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March 4th, 2018, 06:46 PM   #1
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Thumbs up Why all the usual arguments for 0.999... = 1 fail...

Yes 0.999... can equal 1 if the basis for equivalence is chosen very carefully. However, I argue that by all the usual interpretations of what 0.999... means, it cannot equal 1 for the reasons I give here:

https://medium.com/@karmapeny/why-al...d-e9a25441f26c

By the way, please don't attack me for saying real numbers are invalid; attack my reasoning & not merely the fact that it seems unthinkable.

There are a growing number of mathematicians that are now rejecting the validity of real numbers, notably Norman Wildberger, Doron Zeilberger, other Finitists and their many followers. Professor N Wildberger said "statements like 0.999…=1 ought to be taken with a large grain of salt".

Enjoy.
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March 4th, 2018, 07:05 PM   #2
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Don't have the time...gotta go cash my refund cheque for 99.99
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March 4th, 2018, 07:08 PM   #3
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Finitism is a consistent system. So is infinitism. So what? For a finitist, 0 decimal point followed by infinite nines is meaningless so they can have nothing cogent to say about it.
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March 4th, 2018, 07:20 PM   #4
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A few years ago I made a pact with myself to never join another .999... thread again. I've fallen off the wagon a couple of times, but it was pointless as it always is.

The temptation is strong. To help people understand what the real numbers are, and how .999... = 1 can be proven directly from first principles, from the axioms of set theory. What's a limit, what's a convergent infinite series.

But I have never converted a single unbeliever. It is simply an enormous waste of energy to try.

I'll be watching this thread like the slow motion train wreck it's about to become. I ask the Gods of math for the strength to avoid putting in my 1.999... cents worth.

Wildberger. Have fun with this one folks.

Last edited by Maschke; March 4th, 2018 at 07:24 PM.
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March 4th, 2018, 07:29 PM   #5
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Think I'll go get dressed to the nines...
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March 4th, 2018, 07:51 PM   #6
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Quote:
Originally Posted by Karma Peny View Post
There are a growing number of mathematicians that are now rejecting the validity of real numbers
This suggests to me that you labour under the misapprehension that the mathematical world somehow holds some equivalence to the real world. Every mathematical model is a simplification of what it models and is thus not completely accurate. A successful model is one that is accurate enough.

If you are going to use a different set of axioms to the standard ones, that's fine. But don't pretend that it in any way makes mathematics done under those standard axioms wrong or "invalid".

It's easy enough to find perfectly good systems under which $ 0.999\ldots \ne 1$ without resorting to finitist arguments and thus implying that $0.999\ldots$ doesn't exist or have any meaning. The hyperreals, for example.

PS: your refutation of argument 7 doesn't make sense. It betrays a fundamental misunderstanding of real numbers and limits. The limit of every convergent infinite sequence of rationals is a real number by definition. It doesn't matter whether we can calculate that limit. In the case of $0.999\ldots = 9\sum_{n=1}^\infty \frac1{10}$ we can calculate the limit, and the limit is 1.
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Last edited by v8archie; March 4th, 2018 at 08:08 PM.
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March 4th, 2018, 08:50 PM   #7
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One of my minor pet peeves is people who "prove" that .999... = 1 using a nonsense argument. It seems you have captured several of these in your list. Namely, I share your criticism of arguments 1, 2, 4, and especially 7.

However, the real numbers are well founded whether you like it or not. For this reason, arguments 3, 5, and 6 are perfectly legitimate over the reals. In fact, there is no difference in these arguments. Each is nothing more than the definition of what limit is for reals phrased in 3 different ways. You can disagree with the real numbers if you like, and argue that .999....$\neq 1$ if you work in the p-adics, or hyperreals, etc etc.

This doesn't change the fact that when other people say $.999... = 1$, THEY are working in the reals and their claim is completely true.
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March 4th, 2018, 09:42 PM   #8
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Originally Posted by SDK View Post
I share your criticism of arguments 1, 2, 4, and especially 7.
This confused me until I looked again and found that I was talking about the refutation of argument 8, not 7.

There is also a logical flaw in the statement that "By all the usual arguments, 0.999… cannot equal 1.". It may be true that the usual arguments fail to accurately prove that 0.999... = 1, but that doesn't imply that 0.999… cannot equal 1. It just means that the proposition may remain unproven. An absence of proof is not a proof of absence.
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Last edited by v8archie; March 4th, 2018 at 09:58 PM.
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March 5th, 2018, 08:42 AM   #9
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Quote:
Originally Posted by v8archie View Post
The limit of every convergent infinite sequence of rationals is a real number by definition. It doesn't matter whether we can calculate that limit.
It does matter. It is not sufficient to claim that a limit exists that is a constant, but that its value cannot be calculated. If its value cannot be calculated then its value is not well defined. If a limit's value is not well defined, then the limit is not well defined.

