## 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...
10
33%
No, 1 ≠ 0.999...
14
47%
Other
6
20%

Total votes : 30

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

Promethean has no understanding of the finer points of arythmetic.

and barely even the cruder points of arythmetic as well. i told ya i used to huff spray paint after school everyday in the eighth grade, right? i probably destroyed more brain cells in that year alone than i would throughout the rest of my life. but the shit you see while high on inhalents is something you'll never regret. this one time me and matt were in the back yard and we saw this cloud turn into the starship enterprise and engulf a helicopter. pulled it right in with a tractor beam.

but check this out. when the old man put me into the looney bin ('holly hill'), i had to take an entry I.Q. test. thirteen years old and determined NOT to cooperate. i didn't even try. 123. now that's nothing spectacular, but you gotta remember i had a brain saturated with spray paint. what got me was the math. if i'da known anything about that math, i'm sure i woulda broke a buck thirty at least. i mean math's a big chunk of that test, bro.

then i took another one a few years back at the processing camp before they put me in medium security. scored 118 on that one. fucking math again. i swear to god if it weren't for math i'd be a near genius. so, i've spent my entire life compensating for this failure and shame by producing philosophies that prove numbers are dumb. hey, you gotta do what you gotta do, right? think i'm gonna sit here and let you nerdy fucks make me look like a dummy? yeah i got your numbers, buddy. i got your numbers hangin.
promethean75
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### Re: Is 1 = 0.999... ? Really?

but i didn't even tell you about the special powers it gave me. basically what happened was the composite chemical compounds of the paint formed a symbiotic bond with the membranes of my psuedounipolar neurons... essentially preserving them as well as enhancing the conductive element of the axon fibers. which means, positively charged ions that travel across this membrane have a much slower decay rate. and this is why i can see much further into things... why i can process philosophical information at much greater speeds. you could say that the paint ('rustoleum') did to my brain what the adamantium did to wolverine's skeletal structure. well with the exception of the regions responsible for processing mathematical information. but that was a sacrifice i had to make to gain these powers.
promethean75
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### Re: Is 1 = 0.999... ? Really?

obsrvr524 wrote:
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. More at "Wikipedia"

Listing with counting numbers is a demonstrative way to prove 1:1 correspondence for an infinite number of sets.

We’re splitting hairs here.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
Certainly, infinity is not an ordinal number. There is no such thing as infinity-th position. But I don't see a problem with treating it as a cardinal number.
If you have the series 1+1+1+...

then the "last" 1 is at the infinity-th position and total "value" is infinity.

How can it be otherwise?

There is no "last" number. That is where "endless" comes into this. And that is also why maths don't work with infinity. The only thing you know about the value is that it is higher than all numbers, which actually means that it doesn't exist except as a direction.

Ecmandu wrote:Listing with counting numbers is a demonstrative way to prove 1:1 correspondence for an infinite number of sets..

It doesn't work that way. You have to directly relate each item in one set to a unique item in another set with nothing left over. There are only two sets. Counting them doesn't mean anything.
You have been observed.
obsrvr524

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

The counting implies the correspondence without ambiguity.

Like I said, were splitting hairs here.

Counting makes it clearer than an infinity number of sets are in correspondence, meaning that an infinite number of sets can be bijected.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

I'll leave you in your bubble.
You have been observed.
obsrvr524

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

obsrvr524 wrote:I'll leave you in your bubble.

It’s implied. The problem you have with correspondence working on more than 1 list is that it refutes your argument.

I hate to be the one to break it to you:

1:1 Correspondence can work for an infinite number of sets.

You’re the one in the bubble here. I’m trying to pull you out of it.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:If you're saying that my calculations are influenced by my image, you're wrong. The image is nothing but a reflection of my calculations. It's supposed to show how I'm doing my calculations (and it is implied that the way I'm doing them is the way they should be done.)

The first element of the purple rectangle is $$0.9$$ and it's constructed from the second element of the green rectangle which is $$0.09$$. It is not constructed from the first element of the green rectangle. That's why the purple rectangle does not start at the same place as the green rectangle.

