## 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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No, 1 ≠ 0.999...
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Other
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:I don't remember where, but I remember that James defined an infinitesimal as merely something immeasurably small. I am not seeing any contradiction there nor in

"0.999... + an infinitesimal" = 1.0"

Regarding this "+ an infinitesimal" idea:

Let's grant the possibility that $$0.\dot9 + infinitesimal = 1.0$$
Divide by 3 to get:
$$0.\dot3 + \frac{1}3(infinitesimal) = \frac{1}3$$

What the hell is $$\frac{1}3(infinitesimal)$$?
Isn't the infinitesimal the smallest possible real number? How can you then divide it by 3 to make it even smaller?

Or is it that $$0.\dot3 + infinitesimal = \frac{1}3$$?
In which case multiplying by 3 gets:
$$0.\dot9 + 3\times{infinitesimal} = 1.0$$
Now we have the difference betwen $$0.\dot9$$ and $$1.0$$ being 3 times something?
Or do we allow the double standard that infinitesimals are as small as we need them to be, but infinites can all be different sizes?

The ridiculousness of the infinitesimal is infinite.

So the function of $$0.000...1$$ to make $$0.\dot3 = \frac{1}3$$ and also $$0.\dot9 = 1$$ is entirely inconsistent.
$$\frac1{\infty}$$ is as absurd as it is undefined and "convenient" - that's what leads to logical contradictions, not that people "such as Ecmandu, takes a bit too literally".

Magnus Anderson wrote:Most likely, that's not the case. For example, I am pretty sure that you do not really understand my arguments. You merely think that you do.

I can just as easily say that most likely I do understand your arguments, and you merely think I don't.

The problem is neither of us can point out what's beyond the understanding of the other because it's beyond "their" understanding.
Either I lack the understanding to see your understanding, and therefore that's why I don't accept it, or you lack the understanding to see my understanding, and therefore that's why you don't accept it.
Who is right?
Considering these natural restrictions, this rules out the less smart person from legitimately arbitrating this dilemma, hence why multiple other people than me are trying to point you in the less preferable direction - that this less smart person might in fact be you. It might be me - I kinda hope it is because I can tell you that being the smartest guy in the room too often can get a bit annoying, but I'm just going by probability here - and I hope this is just my complacency acting against me. I'm always considering the possibility that I might be wrong - how about you? I've tried to lay down the criteria to follow to prove me wrong, have you? Do you know how to be proven wrong? If not, I advise whole-heartedly to only forward a proposition if you have first understood the conditions under which such a proposition can be falsified. What are necessary conditions for you to accept that your propositions are flawed? Do you have too much pride to admit the possibility that they are?

Magnus Anderson wrote:Numbers are used to represent quantities. "5" can be used to represent any five things in absolutely any order that you can imagine. Numbers have nothing to do with how things are ordered (just as they have nothing to do with their position in space and time.)

That's why we speak of finite and infinite sets, not of finite and infinite sequences. When it comes to sets, there is no such thing as first element, second element, third element, etc. There is also no such thing as first element and last element. Sets have a number of elements and this number is either finite or infinite. There is thus no such thing as "direction of infinity".

There's so much wrong here.

"5" can be used to represent any five things in absolutely any order that you can imagine - if 5 is cardinal only.
Ordinal numbers absolutely have everything to do with how things are ordered. By definition there is a 1st, 2nd, 3rd element etc.
I can only assume you've only looked up cardinality so far and weren't aware of ordinality and ordered sets.

So no, I'm not forgetting that "sets have no dimensions and that their elements are not ordered".
You just didn't know about ordinality.

You keep on explaining to me the concept of "One-to-one correspondence" as though I hadn't mentioned it several times already - stop it. I know what bijection is.
Last edited by Silhouette on Sun Jan 05, 2020 4:52 am, edited 2 times in total. Silhouette
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:Now you went and made me look it up. This whole thing seems like a silly discussion.
-

James S Saint » Sun May 14, 2017 7:14 am wrote:The real definition, intuitively used by just about everyone for thousands of years would be:

1 infinitesimal == the smallest amount 1/x as x approaches infinity = "0+" - used in Calculus all of the time.

When working with a defined standard infinity and higher infinities (such as the hyperreals) an infinitesimal can be expressed as:

1 insml = 1 / infA, wherein infA is the infinity associated with the natural numbers. But that is for a more advanced discussion than this thread needs.
James S Saint » Sun May 14, 2017 9:18 pm wrote:As I said, an infinitesimal is a CLASS of quantities reached by 1/x as x goes to infinity (more properly "goes toward..").

It’s not too silly. His “goes toward” means that he’s using infinity in a temporal sense.