This cuts to the crux of my 'argument 8'. I will now try to clarify the points I made in that argument:


8.1. A 'real number' is a complicated and unwieldy structure that arguably can only exist in the imagination, because there is nothing in the real world with similar properties (an infinite set of infinitely long structures).

There is no way we can do fundamental operations with such a complicated structure. For mathematical operations we must work with a symbol (like pi) or we work with individual sequences.

This point is just an observation and is not critical for my case.

8.2. The modern concept of real numbers relies on the validity of the concept of limits.

8.3. There are many sequences for which the limit cannot be calculated. This includes all divergent sequences (which are said to not have a limit) and all sequences that are said to correspond to irrational numbers.

8.4. It is inadequate to define something as being itself, or as being a group of things that includes itself.

It can be argued that when we use the symbol pi, it refers to the corresponding equivalence class of rational Cauchy sequences, and in that equivalence class is a bunch of sequences, not the symbol pi. Thus the definition is not recursive.

However, this appears to contradict your statement that the limit of a convergent infinite sequence is a real number. Is a real number defined as the limit or is it an equivalence class? In the case of 0.9, 0.99, 0.999, ... when we find the limit we do not find an infinite set of infinite sequences; we just get the finite value 1. When we find the limit, we appear to just be finding one of the infinitely many sequences in the corresponding real number (i.e. in the corresponding equivalence class of rational Cauchy sequences).

It is my understanding that when we write a symbol like 0.999... then we are implying a sequence that further implies a real number (= an infinite set of Cauchy sequences) that contains the aforementioned implied sequence. Hence we have a recursive definition.

This alone should be enough to dismiss the concept of real numbers as not being well defined.

8.5. In terms of a quantitative value, we describe an irrational like pi in terms of the leading terms of one of its sequences, usually the base 10 one. The limit of the pi sequence 3, 3.1, 3.14, ... can only be described quantitatively as being the sequence itself (ok, a different base may be used, but the issue is the same). We are reduced to describing the limit of a pi sequence as being a pi sequence. Effectively we have defined its limit as being itself! This is not an adequate definition.

Worse still, this value cannot be expressed in its entirely, so we have to use our imagination; we have to believe that infinitely many digits can exist, and that all these powers-of-ten quantities all add up to form a constant.

Even with all the complexity that comes with the machinery of limits, convergence and real numbers, we still cannot escape the fact that in order to appreciate quantitative value of an irrational number we must still examine one of its sequences and believe that we can add up its infinitely many static parts.

It is not sufficient to claim that a limit exists that is a constant but that its value cannot be calculated. If its value cannot be calculated, then its value is not well defined. If a limit's value is not well defined, then the limit is not well defined.

8.6. In order for a sequence corresponding to an irrational number to have a constant value, all its infinitely-many terms must exist at the same time. The same should hold for sequences that correspond to rational numbers, otherwise we would have an inconsistency.

8.7. My 'argument 7' shows algebraically why the sum 0.9 + 0.09 + 0.009 + ... cannot equal 1 (because it would result in contradiction).

8.8. Since my 'argument 7' shows that the sum 0.9 + 0.09 + 0.009 + ... cannot equal 1, and since a limit still needs to equal the addition of the infinitely many parts, then the limit argument must be flawed.

8.9. If the limit argument is flawed, then so are real numbers.


Quote:
Originally Posted by SDK View Post
I share your criticism of arguments 1, 2, 4, and especially 7.
Many thanks.


Quote:
Originally Posted by SDK View Post
arguments 3, 5, and 6 are perfectly legitimate over the reals. In fact, there is no difference in these arguments. Each is nothing more than the definition of what limit is for reals phrased in 3 different ways.
If the reals are well defined, then not only are there no differences in those arguments, but they are not arguments at all because the starting position is that 0.999... and 1 are just two different symbolic representations for the same real number. The starting position is that equality is a given by definition.

Many thanks for pointing this out this lack of clarity with my article. To correct this, I have added the following text to my article:

"Please note that arguments 1 to 7 existed before the limit approach to real numbers was conceived, so the modern concept of real numbers is not applicable to these arguments.

It was in the 16th century when Simon Stevin created the basis for modern decimal notation in which he allowed an actual infinity of digits. But it was not until the early 19th century that limits and convergence were introduced. The original idea behind infinite decimals was that they were the sum of their rational parts."