The way in which you're doing the calculations of summation is the way it should be done.
The way in which you're justifying the set-correspondence is a different calculation to this summation, and only seemingly connected by your "reflection of your calculations" via your image. Consider this example:

I have two sets, A and B, which each contain one instance of each integer from 0 upwards, in ascending order $$\{0,1,2,3,...\}$$.
I state however that B can be constructed from A by first inserting 0 to start the list, and consequently populating the rest of B by adding 1 to each element in A.
This construction of B from A is entirely possible, but for each element in the list, there is one-to-one correspondence between the identical sets. The implication that B has an extra element because 0 was added to it separately to the construction of the rest is inconsequential, because for every element in A there is still an identical element in B - literally forever and with no gaps. You never get to any un-matching element in B from its very start and onward forever - it is impossible for there to be one, since the two sets were identical to begin with. It was only the alternative construction of B from A that gave the "appearance" that correspondence was skewed from being one-to-one. Superficial.

In exactly the same way, the working you do to get from Green to Purple is entirely valid summation in itself, and yet this construction goes nowhere to changing the objective bijection between the two sets, with element for element identity from the start and indefinitely onward from there.

Magnus Anderson wrote:
Silhouette wrote:There is no "product itself" of a divergent infinite product - you never get there.

There is. The fact that the product never ends does not mean it does not exist. (Indeed, I can use Gib's argument against you: infinite sums have no temporal dimension, so they can be considered complete.)

What's true is that there may be no finite number equal to an infinite product. In the case of $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots$$, there is no finite number equal to it. It has a limit, which is $$0$$, but that's not the same thing as its result (its result being greater than its limit.)

Infinite sums being "complete" refers to there being no gaps in the set, in the sense that each jump between one element and the next is defined from the representation itself to be constant throughout.
This doesn't mean that any "product" doesn't always have an extra element after it, or to add to it etc. An infinite set can be completely defined, but it still has no product because there's always more than any product. Any "product" is therefore undefined, it's infinite, it doesn't "equal" its limit - this is something I thought you already indicated you had grasped...

Magnus Anderson wrote:
Silhouette wrote:$$1+\infty$$ is almost literally "more boundless" (by the finite quantity of "one").
Something can't be more boundless than boundless, so it's boundless whatever finite thing you do to it along its boundlessness.

Quantities are either finite or they are infinite. There are no degrees. That's where we agree. Where we disagree is that $$\infty + 1$$ means "more infinite". It does not. What it means is "larger infinite quantity".

You're trying to tell me that "larger infinite quantity" does not mean "more infinite"?
"Larger" is a synonym with "more" (or it is at least a type of "more"), and you yourself clarified by "infinite", you mean "infinite quantity".
Is your distinction therefore between the "type" of "moreness" not having anything to do with being "larger"? Adding 1 deals with (finite) quantity, so both "larger" and "more" apply to the specific example you're talking about. Basically nothing in your words distinguishes between "more infinite" and "larger infinite quantity".

The question is whether the output as a whole is distinguishable from the previous input states. I can watch an apple and a banana be put into the same basket and ask someone else who didn't watch the process to describe the product, and they'd be able to accurately describe it as an apple added to a banana. No information is lost upon the completion of the operation.
If you asked the same of someone adding an apple to an infinite line of apples, someone else who didn't know the input process would only recognise the result as an infinite line of apples, and could only guess at any operation previously performed. You yourself can only know the result as "an infinite line of apples plus 1 apple" as a result of your memory of a past state - as opposed to the current state.
It's perhaps a bit of an advanced concept, but the act of operating on an infinite with a finite results in maximum information entropy - with all information guaranteed to be lost. The infinity "added to by 1" already has maximum entropy, though operating on it with the finite quantity of 1 as a separate input at least contains some information that you'd expect to result in a certain outcome - if only the operation of adding 1 wasn't to an infinite quantity. Mixing the two completely eliminates the certainty of the finite operand, and treating the two separately in the result like you would an apple and a banana, which you can't mistake from one another, only serves to hide the fact that you absolutely can mistake infinity from any addition of 1 to it. We're dealing with information here, I'm sure you'll agree, so the information entropy of what you're proposing cannot be disregarded.

Magnus Anderson wrote:
Silhouette wrote:That's why adding one results in boundlessness both before and after you do it

The truth value of the statement $$\infty + 1 = \infty$$ depends on its meaning.

If what we mean is "Some infinite quantity X + 1 = Some infinite quantity that is not necessarily equal to X" then the statement is true.

If what we mean is "Some infinite quantity X + 1 = The same infinite quantity X" then it is false.