Infinity is not subject to temporal logic.
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### Re: Is 1 = 0.999... ? Really?

"Temporal" means "time dependent". What kind of logic is time dependent?
You have been observed.
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:"Temporal" means "time dependent". What kind of logic is time dependent?

Logic that uses the term “moving to” as James did.

“Approaching”, “moving towards”

That’s all temporal.

Infinity requires different terminology because it is NOT!! Spacio-temporal!!

Now, because of this understanding of infinity, people have theorized “bound infinities” (a possible interpretation of this lack of spacio-temporal), but, if you really think about it, “bound infinities i.e... convergent infinities) are a contradiction to the (by definition) boundless nature
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:There's so much wrong here.

"5" can be used to represent any five things in absolutely any order that you can imagine - if 5 is cardinal only.
Ordinal numbers absolutely have everything to do with how things are ordered. By definition there is a 1st, 2nd, 3rd element etc.
I can only assume you've only looked up cardinality so far and weren't aware of ordinality and ordered sets.

So no, I'm not forgetting that "sets have no dimensions and that their elements are not ordered".
You just didn't know about ordinality.

We are not talking about ordinal numbers. $$0.999\dotso$$ is not an ordinal number. $$1$$ is not an ordinal number.

Regarding this "+ an infinitesimal" idea:

Let's grant the possibility that $$0.\dot9 + infinitesimal = 1.0$$
Divide by 3 to get:
$$0.\dot3 + \frac{1}3(infinitesimal) = \frac{1}3$$

What the hell is $$\frac{1}3(infinitesimal)$$?
Isn't the infinitesimal the smallest possible real number? How can you then divide it by 3 to make it even smaller?

The word "infinitesimal" does not refer to a number that is greater than zero but less than any other number one can think of. The word "infinitesimal" refers to a reciprocal of infinite quantity, and since there are multiple infinite quantities, there are also multiple infinitesimal quantities. The standard infinitesimal quantity is $$\frac{1}{\infty}$$. There are infinitesimal quantities smaller than that.

Like Ecmandu, what you're doing here is misunderstanding the position of your interlocutors (in this particular case, you failed to understand what an infinitesimal is) and then, based on that misunderstanding, deriving a (mistaken) conclusion that your interlocutors are contradicting themselves. It's a strawman. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

I understand what you’re saying and I’m sure silhouette does as well.

You are stating that infinities can be added, multiplied, subtracted and divided, powered, rooted etc...

We’re both telling you that it doesn’t work that way.

Here: let me give you an analogy!

God is omnipotent, omniscient and omnipresent?? (I’m an atheist but that’s not the point here)

If there were 2 gods, they would do and be exactly the same thing. Remember Ockham’s razor? Not to multiply entities needlessly?
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:God is omnipotent, omniscient and omnipresent?? (I’m an atheist but that’s not the point here)

If there were 2 gods, they would do and be exactly the same thing.

Well, if you say there are 2 gods, you can't then say there is 1 god. That would be a logical contradiction.

On the other hand, maybe you want to say that the number of gods who are omnipotent, omniscient and omnipresent is necessarily one?

Let's define these terms.

If you're omnipotent, you have unlimited power (i.e. you can do literally anything you want.)

If you're omniscient, you know everything.

If you're omnipresent, you're present everywhere (i.e. you exist at every point in space and time.)

Assuming we agree on the definition of these terms, does it follow that only one person who's omnipotent, omniscience and omnipresent can exist? Of course not.

But regardless of your apparently bad example, what you're trying to say with regard to infinity is pretty clear. You're trying to tell us that, by definition, there can only be ONE infinite set.

And yet, we have the set of natural numbers N, the set of integers Z, the set of real numbers R and so on, all of which are infinite.

And there can be any number of infinite sets. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:God is omnipotent, omniscient and omnipresent?? (I’m an atheist but that’s not the point here)

If there were 2 gods, they would do and be exactly the same thing.

Well, if you say there are 2 gods, you can't then say there is 1 god. That would be a logical contradiction.

On the other hand, maybe you want to say that the number of gods who are omnipotent, omniscient and omnipresent is necessarily one?

Let's define these terms.

If you're omnipotent, you have unlimited power (i.e. you can do literally anything you want.)

If you're omniscient, you know everything.

If you're omnipresent, you're present everywhere (i.e. you exist at every point in space and time.)

Assuming we agree on the definition of these terms, does it follow that only one person who's omnipotent, omniscience and omnipresent can exist? Of course not.

But regardless of your apparently bad example, what you're trying to say with regard to infinity is pretty clear. You're trying to tell us that, by definition, there can only be ONE infinite set.

And yet, we have the set of natural numbers N, the set of integers Z, the set of real numbers R and so on, all of which are infinite.

