## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
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32%
No, 1 ≠ 0.999...
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59%
Other
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:Addition is a binary operation. The only way to define addition of infinitely many summands is to define it as the limit of the sequence of partial sums.

Yes, addition is a binary operation. It is a function that takes two values and outputs a single value. We know that. Do you have anything else to say? Because you are repeating yourself.

You seem to think that an expression such as "2 + 2 + 2" represents an operation of addition that takes three values and outputs a single value. If that's what the expression stands for then it contradicts the definition of the word "addition". As you say, addition takes two inputs; but this expression, you also claim, represents a function that takes three. How can it then be addition? It cannot be -- unless you change the definition of the word "addition" that is. Fortunately for us, that's not what the expression stands for. It stands for a chained series of additions where the result of the first addition (which is "2 + 2") is taken as the input of the second addition. This means that you DO NOT have to redefine the concept of addition. You DO NOT have to extend it so that it can take three (rather than merely two) values as its input.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:It stands for a chained series of additions where the result of the first addition (which is "2 + 2") is taken as the input of the second addition.

Exactly what I explained to you in detail a couple of posts back.

Yes you STILL have no definition of an infinite sum, since you can only use induction to reduce a finite sum to a succession of binary operations.

Well folks it's been fun, I'm done here. All the best.
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:Who said I disagree?

So who would you want to debate?
I can moderate.

I can debate anyone: Certainly Real, phyllo, wtf, Ecmandu, etc. But we'll have to find a set of rules that both me and the other participant are willing to abide by. The job of a moderator (you or anyone else) would be merely to check that the rules are respected and to intercept when violation is suspected.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

No one can agree to rules until they hear them. Why don't you propose some on the General Resolution Debate thread. We can talk about it.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

Hi all, thought I'd contribute the following to this thread.

A) The set of all numbers (if it's a non-contradictory number, then it is included in this set)
B) The set of all numbers except the number 10

It seems that both A and B encompass an endless number of numbers. However, one cannot deny that A is greater than B in terms of quantity. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes, but there is only one infinity (so there aren't infinities of various sizes).

If you tell me "there is no end to the number of numbers that B encompasses", and I ask you "does B encompass the number 10?", you will say "no". To which I will say "clearly, there is an end to the number of numbers that B encompasses. Had you said 'excluding 10, there is no end to the number of numbers that B encompasses' I might have believed you". I say might because I'm not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).

Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).

Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set of numbers.
Certainly real
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### Re: Is 1 = 0.999... ? Really?

Umm...

CR,

What you posted is an extreme lack of understanding of correspondence. Cantor figured out correspondence a couple hundred years ago.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Certainly real wrote:Hi all, thought I'd contribute the following to this thread.

A) The set of all numbers (if it's a non-contradictory number, then it is included in this set)
B) The set of all numbers except the number 10

It seems that both A and B encompass an endless number of numbers. However, one cannot deny that A is greater than B in terms of quantity. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes, but there is only one infinity (so there aren't infinities of various sizes).

If you tell me "there is no end to the number of numbers that B encompasses", and I ask you "does B encompass the number 10?", you will say "no". To which I will say "clearly, there is an end to the number of numbers that B encompasses. Had you said 'excluding 10, there is no end to the number of numbers that B encompasses' I might have believed you". I say might because I'm not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).

Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).

Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set of numbers.

Except for your use of the word "infinity" (instead of "infinite") I can accept all of that but -

Wouldn't the set of two truly infinite sets be "larger" than merely one infinite set?
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:Wouldn't the set of two truly infinite sets be "larger" than merely one infinite set?

I get your point. Here's my reasoning for why a set of two truly infinite sets is contradictory:

Take S to be the set of all sets. Take S' to be the set of all sets except S.

