't disagree, but you cannot then prove that 10 * 9.999... = 9.999...
With what you are saying you can maybe prove 0.333... * 0.333... = 0.9999...
But you cannot prove the first since if you use induction you don't get where you want...

Originally Posted by User Name:

This is hard to do. It could be said that infinite digits means that the number has no exact decimal expansion, and that we successively approximating it, as we do with all expansions of irrational numbers.

The idea of a number being infinitely close is something that shouldn't be possible in the real numbers. One way we can look at it is as a sum of digits to powers of 10. This is the definition of a decimal expansion. Like sqrt(2) = 1 * 10*0 + 4 * 10^-1 + 1 * 10^-2 + ... for example. This same idea carries for all real numbers. So, with this in mind, we can look at the sum that has already been used 10000 times. The only difference is that, this time, there's an explanation of where it came from.

Seeing a decimal with infinite digits as a sum as you say is fine, this implies though that all the functions used, like multiplication, subtraction etc should be done using sums. In which case you need a set of axioms and theorems for sums with infinite digits. So you kind of step out of the classical calculus. If you don't use those axioms, you cannot define infinite sums and you are stuck there.

And that was my initial point. That once you use those axioms you would define the sum equal to its limit, that define 0.999... = 1. So the argument that "it is not a number but a process" or "an infinitesimal is not zero" or other things said, will be resolved. A process is a number as long as it converges, an infinitesimal has a limit of 0, thus can be seen as such etc etc.

In any case you follow the logic of your expanded system compared to your old one. Whereas the proves without doing so imply that you can use classical calculus and the the already made definitions to prove your point.

But I really think by now the point of argument was missed. As most debates since the actual points of arguments are not defined after a couple of days the points are simply missed. Meaning that I am not really sure we are actually disagreeing

Originally Posted by User Name:

The idea of a number being infinitely close is something that shouldn't be possible in the real numbers.

Of course it can.

Infinitesimal - Wikipedia, the free encyclopedia

Don't disagree, but you cannot then prove that 10 * 9.999... = 9.999...
With what you are saying you can maybe prove 0.333... * 0.333... = 0.9999...
But you cannot prove the first since if you use induction you don't get where you want...

You cannot prove these because by any attempt those answers would be wrong.

1/3 * 1/3 = 1/9
Do the basic arithmetic you learned in grade school and you come up with this.

10*10 = 100

It follows that if 99.9... is 100 then it could also be written as 99.9...

What I'm failing to get from arguments to the contrary is why infinity is not properly dealt with (or if indeed we need to deal with these concepts at all) if we have a number like 0.9... People on the other side seem to think there is an apparent paradox here. 0.9... is carried to infinite places. Therefore, doing arithmetic in the normal way, as long as you can reckon enough places, the answer is well defined. This is why it's OK to do conventional math with pi, for example, or e, or any irrational number.

Arguments that 0.9... < 1 seem to come from intuition only, because it hasn't been proven rigorously enough apparently, even if they are willing to accept that 0.9... is 1.

Mario may be onto something when he talks about other sets of numbers.

But I think C_ntua is just rather confused by infinity from the start.

Originally Posted by C_ntua

And that was my initial point. That once you use those axioms you would define the sum equal to its limit, that define 0.999... = 1. So the argument that "it is not a number but a process" or "an infinitesimal is not zero" or other things said, will be resolved.

Cauchy himself used infinitesimals as you can read on the same link I provided above. In there, towards the top of the article, there's also a quote that validates your insistence in treating infinitesimals as a process:

[...]Thus, Fermat thought of them as adequal to zero, while Cauchy defined infinitesimals in terms of variable quantities (such as sequences) tending to zero.

They do exist, hidden in traditional calculus. And they are not zero. Merely treated as zero, either through adequality early on, or more recently as a process, as you -- I believe, well -- define.

Originally Posted by whiteflags

Mario may be onto something when he talks about other sets of numbers.

More or less. Not so much other sets of numbers, but the exact same set with different axioms that aim to make a distinction between infinitesimals and other numbers of the Real set; Standard and nonstandard reals.

Much like we have rationals and irrationals to describe some of R numbers properties, we can also have, say, nonstandard numbers which include infinitesimals (but also such quantities as 0.999... and the whole of the irrationals), and standard numbers which are those that can be precisely defined, such as naturals or terminating decimals.