And there can be any number of infinite sets.

Not if there’s correspondence. We know that they’re the same size. Not only do we not just know that, we know they are equal in size to the “highest possible infinity.

I showed you already, that assuming there are “uncountable infinities”, that all you have to do to list the reals is make a cheat list....

1.) rational number
2.) uncountable infinity
3.) rational number
4.) uncountable infinity

Etc....

That’s the superset and they can be listed in correspondence using this cheat.

Since the superset is in correspondence it’s equal in size to all the correspondence subsets.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:The word "infinitesimal" does not refer to a number that is greater than zero but less than any other number one can think of. The word "infinitesimal" refers to a reciprocal of infinite quantity, and since there are multiple infinite quantities, there are also multiple infinitesimal quantities. The standard infinitesimal quantity is $$\frac{1}{\infty}$$. There are infinitesimal quantities smaller than that.

Like Ecmandu, what you're doing here is misunderstanding the position of your interlocutors (in this particular case, you failed to understand what an infinitesimal is) and then, based on that misunderstanding, deriving a (mistaken) conclusion that your interlocutors are contradicting themselves. It's a strawman.

I'm not misunderstanding your position, I'm showing you how ridiculous it is. You don't understand your position as ridiculous for some reason, so you think a representation of it that shows its ridiculousness must be something different to it.

I've just shown you that as soon as you put it in an equation and operate on the equation, it conveniently shrinks and inflates to whatever you need it to be to "fill" this mysterious gap that isn't even there.
Wiki says Archimedes originated "what came to be known as the method of indivisibles", when divisibility is exactly what I just showed to be necessary to operating on equations with an "infinitesimal" with any consistency.
Ways that the article describes "Infinitesimal" include "indistinguishable from 0".... I wonder why. Yet in the same breath somehow also "not zero". Convenient? You use "taking it literally" as though it's a bad thing. Math isn't some metaphorical mysticism you know?
Another description is "so small you can't measure it", in which case - why is it given the measurable quanta of ε?

Like its reciprocal, ω, it's just another example of "this is impossible/contradictory, but let's give it a symbol and treat it as possible". Bury the contradictions & formalise a bunch of logic around it for legitimacy = new math.

Why can't you see that it's perfectly possible to understand the subject matter and reject its logical legitimacy? I can't be clearer, and still you hear none of it and just equivocate and accuse incredulity.

And for this "2 gods" analogy of Ecmandu's, "even if there were 2 there would be 1" being a logical contradiction is the whole reason there can't be 2 (or more) - it's a proof by contradiction and makes perfect sense. For omnipresence it's obvious that "being" literally everywhere literally makes it impossible for "being" to be shared with another being - there's no room. Add in omnipotence and omniscience and you're doing the same thing as well as being the same being everywhere - what's the difference? There is literally nothing to distinguish.

The same goes for what I was calling "true" Infinity in a post that you've completely ignored. There's only 1 way in which extension can be infinite, but there's many ways in which extensions can be restricted in ways that they're not infinite e.g. Natural numbers being bounded from 1 and for other dimensions than the number line it follows infinitely: there are bounds to this "infinite" progression on all sides except one, like a box that only has 1 side open and with the walls around it extending infinitely in the one way that extension is infinite. By contrast, integers are bounded for other dimensions than the number line they follow, but the opposite side of the box to the one already open for Natural numbers has now been opened with the walls extending infinitely in the one way that extension is infinite throughout the box. If you know anything about topology, you will know that a straw has 1 hole (not 2 or 0). In the same way, the box doesn't now have twice as many holes for the integers as the natural numbers - the hole is infinitely long in 1 way only. Complex numbers open up 2 more opposite sides to the box, still leaving two last finite opposite sides in place. In the same way as the straw, there's still only one way in which extension is infinite - it's not squared when you go from a number line to a number plane with area. And as I mentioned, even removing the remaining two sides isn't touching on more than three dimensions, and even for infinite dimensions there's a sense in which any definition violates infinity - but even then there's only 1 way in which infinity is infinite!

This proves that there is only 1 infinity in the same way as there could only be 1 god if there was one. But there are many finite possibilities for finite restriction around this one way of being infinite.
In the same way, none of the sets $$\Bbb{N}$$, $$\Bbb{Z}$$, $$\Bbb{Q}$$, $$\Bbb{R}$$, $$\Bbb{C}$$ etc. are "more" or "less" infinite than infinite, but they are more or less finitely restricted around the one way in which they're each infinite.

