Is 1 = 0.999... ? Really?

Silhouette · January 5, 2020, 10:05pm

You can steal my disproof of Cantor’s diagonal argument if you want.

Ecmandu · January 5, 2020, 10:12pm

Nice for you to look out for me prom.

So… please read silhouettes post above.

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.

Magnus_Anderson · January 5, 2020, 10:31pm

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

Silhouette:

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.

Ecmandu · January 5, 2020, 10:42pm

Magnus Anderson:

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

Silhouette:

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!

I’ve had this conversation before.

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!

Magnus_Anderson · January 6, 2020, 1:28am

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.

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.

Hard to see how any of this is relevant.

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.

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

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.

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.

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.

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.)

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”.

Silhouette · January 6, 2020, 1:39am

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?

Ecmandu · January 6, 2020, 2:12am

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.

Magnus_Anderson · January 6, 2020, 2:20am

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

Ecmandu · January 6, 2020, 2:35am

Magnus Anderson:

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 …

Silhouette · January 6, 2020, 2:56am

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.

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?

obsrvr524 · January 6, 2020, 10:29am

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.

Magnus_Anderson · January 6, 2020, 12:53pm

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 · January 6, 2020, 1:16pm

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.

And “numerical representation” means exactly what? How is (2\times\infty) not a “numerical representation”?

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 · January 6, 2020, 2:12pm

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.

promethean75 · January 6, 2020, 2:31pm

Thank you for that edifying discourse, E. I was intrigued, and feel I should now risk the following hypothesis; all the great mathematicians were great because they were gay.

You heard em, Andy. Either make some significant changes to your sexual orientations, or continue to flunk this thread. Makes no difference to me.

obsrvr524 · January 6, 2020, 4:34pm

I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That’s where his infA originated (as opposed to infB). And his reasoning was simply that trying to say “2 * infinity” is like saying “2 * length”. For it to make any sense, you have to be more specific.

Magnus_Anderson · January 6, 2020, 5:06pm

Ecmandu and SIlhouette are telling us that even if you define a standard infinity, you cannot add two infinities together to get a bigger infinity.

Ecmandu · January 6, 2020, 5:34pm

Magnus. Good. We’re still on the same page.

I thought it was an interesting argument that there are more numbers than decimals; however, anytime you use a ratio, decimals are implied. Can you contradict this?

Silhouette and I have provided a few compelling arguments already that you haven’t addressed as to WHY 2*infinity is a contradiction.

You keep stating it, but you avoided our proofs through contradiction.

Maybe you think there’s a backdoor that circumvents our “proofs”, that’s fine, but I haven’t seen it yet.

I personally used the “cheat” argument and the “2 gods” argument and the “quantum flux” argument.

I can let silhouette speak for his own arguments, the “straw/box” argument of his is really good.

Magnus_Anderson · January 6, 2020, 6:10pm

There is no decimal representation of (\frac{1}{3}), for example. (0.333\dots) isn’t it. That’s a different number.

If you’re asking for a rigorous proof, I don’t have one. And I don’t think it’s necessary. You can’t demand of me to prove more than it’s necessary to prove in order to show that (0.999\dotso) does not equal (1). I sometimes do so, but it’s only because it’s fun, not because it’s necessary.

On the other hand, since you are the one claiming that the set of decimal numbers is the set of all numbers, it’s up to you to prove it.

You provided no compelling arguments, no proofs. You merely defined the word “infinity” in your own way and used the logical implications of such a concept to “prove” that I am wrong even though that’s not the way I define the word “infinity”. (In fact, that’s not the standard definition of the word.)

The “straw/box” argument merely shows that he does not understand the simple fact that there are no inherently correct definitions of words.

Ecmandu · January 6, 2020, 6:42pm

So here’s the deal Magnus !

You stated outright that you don’t believe that 1=0.999…

So what are we arguing about?

You don’t believe in convergence theory and convergence theory is necessary to quantify infinity.

You know? There’s a big difference between infinity and infinite sequences.

1/3 * 3 equals nine in base 10; 10/3 equals 3*3 which equals 9 to get back to ten.

So… really… we have an operator trick.

Now, if as you state, infinity doesn’t converge, then there are no orders of infinity that can be used, because as non convergence infinity ceases to be applied as a unit of measurement which is subject to operators such as additions and powers.