So as above, the problem is not only that the equality or inequality of "X+1" and "X" is indeterminable, undefined and undefinable when X represents an infinity. "X" isn't even determinable as different from any other "X" when X represents an infinity, because in in each case the quantity is undefined - no matter how well defined the finites are in its construction or representation, the infinity in the construction or representation is also undefined. There is no "the same infinite quantity X" because neither X is defined. This is why you have to revert back to "Some infinite quantity X + 1 = Some infinite quantity that is not necessarily equal to X", therefore $$\infty + 1 = \infty$$ is true.

Magnus Anderson wrote:
Silhouette wrote:It's literally impossible to test whether you added, took away or did whatever bounded thing to a boundless length after you've done it and are therefore able to make an equation about it. You can only validly comment on what you did with the finite quantity of 1 apple before it was absorbed into the boundless non-finite mass, you cannot validly comment on that resulting infinity that stays infinite in the same and only way that infinity can be infinite. So with no change in the result, there is therefore no valid equation or statement to make about the result as changed.

It seems to me that like Gib you're not able to distinguish between conceptual and empirical matters.

Gib keeps asking questions such as "How can we empirically determine (specifically, through direct observation) whether any two infinitely long physical objects are equal in length or not?"

The question is irrelevant and it is so precisely because it's empirical and not conceptual.

We're talking about concepts here, and to concepts we should stick. A conceptual matter cannot be resolved empirically.

You keep jumping to the conclusion of inability when people disagree with you.

Whether empirical or conceptual is a matter of Epistemology either way: you are championing the conceptual over the empirical in order to gain knowledge from what you're conceptually attempting. This goes as far as to present situations that seem like they ought to be reasonable, but whether or not they actually are reasonable requires testing. You aren't gaining knowledge from presenting situations conceptually until they've been examined. It turns out that when one tests what you judge ought to be conceptually viable, it can't be done.
This is not insignificant and certainly isn't a mistake by those who offer empirical testing to conceptual matters. Most of my objections are actually logical ones, and occasionally I back them up with empirical objections - either way, in order for you to forward your conceptual propositions as knowledge, they need to be testable and tested. They fail both logically and empirically - both of which are valid approaches to evaluating the conceptual.

Magnus Anderson wrote:
Silhouette wrote:Your error is to say that since 1=0 is a contradiction, we ought to be able to treat infinites alongside finites as though they were compatible.

That's not what I'm saying.

The contradiction that you speak of arises as a result of not understanding the implications of the concept of infinity that is being employed.

I've previously said that $$\infty + 1 = \infty$$ is true. There is no doubt about it. But that's only the case if the symbol $$\infty$$ means "some infinite quantity, not necessarily the same as the one represented by the same symbol elsewhere". In such a case, you can't subtract $$\infty$$ from both sides and get $$1 = 0$$. This is because $$\infty - \infty$$ does not equal $$0$$ given that what that expression means is "Take some infinity quantity and subtract from it some other infinite quantity that is not necessarily equal to it". If the two symbols do not necessarily represent one and the same quantity, then the difference between them is not necessarily $$0$$.

Some progress - this is what I've been saying all this time.

Magnus Anderson wrote:But that's PRECISELY what mathematicians do when they try to prove that $$0.\dot9 = 1$$. There is literally no difference between people proving that $$0 = 1$$ by subtracting $$\infty$$ from both sides of the obviously true equation $$\infty + 1 = \infty$$ and various Wikipedia proofs that $$0.\dot9 = 1$$ except that it's much easier to see that the former conclusion is nonsense and that the proof must be invalid.

The terms that I've brought up before of "divergent" and "convergent" make a difference here.

If we compare the two similar looking infinite sums of $$\sum_{i=1}^\infty\frac9{10_i}$$ and $$\sum_{i=1}^\infty\frac9{10^i}$$, the former indefinitely increases linearly with no finite limit and the latter indefinitely increases harmonically with a finite limit (of 1). Both are "complete" in that their progressions are constant and regular so we can project their trajectories very clearly, but while both series have infinitely many terms, the former also tends to infinity whilst the latter tends to finity. This difference is why convergent series are useful for things like calculus, and divergent series are not. We can surmise logically from their respective trajectories that:
1) the former tends to any arbitrarily large infinite quantity, for any arbitrarily large infinite quantity of iterations, that cannot be said to equal any other infinite quantity with an infinite margin for error, and
2) the latter tends towards no other number than 1 (the limit that it never actually gets to) though the margin for error for any arbitrarily large infinite quantity of iterations tends to no other number than zero (any number than zero can be divided smaller, so logically it can only be zero).
With infinite margin for error to to subtract an indefinite number of corresponding digits (or elements in a set), versus logically zero margin for error to accompany the problem of subtracting an indefinite number of decimal places (or elements in a set), we can accept the latter but not the former.