There is nothing wrong with any of this logic, I'm sorry. "You just don't understand me" isn't going to cut it.
To prove me wrong you need infinity to have finite bounds so that its boundlessness can be bounded from another boundlessness so that they can be distinguished. You need the quality of having no quantity to be a quantity to treat infinites and infinitesimals as measurable quanta and thereby legtimately "be" quantities despite being quantity-less by definition and derivation of "infinite".
Repeating the same mistakes over and over isn't doing this, nor is it ever going to. If you're not getting the core that I just laid out, you're just wasting time, keystrokes and computer bits. Silhouette
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### Re: Is 1 = 0.999... ? Really?

I just wanna say thanks to all who are contributing to this thread... specifically because it's keeping ecmandu occupied and off the philosophical streets.
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### Re: Is 1 = 0.999... ? Really?

When you guys figure out if 1 does equal 0.999, please inform me immediately in a PM.... because I intend to steal the material of whoever solved this problem and get it published.
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### Re: Is 1 = 0.999... ? Really?

promethean75 wrote:When you guys figure out if 1 does equal 0.999, please inform me immediately in a PM.... because I intend to steal the material of whoever solved this problem and get it published.

You can steal my disproof of Cantor's diagonal argument if you want. Silhouette
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### Re: Is 1 = 0.999... ? Really?

Nice for you to look out for me prom.

I’m going to lay down some Freudian/Jungian psychology that’s on topic here (inspired by silhouette)

The phallic is any protrusion. The feminine is space itself.

The feminine is considered primordial, the great mystery.

For example, the phallic symbol is a house, the feminine symbol is an open door of that house.

The feminine is considered the open box in silhouettes analogy.

Many men have vagina envy. They want to be the space that allows everything to settle and be. No matter how hard they try, the feminine laughs at them, you can’t damage space, but! You can always damage the phallic (material).

In these disciplines, the feminine reigns supreme.

Magnus is trying his hardest to square space. He wants to be supreme as a phallic. But he’ll never be that.

As this thread continues, he’s slowly realizing that, and it frustrates him.

So for what it’s worth, there’s some Freudian/Jungian psychology for the day.
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### Re: Is 1 = 0.999... ? Really?

One more go at this (repeating myself to some extent):

Silhouette wrote:Regarding this "+ an infinitesimal" idea:

Let's grant the possibility that $$0.\dot9 + infinitesimal = 1.0$$
Divide by 3 to get:
$$0.\dot3 + \frac{1}3(infinitesimal) = \frac{1}3$$

What the hell is $$\frac{1}3(infinitesimal)$$?
Isn't the infinitesimal the smallest possible real number? How can you then divide it by 3 to make it even smaller?

Or is it that $$0.\dot3 + infinitesimal = \frac{1}3$$?
In which case multiplying by 3 gets:
$$0.\dot9 + 3\times{infinitesimal} = 1.0$$
Now we have the difference betwen $$0.\dot9$$ and $$1.0$$ being 3 times something?
Or do we allow the double standard that infinitesimals are as small as we need them to be, but infinites can all be different sizes?

The ridiculousness of the infinitesimal is infinite.

So the function of $$0.000...1$$ to make $$0.\dot3 = \frac{1}3$$ and also $$0.\dot9 = 1$$ is entirely inconsistent.
$$\frac1{\infty}$$ is as absurd as it is undefined and "convenient" - that's what leads to logical contradictions, not that people "such as Ecmandu, takes a bit too literally".

Suppose that $$0.\dot9 + \frac{1}{\infty} = 1$$. (I say "suppose" because I am not really sure about the exact infintesimal value. I am sure that $$1 - 0.\dot9$$ is equal to some infinitesimal quantity but I do not know exactly which one.)

Divide it by $$3$$. We get:

$$0.\dot3 + \frac{1}{3}\times\frac{1}{\infty} = \frac{1}{3}$$

Your claim is that $$\frac{1}{3\times\infty}$$ is a contradiction in terms. You think so because you think that the word "infinitesimal" refers to the smallest possible real number. That's not what the word "infinitesimal" means and that's your mistake.

Let us now suppose that $$0.\dot3 + \frac{1}{\infty} = \frac{1}{3}$$.

Multiply it by $$3$$. We get:

$$0.\dot9 + 3\times\frac{1}{\infty} = \frac{1}{3}$$

Your claim is that the difference between $$0.\dot9$$ and $$1$$ is $$3$$ times something. That's true. But we are supposed to believe that this is a problem. You did not tell us why, so we have to figure it out on our own.

I can only guess.

Perhaps you are trying to tell us that $$\frac{3}{\infty}$$ is a contradiction in terms because, by definition, infinitesimals cannot be bigger than other infinitesimals. But that's not true. There are bigger infinitesimals. So if that's what you think is problematic, that would be your mistake.