Quote:
Originally Posted by v8archie View Post
There is also a logical flaw in the statement that "By all the usual arguments, 0.999… cannot equal 1.". It may be true that the usual arguments fail to accurately prove that 0.999... = 1, but that doesn't imply that 0.999… cannot equal 1. It just means that the proposition may remain unproven. An absence of proof is not a proof of absence.
Fair point, especially for cases like my 'argument 1'. I have taken this on-board and amended my conclusion accordingly. Many thanks.

It now says:

"None of the usual arguments for 0.999… = 1 are valid; indeed, after the flaws are removed, many of them prove 0.999… cannot equal 1 (where 0.999… is considered to be the sum of its parts)."

Last edited by skipjack; March 5th, 2018 at 05:57 PM.
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March 5th, 2018, 11:13 AM   #10
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Quote:
Originally Posted by Karma Peny View Post
It does matter. It is not sufficient to claim that a limit exists that is a constant, but that its value cannot be calculated. If its value cannot be calculated, then its value is not well defined. If a limit's value is not well defined, then the limit is not well defined.
This doesn't make any sense. Take $\pi$ for example. It is perfectly well defined as the ratio of a circle's circumference to its diameter, but we can't calculate it exactly. The same goes for $\sqrt2$ (the length of the diagonal of the unit square).

Quote:
Originally Posted by Karma Peny View Post
A 'real number' is a complicated and unwieldy structure
No it isn't. It's a number, just like the natural numbers. It is abstract, but so are many things. You are focusing on the most complicated definition you've found. Perhaps because it suits your purpose, perhaps because you know no better. But a better (and equivalent) definition is that every real number is the limit of at least one sequence of rationals.

The good news is that every such real number can be described by at least one monotonic sequence of rationals, and that we can use such a sequence to determine its value to any desired accuracy.

Moreover, we can operate on such numbers using the theory of limits. The symbol $\pi$ is no different from the symbols $2$ or $\frac17$. All are just symbols that represent a single number.

The validity of the concept of limits should not be a problem because they are defined entirely in terms of finite structures.

Quote:
Originally Posted by Karma Peny View Post
Is a real number defined as the limit or is it an equivalence class?
There are many equivalent definitions of the real numbers. This is not unusual in mathematics. Selecting a definition that makes your ensuing analysis difficult is an act of folly.
Quote:
Originally Posted by Karma Peny View Post
When we find the limit, we appear to just be finding one of the infinitely many sequences in the corresponding real number (i.e. in the corresponding equivalence class of rational Cauchy sequences).
This is trivially not a problem because the equivalence class is defined as those sequences having the same limit.

Quote:
Originally Posted by Karma Peny View Post
It is my understanding that when we write a symbol like 0.999... then we are implying a sequence that further implies a real number (= an infinite set of Cauchy sequences) that contains the aforementioned implied sequence. Hence we have a recursive definition.
This can happen with the wrong choice of definition. Numbers are abstract objects. The fact that they can be viewed as sets or sequences doesn't mean that they are those things.

Quote:
Originally Posted by Karma Peny View Post
This is not an adequate definition [of $\pi$]
Perhaps that's why nobody defines $\pi$ like that. In general, I think you have all this back-to-front. You don't (necessarily) have to start with a sequence - far less a particular one. If you have a definition of your number and you can find any convergent sequence having that number as a limit, then the number is a real number.

Quote:
Originally Posted by Karma Peny View Post
we have to believe that infinitely many digits can exist, and that all these powers-of-ten quantities all add up to form a constant.
This is why the theory of limits exists, defined in terms of finite objects. If you don't believe that infinitely many digits exist you probably ought to have a good answer to what the smallest "power-of-ten" quantity is, but it's not necessary because the theory of limits uses only finite numbers.

Moreover, there is (should be) no suggestion that a real number is the sum of infinitely many elements. It's not. A limit is a limit, not a sum.

Quote:
Originally Posted by Karma Peny View Post
The same should hold for sequences that correspond to rational numbers, otherwise we would have an inconsistency.
It does. There are obviously infinite sequences with rational limits.

[QUOTE=Karma Peny;589536]The original idea behind infinite decimals was that they were the sum of their rational parts.[\quote]
You could equally well discard all of science on the basis that old theories have been superseded. It's not an argument that holds any water. It's also important to remember that the decimal representation is not the number. Even if infinite decimals didn't make sense, it wouldn't stop real numbers from existing.

Quote:
Originally Posted by Karma Peny View Post
(where 0.999… is considered to be the sum of its parts)
As I say, it's not. So this is rather misleading.

Note: in an attempt to keep the length of this down, I have avoided quoting every one of your points, but I think that all are covered if not explicitly. If you'd like me to address some specific point, let me know.

Last edited by skipjack; March 5th, 2018 at 05:58 PM.
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