Magnus Anderson wrote:
Silhouette wrote:Sure, if saying what something "isn't" gives it a meaning about what it "is".
You don't seem to understand the extent to which "ends" apply when it comes to definition.
Full definitions that include what something "is" as well as what it "isn't" are separating what it "is" from what it "isn't" by a bound (i.e. an "end") that's as clear as possible.

To define some symbol S is to verbally (or non-verbally) describe the meaning of that symbol S. The usual aim is to communicate to others what meaning is assigned to the symbol.

In other words, the meaning of a symbol precedes its verbal (or non-verbal) representation (which is what definitions are.) They are superficial things, very much in the Freudian "tip of the iceberg" sense.

The meaning of a word does not have to be described in order for it to have a meaning. This means the word "infinity" is a meaningful word so as long there is some kind of meaning assigned to it regardless of what kind and how many descriptions of its meaning exist.

You can describe the meaning of words any way you want, so as long your descriptions do the job they are supposed to do (which is to help others understand what your words mean.) If you can define words using "is not" statements and get the job done, then your definitions are good.

Using a transistor "NOT" gate as an analogy, let's call our input the word/symbol "infinite" and any output is any meaning we understand.
If we supply our input into the circuit, the transistor "switch" is turned on (allowing current to run through it) and the power supply and everything just runs straight to ground with no output provided.
If we turn our input "infinite" off, the transistor no longer completes the circuit between power and the ground, and power can now only flow out the output.
In both cases the input "infinite" is never connected to the output "meaning". The only way we can get an output is to turn "infinite" off and the meaning we understand is just the default: "NOT" that input. We're recognising and processing the word by throwing it in the bin and switching to what we do know which is the opposite of that i.e. "finite".

It's just an analogy, of course, but our brain circuits - just like our dictionary definitions - are "getting the job done" by defining something else than "infinite" and just saying "not that". Describe that input "any way you want" just as you say. Any meaning we interpret is what's assigned to "not that" i.e. the meaning that "finite" has for us. The conflation of this with "infinity is a meaningful word" is that Freudian "tip of the iceberg" that you were talking about - it's the recognition of the symbol, verbal or non-verbal representation only. The shape/sound of the signifier is defined for us, but the meaning we get from detecting it is the meaning of its opposite, not the meaning of the signified.

I'm just saying don't conflate.

Magnus Anderson wrote:
Silhouette wrote:The reason that expressions that include e.g. division by zero are undefined is because you can't clearly bound what it "is" from what it "isn't", since in this case any answer is no more valid than any other:

$$\frac{1}{0}$$ has no valid answer (not a single one) since there is no number that gives you $$1$$ when you multiply it by $$0$$.

The limit of $$\frac1{0}$$ as the denominator tends towards zero is infinity: it's undefined. There's no way to clearly bound exactly what it "is" from what it "isn't". Like I just explained, you know it doesn't work for anything finite, and it's only to that extent that you can know its infinitude. Obviously there's no finite number that gives you $$1$$ when you multiply it by $$0$$, and even extending these finite numbers as far as you can conceive (which is only for finites, by definition) it makes no sense how extending it even further than that could give you $$1$$. All we can know for sure is what happens when the finite limit of the denominator tends towards zero - and that can't be pinpointed to tend towards anything.

Magnus Anderson wrote:Infinite series and limits is what they taught you at school (and thus what you're familiar with) but they are not relevant to the subject at hand.

Not seeing the relevance to the subject at hand is a result of you needing to go to school.
I'll just leave that non-comment there to match the non-comment content of your baseless and unelaborated quip.
It's adorable that you think I've not continued my learning and critical thinking way beyond school level - I can't wait to see all this next-level stuff that you've obviously been holding back from the discussion so far. So far it's just been weak as shit childlike speculation, personal incredulity and your denial of any need to prove it better than that kind of level. It's not surprising that you have such a negative attitude to what people learn at school, because this unworkable attitude that you have against learning like people do at school. I loved school, and learning - I still love learning and can't wait for anyone to show me something I've not learned or thought of before. I love being a student. You look down on that as though you never learned to learn a thing. Where's that getting you, exactly? Do you feel competent and confident with your closed mind?