Or maybe you are trying to tell us that there is a contradiction between the earlier conclusion (that the difference is $$1$$ infinitesimal) and this subsequent conclusion (that the difference is $$3$$ infinitesimals.) But there is no such a contradiction. To make it clear, I'll give you an analogous argument:

Suppose that $$0.9 + a = 1$$. This means the difference between $$0.9$$ and $$1$$ is $$a$$. Divide by $$3$$. What do we get? We get $$0.3 + \frac{a}{3} = \frac{1}{3}$$.

Now suppose that $$0.3 + a = \frac{1}{3}$$. Multiply by $$3$$. What do we get? We get $$0.9 + 3\times a = 1$$. The difference between $$0.9$$ and $$1$$ is now $$3\times a$$.

Can we now conclude that arithmetic is "ridiculous"? Of course not. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:One more go at this (repeating myself to some extent):

Silhouette wrote:Regarding this "+ an infinitesimal" idea:

Let's grant the possibility that $$0.\dot9 + infinitesimal = 1.0$$
Divide by 3 to get:
$$0.\dot3 + \frac{1}3(infinitesimal) = \frac{1}3$$

What the hell is $$\frac{1}3(infinitesimal)$$?
Isn't the infinitesimal the smallest possible real number? How can you then divide it by 3 to make it even smaller?

Or is it that $$0.\dot3 + infinitesimal = \frac{1}3$$?
In which case multiplying by 3 gets:
$$0.\dot9 + 3\times{infinitesimal} = 1.0$$
Now we have the difference betwen $$0.\dot9$$ and $$1.0$$ being 3 times something?
Or do we allow the double standard that infinitesimals are as small as we need them to be, but infinites can all be different sizes?

The ridiculousness of the infinitesimal is infinite.

So the function of $$0.000...1$$ to make $$0.\dot3 = \frac{1}3$$ and also $$0.\dot9 = 1$$ is entirely inconsistent.
$$\frac1{\infty}$$ is as absurd as it is undefined and "convenient" - that's what leads to logical contradictions, not that people "such as Ecmandu, takes a bit too literally".

Suppose that $$0.\dot9 + \frac{1}{\infty} = 1$$. (I say "suppose" because I am not really sure about the exact infintesimal value. I am sure that $$1 - 0.\dot9$$ is equal to some infinitesimal quantity but I do not know exactly which one.)

Divide it by $$3$$. We get:

$$0.\dot3 + \frac{1}{3}\times\frac{1}{\infty} = \frac{1}{3}$$

Your claim is that $$\frac{1}{3\times\infty}$$ is a contradiction in terms. You think so because you think that the word "infinitesimal" refers to the smallest possible real number. That's not what the word "infinitesimal" means and that's your mistake.

Let us now suppose that $$0.\dot3 + \frac{1}{\infty} = \frac{1}{3}$$.

Multiply it by $$3$$. We get:

$$0.\dot9 + 3\times\frac{1}{\infty} = \frac{1}{3}$$

Your claim is that the difference between $$0.\dot9$$ and $$1$$ is $$3$$ times something. That's true. But we are supposed to believe that this is a problem. You did not tell us why, so we have to figure it out on our own.

I can only guess.

Perhaps you are trying to tell us that $$\frac{3}{\infty}$$ is a contradiction in terms because, by definition, infinitesimals cannot be bigger than other infinitesimals. But that's not true. There are bigger infinitesimals. So if that's what you think is problematic, that would be your mistake.

Or maybe you are trying to tell us that there is a contradiction between the earlier conclusion (that the difference is $$1$$ infinitesimal) and this subsequent conclusion (that the difference is $$3$$ infinitesimals.) But there is no such a contradiction. To make it clear, I'll give you an analogous argument:

Suppose that $$0.9 + a = 1$$. This means the difference between $$0.9$$ and $$1$$ is $$a$$. Divide by $$3$$. What do we get? We get $$0.3 + \frac{a}{3} = \frac{1}{3}$$.

Now suppose that $$0.3 + a = \frac{1}{3}$$. Multiply by $$3$$. What do we get? We get $$0.9 + 3\times a = 1$$. The difference between $$0.9$$ and $$1$$ is now $$3\times a$$.

Can we now conclude that arithmetic is "ridiculous"? Of course not.

Magnus!

The infinitesimal that creates the equality has to be invariable (thus not being able to create the equality)

What I mean by that is that the infinitesimal is in a “quantum state”. It’s either 0-9, but not one in particular.

What you’re doing here is stating that it’s NOT 0 through 9, but in every instance, just exactly the one you want it to be. You can’t have it both ways!
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:You use "taking it literally" as though it's a bad thing. Math isn't some metaphorical mysticism you know?