Clearly, neither S nor S' are finite in the number of sets they encompass. Given that you've accepted a distinction between truly infinite and infinite (or infinite and semi-infinite, whichever label you choose for the semantics we are focused on), it follows that you accept that S is truly infinite whilst S' is not truly infinite. To me, S is an infinite set whilst S' is a semi-infinite set. If I'm understanding you right, to you, S is a truly infinite set, whilst S' is an infinite set. What matters is that this semantical/meaningful distinction in the quantity that S encompasses and the quantity that S' encompasses is clear.

So, S is a truly infinite set, whilst S' is not. You ask, wouldn't the set of two truly infinite sets be "larger" than merely one infinite set?

My answer: there is only one set of all sets. Since there cannot be two sets of all sets, it follows that only S is a truly infinite set.
Certainly real
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### Re: Is 1 = 0.999... ? Really?

CR,

I know why you’re trying to make these arguments.

Your arguments themselves are fallacious, but we need not get into that.

If you don’t understand what I’m saying, it’s not your fault.

We are all beings who were never born and never die.

This current plan that we’re in was our best idea at the time from keeping us from being eternally bored.

God isn’t real CR. We all made this together.

Let me explain something to you. Blasphemy doesn’t exist. What does exist is your reputation in the spirit world. Good reputation? Decent life. Bad reputation? Not so much.

This is the way current existence as a plan works. You can have the best karma (karma is your spirit reputation) and crash and burn all that good reputation (karma) really quickly. You can also be a complete piece of shit and build extremely great karma really quickly.

Now here’s the deal dude. I found really serious flaws in the current plan. Not everyone can do that. Let’s go even one step further... I’m making a new plan.

When you’re talking to me, you’re talking to something you weren’t prepared to understand. And you might not understand it at this stage of your forever.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Certainly real wrote:Take S to be the set of all sets. Take S' to be the set of all sets except S.

That takes us back to an earlier fallacy - what you just said is the same as saying -
"Take a square that is bigger than itself = S.
Take a square that is not bigger than itself = S' "

I said "truly" only to distinguish from your "semi-". Perhaps "completely" would have been a better choice.

Certainly real wrote:there is only one set of all sets..

No - there is not even one.

The "set of all sets" is an oxymoron.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:No - there is not even one.

The "set of all sets" is an oxymoron.

Truth.
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Consider the following:

A = The set of all numbers (if it's a non-contradictory number, then it is included in this set)
B = The set of all numbers except the number 19

It seems that both A and B encompass an endless number of numbers. However, one cannot deny that A is greater than B in terms of quantity or cardinality. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes, but there is only one infinity (so there aren't infinities of various sizes).

If you tell me "there is no end to the number of numbers that B encompasses", and I ask you "does B encompass the number 19?", you will say "no". To which I will say "clearly, there is an end to the number of numbers that B encompasses. Had you said 'excluding 19, there is no end to the number of numbers that B encompasses' I might have believed you". I say might because I'm not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).

Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).

Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set.
Certainly real
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### Re: Is 1 = 0.999... ? Really?

It's important to note that you're using known words in a way other people don't.

Normally, a set is said to be infinite if and only if it contains a number of elements greater than every integer. That's it. Thus, both $$\{1, 2, 3, \dotso\}$$ and $$\{2, 3, 4, \dotso\}$$ are infinite because the cardinality of both sets is greater than every integer. That one is greater than the other is irrelevant.

Also, the distinction between your "infinite" and your "semi-infinite" isn't entirely clear. (The relevance of that distinction isn't clear either but let's ignore that for now.) If the set of all numbers is "infinite", what is the set of all numbers and all letters given that it's bigger?
Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Certainly real wrote:Consider the following:

A = The set of all numbers (if it's a non-contradictory number, then it is included in this set)
B = The set of all numbers except the number 19

It seems that both A and B encompass an endless number of numbers.

In the English language that means that both are infinite (or "endless"). Saying that a set is infinite is merely saying that it has no last element.

Certainly real wrote:However, one cannot deny that A is greater than B in terms of quantity or cardinality. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes,

We don't have a problem with you calling some of them "semi-infinite" - meaning that the set is less than the set of all integers. We don't see why you want to say that but it's logically ok.