With this as a basis, it could be possible to establish a set of axioms built on top of existing axioms and that, without changing the number set, allow for a more rigorous formulation of infinitesimals and other quantities. I'm talking of course of something that has been already done. I'm talking of IST.

This effort was done exactly because the Compactness Theorem proved that it was possible to formalize infinitesimals. It's for this reason that I insist that the discussion of whether an infinitesimal is or not in R is a mistake. This is merely an issue of axioms, not of number sets.

Originally Posted by Mario F.

Of course it can.

Infinitesimal - Wikipedia, the free encyclopedia

In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard.
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number which is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels i.e, in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level and there are also infinitesimals with respect to this new level and so on.

[strike through]You remain irrelevant.[/strike through]

EDIT: That's too strong.

I'll admit there is some use in the fact that, under special circumstances, 1 != .999.... (I admitted this much earlier, but I'll do it again anyway.) My argument is that there is an equal amount of usefulness in admitting, that under the standard axioms(those I've linked to 10 times), .999... == 1.

Originally Posted by C_ntua

Don't disagree, but you cannot then prove that 10 * 9.999... = 9.999...
With what you are saying you can maybe prove 0.333... * 0.333... = 0.9999...
But you cannot prove the first since if you use induction you don't get where you want...

10 * (9 * 10^-1 + 10 * 9 * 10^-2 + ...) = 10 * 9 * 10^-1 + 10 * 9 * 10^-2 + ... = 9 * 10^0 + 9 * 10^-1 + 9 * 10^-2 + ...

Originally Posted by C_ntua

Seeing a decimal with infinite digits as a sum as you say is fine, this implies though that all the functions used, like multiplication, subtraction etc should be done using sums. In which case you need a set of axioms and theorems for sums with infinite digits. So you kind of step out of the classical calculus. If you don't use those axioms, you cannot define infinite sums and you are stuck there.

And that was my initial point. That once you use those axioms you would define the sum equal to its limit, that define 0.999... = 1. So the argument that "it is not a number but a process" or "an infinitesimal is not zero" or other things said, will be resolved. A process is a number as long as it converges, an infinitesimal has a limit of 0, thus can be seen as such etc etc.

There's no such thing as a process number. A limit gets it value from a rigorous definition and an infinite sum inherits that rigor because it is a limit. That's what I've been trying to say. It's only equal because you must take a limit, and that limit has a value. If we had infinitesimals, and in cases we do, there would be an alternative to the limit. But, in standard analysis, there isn't. They are equal for the very fact we don't, in standard analysis, have an infinitesimal.

I guess, subjectively, this could be seen as a limitation, but objectively, it is merely a conclusion drawn from implications from axioms and theorems. Think of math as an arbitrary system with its own laws, defined by its axioms, and it no longer seems contradictory. It just is.

Originally Posted by User Name:

My argument is that there is an equal amount of usefulness in admitting, that under the standard axioms(those I've linked to 10 times), .999... == 1.

No. Your argument was "The idea of a number being infinitely close is something that shouldn't be possible in the real numbers." (sic)
I was answering just that. And you were demonstrably, entirely, undoubtedly, proved wrong (by axioms in fact I have been linking too many times to count).

It's funny that you didn't even bother to read what exactly "nonstandard" means in IST. It's not what you think. It's instead a similar distinction to irrationals and rationals. You keep producing posts without properly researching the issue at hand (or even bothering reading posts in between that in the meantime could have spared you the embarrassment, like post #248)", draw the wrong conclusions and, if that is not enough, then come with the pearls like this:

[strike through]You remain irrelevant.[/strike through]

EDIT: That's too strong.

You think? Let me tell you about relevancy... Do you still insist that "The idea of a number being infinitely close is something that shouldn't be possible in the real numbers"?

Originally Posted by Mario F.

No. Your argument was "The idea of a number being infinitely close is something that shouldn't be possible in the real numbers." (sic)
I was answering just that. And you were demonstrably, entirely, undoubtedly, proved wrong (by axioms in fact I have been linking too many times to count).

It's called "compromise." It makes you seem less like an ...............

Could you tell me which axiom proves nested interval theorem wrong?

Have you even mentioned any theorems or axioms, specifically? All I've read are general terms and theories.

Originally Posted by User Name:

It's called "compromise." It makes you seem less like an ...............

Really?

Could you tell me which axiom proves nested interval theorem wrong?