Magnus Anderson wrote:
Silhouette wrote:See how you're only going for one side of the "undefined" and concluding that the other side therefore doesn't exist?
$$0.\dot01$$ never gets to $$0$$ also means no gap can ever come into existence between $$0.\dot01$$ and $$0$$.

That's not true. It simply means the gap cannot be represented using one of those numbers you can understand.

$$0.\dot01$$ must attain $$0$$ in order for there to be no gap.

You saying there ought to be some other number/concept/anything without being able to say anything about it doesn't make it true.

I'll grant you anything at all to fill this "gap" and there can always be something smaller, and $$0.\dot01$$ will still be a contradiction. I'm waiting for substance here beyond "there must be something" based on that contradiction. Nothing can be close enough to zero to suffice here, numbers or otherwise, and logically only 0 cannot be divided smaller and thus satisfy this "gap" - which just so happens to be the limit that it must tend towards even if it can't literally get there. The only thing you have against 0 is incredulity. You need more!!! I'm sorry, that's just how Epistemology works.

Magnus Anderson wrote:
Silhouette wrote:Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.

This is the problematic statement. It's a non-sequitir.

A non-existent gap would be a gap that has zero size. The size of an infinitely small gap is greater than zero -- by definition. No partial product of $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$ is equal to $$0$$. Note that this is different from $$0 \times 0 \times 0 \times \cdots$$ where every partial product is equal to $$0$$. Both are infinite, never-ending, products but one evaluates to a finite number that is $$0$$ (is equal to it) whereas the other doesn't evaluate to any finite number and is provably greater than $$0$$.

*Sequitur. An infinitely small gap must be even smaller than that, if it has any size at all, in order to satisfy being small enough. If it has no size at all, it is indistinguishable from zero. Logically only zero can fit the bill by being literally indivisible. If only zero can be small enough for this gap, then there's no gap. Partial products are all fair and lovely, but they're only partial. It's only by being infinite that this can happen as for finites only products of at least one zero would fit the bill. But we're not talking merely partial products or finite progressions are we - as I keep saying and as you keep denying with one hand while you appeal to treatment as/like finites with the other. If it's impossible to prove the "gap" is greater than zero, then there's no grounds to say there's any gap nor to distinguish it from zero in any possible way. It can therefore be treated as 0 and all the other proofs fall into place like a complete jigsaw. Give yourself a break.

obsrvr524 wrote:
Silhouette wrote:$$0.\dot0{1}$$ never gets to $$0$$ also means no gap can ever come into existence between $$0.\dot0{1}$$ and $$0$$.

The undefinability of any "gap" means there's no grounds to distinguish $$0.\dot9$$ from $$1$$, nor $$0.\dot0{1}$$ from $$0$$

Neither of those claims make any sense to me. So you haven't proven anything to me. Why would your first claim be true?

Silhouette wrote:Saying there's always a smaller gap, therefore there always is "a" gap that never fully vanishes, is no different from saying there's never any point at which the gap can be said to come into existence.

That one doesn't make any sense either.

Fortunately making sense to you isn't a necessary criteria for such things to have already been proven, but I can try to convince you all the same if you're actually open to the possibility. Magnus isn't, are you?

The first claim is rephrasing one side of the argument as the other side.
$$0.\dot0{1}$$ is a contradiction, but even if what it's trying to represent wasn't a "fully bounded boundlessness", it's a quantity that is always smaller than it is: it always has to be divided more to suffice as small enough. Logically the only quantity that is small enough to not be divisible further is zero, which not incidentally is the limit of what this gap has to be. So even though we can't get to the limit, we can confirm through logic that the answer is in fact zero, as though we actually could get there by means of the limit.

Therefore, inasfar as one way of thinking can say there's always a gap because the limit never reaches 0, the other way of thinking shows that such a gap could never be small enough to come into any existence.
It was an attempt to get you guys to try and see both sides of the argument and not just your own - so it's fair enough if that kind of thinking didn't make sense. Maybe it was just my wording and you are actually able to see both sides, in which case the explanation I just gave should have straightened things out.

obsrvr524 wrote:
Silhouette wrote:"They" have a proof that if $$0\times{n}=0$$, "n" can be any number you want since all numbers multiplied by zero equal zero. Therefore dividing both sides by 0 to get $$\frac0{0}$$ it's not $$\infty$$ that this equals, but "undefined".