Certainly real wrote:but there is only one infinity (so there aren't infinities of various sizes).

But that is not really ok.

First "infinity" isn't a quantity. "Infinity" means the first step beyond the endless. It is a target to aim for. But it is not on the map. No set ever reaches or encompasses infinity.

But also there is the issue of having more than one completely infinite set (set of all A1, A2, A3... and the set of all B1, B2, B3...). Combined infinite sets have more than merely one infinite set (obviously). And there is no limit as to how many infinite sets can be combined so there is no "infinity".

Certainly real wrote:If you tell me "there is no end to the number of numbers that B encompasses", and I ask you "does B encompass the number 19?", you will say "no". To which I will say "clearly, there is an end to the number of numbers that B encompasses. Had you said 'excluding 19, there is no end to the number of numbers that B encompasses' I might have believed you". I say might because I'm not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).

Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).

Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set.

That is all merely semantics after you invented a word. But why do we care?

There are obviously sets that are more than merely the infinite set of integers. That is why James defined his "$$InfA$$" - to distinguish the set of all integers (or naturals) from any shorter or longer set. Your "set B" above would be $$InfA-1$$.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:what is the set of all numbers and all letters given that it's bigger?

It may be that the set of all numbers is not infinity because we have to account for the set of all letters as well. Where we have to do this, then the set of all numbers is a semi-infinite to me. To me, if it can be greater in size, then it is not infinite. Infinity = the set of all things. Infinity = that which is omnipresent. Nothing is greater than this in size.

Since you do not call air on our planet as being omnipresent (there is no air in space, therefore air is not omnipresent), consistency would have you not call that which can be greater in size as being infinite. Thus anything other than Existence or the omnipresent, or the set of all things/existents, is non-infinite and non-omnipresent in my opinion.
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:We don't have a problem with you calling some of them "semi-infinite" - meaning that the set is less than the set of all integers. We don't see why you want to say that but it's logically ok.

Since we do not call air on our planet as being omnipresent (there is no air in space, therefore air is not omnipresent), consistency would have us not call that which can be greater in size as being infinite. Thus, anything other than Existence or the omnipresent, or the set of all things/existents, is non-infinite and non-omnipresent in my opinion. I'm taking an absolute approach regarding this matter. By this I mean if it's not perfectly/truly triangular then I would call it a semi-triangle. Similarly, if it's not truly omnipresent or infinite, then I would call it semi-omnipresent or semi-infinite.

First "infinity" isn't a quantity. "Infinity" means the first step beyond the endless. It is a target to aim for. But it is not on the map. No set ever reaches or encompasses infinity

I think it is a quantity. I see infinity as the quantity which encompasses all semi-infinites and finites. When you put infinity into the same class as semi-infinities, you end up with an inadequacy. You could say there are many infinites and the biggest one encompasses all the smaller ones, but I prefer to say there are many semi-infinities and finites, and they are all encompassed by infinity.

But also there is the issue of having more than one completely infinite set (set of all A1, A2, A3... and the set of all B1, B2, B3...). Combined infinite sets have more than merely one infinite set (obviously). And there is no limit as to how many infinite sets can be combined so there is no "infinity".

That is all merely semantics after you invented a word. But why do we care?
[/quote]

Because it addresses semantical inconsistencies I believe to be present today in maths.

Call the set of all things/existents V. Call the quantity of all things/existents infinity. Call any quantity that is not finite or infinite: semi-infinite. You can account for all semi-infinites in this way (of which there will be an endless number of, and all of which are encompassed by V or infinity). Take infinity out of the equation, and it's like you have no all-encompassing quantity (which to me, is madness. Yet, to my understanding, currently maths is like this. As in it has no all-encompassing quantity).
Certainly real
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### Re: Is 1 = 0.999... ? Really?

So you are just making up your own definitions for existing already defined words.