When James uses $$0.000\dotso1$$ to represent the difference between $$0.999\dotso$$ and $$1$$ he's not saying there is an infinite number of 0s that are followed by 1. There is no 1 at the end of those 0s. But Ecmandu thinks there is because he's not bothering to understand what James is trying to say.

For omnipresence it's obvious that "being" literally everywhere literally makes it impossible for "being" to be shared with another being - there's no room.

Depends on what is meant by the word "omnipresence". If to be omnipresent means to occupy the entire universe such that there is no room for anything else, then yes, there can only be one omnipresent being. But I am not really sure that's what the word means. Either way, the word "infinite" does not imply what Ecmandu thinks it implies. If there is a train made out of an infinite number of wagons, it does not mean there can't be a similar train somewhere else. Not according to the standard definition of the word "infinite". You can work with your own definitions, of course, but if you do, please understand that other people might not be working with the same definitions.

The same goes for what I was calling "true" Infinity in a post that you've completely ignored.

There are no finite bounds to "true" Infinity to allow the placement of "another infinity" next to it for the purposes of comparison of e.g. "size", because there are no bounds to "true" Infinity to be "next to".

Hard to see how any of this is relevant.

i) The sense in which this set is "infinite" is the sense in which the addition of more 1s does not end.
ii) However, the progression starts from "1" so it does end if you follow it backwards - and in that sense the qualification of "infinite" is thrown into question.

This is very hard to understand. What is it that you're saying? Are you saying that the set of natural numbers is not infinite because it has a beginning i.e. becuase there is a first number? First of all, the set of natural number is a set, not a sequence, which means its elements are not ordered i.e. there is no first element. We can list its elements any way we want. We normally start with number 1 but we can actually start with any other number. There is also no "backward" listing of elements. They are always listed in one direction and this direction is always infinite when it comes to infinite sets.

Now take the numerical progression of subtracting "1" from the "starting end" of the natural numbers an infinite number of times.
This completes the set of integers, $$\Bbb{Z}$$
The sense in which this set is "infinite" is now bi-directional, but still only along 1 dimension.

You're treating the set of integers as if it were a sequence of integers. It is not.

"Camp 2" points out that the idea that this extension has been bolted onto the "starting end" of an "infinite" set is a contradiction in terms.

This is why I'm with camp 2, because this "true" Infinity really stays true to its definition and derivation by really being endless/really having no ends.
By contrast, camp 1 seems more lazy in attributing "infinity" to sets in only a finite number of respects, thus allowing "infinite" sets to have cardinality/sizes contrary to what "infinity" itself actually means.

Basically, you're operating with a non-standard definition of the word "infinite". The word "infinite" does not mean "endless in all possible ways". It simply means "endless". A sequence that has a beginning but no end is infinite (it's merely not infinite in both directions.)

Not to mention that the subject of this thread is whether two quantities are equal to each other or not. Quantities can be represented using sets. There is no need to talk about sequences.

If you know anything about topology, you will know that a straw has 1 hole (not 2 or 0).

And if you know anything about logic, you'd know that how many holes a straw has depends on the definition of the word "hole". Depending on how you define the word "hole", a straw can be said to have 0, 1 or 2 holes. There is no inherently correct answer.

In the same way, the box doesn't now have twice as many holes for the integers as the natural numbers - the hole is infinitely long in 1 way only.

Again, depends on how you define the word "hole". There is a meaning of the word "hole" in which the correct answer is 2 holes.

This proves that there is only 1 infinity

Whatever it proves, it certainly does not prove that my claim that there are multiple infinities is false. I suspect you can't comprehend this because you're way too fixated on your own concept of infinity (making it difficult for you to understand what other people mean when they use that word.)

In the same way, none of the sets $$\Bbb{N}$$, $$\Bbb{Z}$$, $$\Bbb{Q}$$, $$\Bbb{R}$$, $$\Bbb{C}$$ etc. are "more" or "less" infinite than infinite, but they are more or less finitely restricted around the one way in which they're each infinite.

First of all, you are not talking about sets. It can't be the case because sets can neither be more/less infinite nor more/less finitely restricted. They are either finite or infinite.

You're talking about other, more complicated, mathematical objects such as sequences (possibly without a real reason.)

Sequences are like sets expect 1) repetition is allowed and 2) order matters. Whereas sets can only be finite or infinite, sequences can be finite, singly-infinite or bi-infinite.

A singly-infinite sequence is infinite in one way. A bi-infinite sequence is infinite in two ways. Note that the second statement claims that sequences can be infinite in more than one way. Yet, the statement is not a contradiction in terms. But you can also describe the difference between the two sets in terms of how many boundaries they have. You can say that a singly-infinite sequence is more finitely-restricted (because it has one boundary more) than a bi-infinite sequence. There is nothing wrong with such a description either.