I don't think you know what you are talking about anymore. This is becoming beyond ridiculous. But your answers have become a good source of knowledge about your character. So I've been entertaining it.

No theorem proves anything wrong. As I've told you before at least two times that I can recall, these axioms are a conservative extensions to ZFC. They don't pretend to invalidate prior axioms, they don't even add anything to prior axioms. Those remain safe. They are a new set of axioms that remain valid within ZFC, while extending it.

Have you even mentioned any theorems or axioms, specifically? All I've read are general terms and theories.

I can see you still haven't read about the Compactness Theorem not even the three letter acronym I've introduced in this debate countless times. Since you insist in not doing so, I prefer to not answer you, because clearly you don't want to know.

And that's all the relevancy you display on this debate.

One day I'm sure you will understand that the issue of an infinitesimal can be entirely axiomatic and have nothing to do with the number system. On that day, you'll remember this debate and the sad picture you drew in here. But as you probably guessed, you went over the mark, and for your remarks and idiocy I lost any interest in discussing with you any further. One could say, I grew bored of you.

Edit: I do have to thank you though. Your attitude, your refusal in researching concepts that were being discussed in here while sticking to your teachings as if a doctrine they were, while I spent the last 3 weeks or so researching and studying things I didn't for the last 20, made me feel better about myself. It confirmed that in the end of the day, all that separates those who know from those who don't is not how much they know, but how much they are willing to learn.

Originally Posted by Mario F.

No theorem proves anything wrong. As I've told you before at least two times that I can recall, these axioms are a conservative extensions to ZFC. They don't pretend to invalidate prior axioms, they don't even add anything to prior axioms. Those remain safe. They are a new set of axioms that remain valid within ZFC, while extending it.

Exactly, no theorem proves anything wrong. So, how is nested interval theorem wrong?

Originally Posted by Mario F.

One day I'm sure you will understand that the issue of an infinitesimal can be entirely axiomatic and have nothing to do with the number system. On that day, you'll remember this debate and the sad picture you drew in here. But as you probably guessed, you went over the mark, and for your remarks and idiocy I lost any interest in discussing with you any further. One could say, I grew bored of you.

Funny to hear claims of idiocy from [largely] the only person disagreeing.

I spent some free time today learning about IST. I have to admit, it is much more credible and rigorous than I had assumed. It also surprised me with how it actually makes sense in an intuitive way.

But, how is it that IST proves that .999... != 1? I'm looking for an axiom of IST or (a) theorem(s) drawn from the implications of the axioms provided by IST.

Also, I have discovered that there are cases in which I would've been wrong in saying that IST is irrelevant. The Transfer axiom (correct name?) allows for those cases.

No more sarcasm from me, for now.

Originally Posted by whiteflags

What I'm failing to get from arguments to the contrary is why infinity is not properly dealt with (or if indeed we need to deal with these concepts at all) if we have a number like 0.9... People on the other side seem to think there is an apparent paradox here. 0.9... is carried to infinite places. Therefore, doing arithmetic in the normal way, as long as you can reckon enough places, the answer is well defined. This is why it's OK to do conventional math with pi, for example, or e, or any irrational number.

Arguments that 0.9... < 1 seem to come from intuition only, because it hasn't been proven rigorously enough apparently, even if they are willing to accept that 0.9... is 1.

Mario may be onto something when he talks about other sets of numbers.

But I think C_ntua is just rather confused by infinity from the start.

I can be well defined, but cannot be proven with conventional math as you call it. So you can use it with some specific rules, you just cannot prove the rules. In other words you have to expand your system.

But people didn't create math just for the fun of it. So every system might have indeed different use and some systems are not applicable in certain uses of math. So the separation is important. Furthermore, even if practically you know the theory, for philosophical reasons understanding where they are based on is interesting.

The paradoxes come if you use a classical system without infinity and add the meaning of infinity and allow the use of decimal numbers with infinite digits. You just have to go a little bit further.

I already gave different definitions of infinity, there is no real confusion, people just use it without really expressing its meaning.
--------
So to clarify one more time, 0.999... cannot be proven to be equal to 1
a) Without axioms about infinite sums
b) Without axioms about decimals with infinite digits
c) Without something else

AND if this is attempted the definition of infinity will save 90% of arguing, so people can give their definition of infinity at least in the sense of "infinite decimal digits"