And you expect that to cause anyone to believe in God?
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
Philosopher

Posts: 2932
Joined: Thu Jul 11, 2019 9:03 am

### Re: Is 1 = 0.999... ? Really?

Certainly real wrote:To me, if it can be greater in size, then it is not infinite. Infinity = the set of all things. Infinity = that which is omnipresent. Nothing is greater than this in size.

It is confusing to say that infinity is a set if at the same time you're claiming that it is a quantity. You might want to say that infinity is a number equal to the number of elements contained within the set of all things (albeit the concept of a set of all things isn't entirely clear and requires further clarification.)

If there is no quantity greater than infinity then what you mean by infinity is perhaps "the largest number". That, however, would raise the question whether the cardinality of the set of all things is equal to this number.

Ultimately, your definition of infinity is non-standard and I don't see why we should adopt it. More importantly, I don't see how any of this answers the question posed in the OP. It looks very much off-topic.
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### Re: Is 1 = 0.999... ? Really?

"The largest number" isn't anywhere close to being the largest quantity. That is what the Cantor set theories were about. That is how we got cardinalities. And as Cantor stated - "there can be no highest cardinality" - no largest quantity.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
Philosopher

Posts: 2932
Joined: Thu Jul 11, 2019 9:03 am

### Re: Is 1 = 0.999... ? Really?

You might want to consider presenting an argument in favor of that claim but somewhere else (:

Right now, I am trying to understand Certainly real's claims. Though, it might be better for me to first do a check on whether what he's saying is relevant to this thread or not. So far, it does not seem to be the case.
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### Re: Is 1 = 0.999... ? Really?

It is a discussion about infinity but it really does belong on one of his threads.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
Philosopher

Posts: 2932
Joined: Thu Jul 11, 2019 9:03 am

### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:So you are just making up your own definitions for existing already defined words.

And you expect that to cause anyone to believe in God?

I gave my reasoning for why I distinguish between the semi-infinite and the infinite. If someone wants to make a distinction between these two semantics by calling one infinite whilst calling the other truly infinite, that's also fine in my opinion. But when no distinction is made between the truly infinite and that which is not truly infinite, then I think confusion can occur.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:It is confusing to say that infinity is a set if at the same time you're claiming that it is a quantity.

Existence is the set of all existents. It is also a meaning. It is also an existent. Its quantitative capacity is infinite, and by this I mean, the quantity of things it encompasses is infinite. So as a set it encompasses an infinity of things, hence why it semantically qualifies as being the infinite set, and why it is also a number. Alternatively you can put it this way:

Infinity = a number
The infinite set = the set that encompasses an infinity of things (the set of all sets: Existence).

You might want to say that infinity is a number equal to the number of elements contained within the set of all things (albeit the concept of a set of all things isn't entirely clear and requires further clarification.)

Yes I think that's what I meant. Maybe I should say the set of all existents instead of the set of all sets. But they both amount to the same thing. Existence, the infinite set. The universal set.

If there is no quantity greater than infinity then what you mean by infinity is perhaps "the largest number". That, however, would raise the question whether the cardinality of the set of all things is equal to this number.

Yes, I see it as the largest number. And yes, I see the cardinality of the set of all things as being equal to infinity.

Ultimately, your definition of infinity is non-standard and I don't see why we should adopt it. More importantly, I don't see how any of this answers the question posed in the OP. It looks very much off-topic.

To be fair, I did not focus on the OP beyond the fact that it is about infinity. So to that end, I will stop discussing infinity and semi-infinity here.
Certainly real
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### Re: Is 1 = 0.999... ? Really?

Infinity is not a noun. It’s a verb. It’s a process. If infinity were a thing it’d have borders.

CR,

Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Infinity is not a noun. It’s a verb. It’s a process. If infinity were a thing it’d have borders.

Who are you talking to? If you're talking to me, you are needlessly repeating yourself. That's not welcome.

I see.
Magnus Anderson
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