Your argument basically amounts to "You're not working with my definitions of words, so you're wrong". Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:Suppose that $$0.\dot9 + \frac{1}{\infty} = 1$$.

Divide it by $$3$$. We get:

$$0.\dot3 + \frac{1}{3}\times\frac{1}{\infty} = \frac{1}{3}$$

Magnus.

You are a reasonable man, yes?

I will say the following with "you think the word infinitesimal refer to the smallest possible real number" in mind (in reference to me):

Asking the question of "what could possibly be the difference between $$1-0.\dot9$$?" one must consider how a quantity of any kind could possibly be smaller than this difference.
Certainly no decimal can do this job.
Even if one offered $$0.000...1$$ as a "literal" possibility, there is certainly no decimal quantity smaller than this - otherwise it wouldn't be small enough to be the difference between $$1-0.\dot9$$.
And yet, I proposed to you the difference between $$\frac1{3}-0.\dot3$$ as having to be 3 times smaller than the quantity that could not possibly be smaller to do this job.

You must ask yourself to what degree of abstraction and metaphor we are forced to resort here.
Without any "literal" possible quantity able to do the job we have to resort to something beyond quantity - hence the formulation of the concept "hyperreal".
Hyperreals formalise this domain of "meta-quantity" - I need you to understand that I understand this and have done since long before this discussion began.

Hyperreals none-the-less quantise what is beyond quantity.
Can you at least do me the courtesy of acknowledging the possibility of an objection to this stunt?
You don't need to agree with me, but c'mon man.
You are reasonable, yes?

What then are we left with when we are forced to deal with "bigger" and "smaller" (obviously quantitatively) than something that is beyond quantity?
Every single thing you've said, when replacing infinites/infinitesimals/hyperreals with finites would have been perfectly absolutely reasonable. Know that I absolutely and unequivocally accept this.
We are both being asked to suspend disbelief when considering arithmetic and algebra that denotes the "beyond quantity" with some definite symbol.
"Here is the beyond-quantity, now let's operate on it as though it was a quantity".

You accept this, I have no doubt.
How reasonable, let's say on a quantitative finite scale of 1-10 is this stunt?
Note that I'm not asking how useful it is - from the start I have said that the better question is "how useful is it to consider this stunt to be true" - don't forget this.
I'm asking how true it is - how valid and sound it is to treat what is beyond quantity quantitatively.
Do you at least see the issue here, even if you are willing to accept it? Silhouette
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### Re: Is 1 = 0.999... ? Really?

I realized that I didn’t actually describe the quantum fluctuation of infinitesimals.

So let’s say 1/3: the infinitesimal on one of those thirds must be a 4 right? What if it’s a negative 3???

Same result: they both equal one!!

1/3+1/3+1/4 = 1

Since the last digit is undetermined (quantum), any of the last digits can be undetermined!

1/1 + 1/1 + 1/1 = 1

This is the crazy math of infinitesimals.

If you can quantumly fluctuate any last decimal by 1, why not other ones? Well, there’s no rule for this.

That means that 0.999... can be 1.000...2 >=< 0.999...7

All sorts of weird shit comes up when you use infinitesimals as a given, and then they become quantum.
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### Re: Is 1 = 0.999... ? Really?

Asking the question of "what could possibly be the difference between $$1-0.\dot9$$?" one must consider how a quantity of any kind could possibly be smaller than this difference.
Certainly no decimal can do this job.
Even if one offered $$0.000...1$$ as a "literal" possibility, there is certainly no decimal quantity smaller than this - otherwise it wouldn't be small enough to be the difference between $$1-0.\dot9$$.
And yet, I proposed to you the difference between $$\frac1{3}-0.\dot3$$ as having to be 3 times smaller than the quantity that could not possibly be smaller to do this job.

You're assuming that decimal numbers can represent all numbers. So if there is no decimal number smaller than $$1 - 0.\dot9$$, you are led to believe there is no number smaller than $$1 - 0.\dot9$$.

I'm asking how true it is - how valid and sound it is to treat what is beyond quantity quantitatively.
Do you at least see the issue here, even if you are willing to accept it?

I have no idea what "beyond-quantity" means. This is one of those terms of yours that you never bother to define. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Asking the question of "what could possibly be the difference between $$1-0.\dot9$$?" one must consider how a quantity of any kind could possibly be smaller than this difference.
Certainly no decimal can do this job.
Even if one offered $$0.000...1$$ as a "literal" possibility, there is certainly no decimal quantity smaller than this - otherwise it wouldn't be small enough to be the difference between $$1-0.\dot9$$.
And yet, I proposed to you the difference between $$\frac1{3}-0.\dot3$$ as having to be 3 times smaller than the quantity that could not possibly be smaller to do this job.

You're assuming that decimal numbers can represent all numbers. So if there is no decimal number smaller than $$1 - 0.\dot9$$, you are led to believe there is no number smaller than $$1 - 0.\dot9$$.

I'm asking how true it is - how valid and sound it is to treat what is beyond quantity quantitatively.
Do you at least see the issue here, even if you are willing to accept it?

I have no idea what "beyond-quantity" means. This is one of those terms of yours that you never bother to define.

“Beyond quantity” means that there are no additives, subtractions, divisions, multiplications, powers, roots etc ...
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:You're assuming that decimal numbers can represent all numbers. So if there is no decimal number smaller than $$1 - 0.\dot9$$, you are led to believe there is no number smaller than $$1 - 0.\dot9$$.

This is why I bring up the notion of abstract and metaphorical for what you're saying.

Numbers are needed to represent numbers. It doesn't matter if they're decimal, binary, whatever.
Is it a number if you can't represent it with a number?!!!

Do you or do you not see the logic in this question?

You can't represent $$1 - 0.\dot9$$ with a number, therefore it's not a number.
Not hard.

Magnus Anderson wrote:I have no idea what "beyond-quantity" means. This is one of those terms of yours that you never bother to define.

Numbers that are beyond numerical representation are "beyond quantity."

I thought that was obvious, but you need me to explicitly say these words as a definition, fine.

Your requirement to treat concepts as numbers even though they're not expressable by numbers is what makes this treatment of numbers abstract and metaphorical.

Numbers denoting numbers is literal and fine.

Do you understand the words that I am saying? Silhouette
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### Re: Is 1 = 0.999... ? Really?

Isn't it by definition that there is always a number smaller than 1-0.999...? Infinity isn't a destination. Infinite means there is always more. That "always more" is the exact difference between 1 and 0.999....

It is part of the very definition of 0.999... that it is not 1.0.
You have been observed.
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:“Beyond quantity” means that there are no additives, subtractions, divisions, multiplications, powers, roots etc ...

So "beyond quantity" refers to anything that cannot be added, subtracted, divided, multiplied, powered, rooted, etc? That's something I can work with it. Tell me, why is it that two infinite quantities cannot be added together to get a bigger infinite quantity? Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:Numbers are needed to represent numbers. It doesn't matter if they're decimal, binary, whatever.
Is it a number if you can't represent it with a number?!!!

The difference between $$1 - 0.\dot9$$ can be represented using a number. James's $$0.000...1$$ is a number. Indeed, the very expression $$1 - 0.\dot9$$ can be considered a number in the same way that a fraction of two integers is considered a number (a rational number.)

What you cannot do is you cannot represent the difference using a decimal number. And that's exactly what your so-called argument against the existence of numbers smaller than $$0.000...1$$ boils down to. You're basically saying that if a quantity cannot be represented as a decimal number, it does not exist. But that's simply not true. "Decimal number" is not synonymous with "number".

Consider that there is no decimal representation of $$\frac{1}{3}$$. Yet, $$\frac{1}{3}$$ represents a real quantity. It's a quantity that can be represented as a rational number ($$\frac{1}{3}$$) as well as a base-3 number ($$0.1$$). The fact that there is no decimal representation of $$\frac{1}{3}$$ does not mean it's not a representation of a quantity.

What makes you think that hyperreal numbers aren't numbers? What makes you think that $$2\times\infty$$ is not a number? Stating that these symbols aren't numbers, expressing your dislike for them and arguing they aren't numbers because you can't represent them using decimal representation does not make them so.

Numbers that are beyond numerical representation are "beyond quantity."

And "numerical representation" means exactly what? How is $$2\times\infty$$ not a "numerical representation"?

Your requirement to treat concepts as numbers even though they're not expressable by numbers is what makes this treatment of numbers abstract and metaphorical.

How is $$2\times\infty$$ not a number?

Let me guess: you're probably operating with your own definition of what numbers are, one that is most likely completely irrelevant to the subject at hand. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:It is part of the very definition of 0.999... that it is not 1.0.

Precisely. By definition, there is a difference between $$0.\dot9$$ and $$1$$. The problem is that some people cannot accept the obvious because they are confused by the complicated. Specifically, what confuses them is the fact that there is no decimal number that can express this difference. This is based on the erroneous notion that the set of decimal numbers is the set of all numbers. That's not true.

I'm sure the difference between the two numbers can be expressed with a hyperreal number. But then, there are people who do not think that hyperreal numbers are numbers . . .

Either way, you don't have to find a numerical representation of a quantity in order to know that such a quantity exists. Magnus Anderson
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