Except that ∞, by definition is NOT a number ("n").

So the "proof" using n/o is invalid.

If $$\infty$$ is not a number then you can't equate it with the number $$\frac0{0}$$
If it is a number, then $$0\times{n}=0$$ is any possible number.
Take your pick - either way the answer is undefined/invalid.

Ecmandu wrote:The reason Silhouette sees no difference between .000...1 and zero is because the 1 in 0.000...1 is never arrived at. It’s ALWAYS zero!

So you are saying that you never get to the end??

And that would mean that there is always a "9", never a "0" in 0.999..., wouldn't it.[/quote]
You never get to the end of a limit, but as I just explained above, logically the answer is the limit anyway.

And yes, there's always another "9" and never a "0" (after the decimal point) in $$0.\dot9$$ never allowing any gap to emerge between it and $$1.\dot0$$.
The whole issue for non-mathematicians is that they don't look the same because they're represented with different digits. Turns out it's true despite appearances - is that so very hard to accept?
Last edited by Silhouette on Wed Jan 15, 2020 6:18 pm, edited 1 time in total.

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

I’m still paying attention Silhouette.

We’re going to have this debate after Magnus “suffocates” from his order of infinity argument.

This turned out to be a dual thread.
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

So Magnus...

Here’s my argument:

The “highest order of infinity”, is my cheat:

1.) rational number
2.) uncounted infinity
3.) different rational number
4.) different uncounted infinity
Etc...

I’m not listing the numbers here, just the concepts.

When you divide the interpolated orders of numbers, you get:

1.) rational number
2.) 0
3.) rational number
4.) 0

Etc...

Also!

1.) 0
2.) uncounted infinity
3.) 0
4.) uncounted infinity

Etc...

No matter how many times you divide it, they are still in correspondence. Mathematicians use abstractions to make their proofs. This is a symbolic abstraction (I’m not listing numbers themselves). Symbolic abstractions are fair game in mathematics!

My other argument against you is the dimensionality argument.

You claim planes are infinitely larger than lines.

That’s not true.

A 2D image can not only appear to be moving (3D), but can also render 3D images. Unary can do the same for 2D images. I can link some of these “optical illusions” if you like.

What this means is that a plane is not bigger than a line.

Those are my two arguments against orders of infinity.

Once you “cry uncle” submit, I can start debating Silhouette on 0.999... /= 1
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:If you have the series 1+1+1+...

then the "last" 1 is at the infinity-th position and total "value" is infinity.

How can it be otherwise?

You are probably right.

$$N = \{1, 2, 3, \dotso\}$$

Sets have no order but we can take a set and arrange its elements in a sequence.

$$(1, 2, 3, \dotso)$$

This is a one-directional sequence that has a beginning but no end. So here, there's no last element, and hence, no infinity-th element.

But this isn't the only kind of sequence there is. Here's a one-directional sequence that has an end but no beginning.

$$(\dotso, 3, 2, 1)$$

There is no first element but there is a last element and this last element occupies infinity-th position in the sequence.

How about a two-directional sequence that has both a beginning and an end?

$$(1, 3, 5, \dotso, 6, 4, 2)$$

First, second, third, etc. But also infinity-th, (infinity - 1)-th, (infinity - 2)-th, etc.

The position that the number $$2$$ occupies in this sequence is an infinite number of positions away from the first position, so we say it occupies infinity-th position.

What I believe this illustrates is that an end in one regard is not an end in all regards. Every single one of these sequences is an infinite one in the sence that it is made out of an infinite number of elements (the number of elements has no end) but some of them have an end in the sense that there's a last element in the sequence and some don't.

So yeah, I'd say that infinity can also be used as an ordinal number.

But what about $$1 + 1 + 1 + \cdots$$? Does it have an end? The visual representation suggests that it does not.
Last edited by Magnus Anderson on Wed Jan 15, 2020 7:29 pm, edited 1 time in total.

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

I don't like that this thread is making me want to sniff paint.

Enough excuses! With philosophy it's "it's all pointless," with chess it's "I don't have time to practice," with math it's "I sniffed glue."

I was frankly gifted at math in school. As the topics got more complex though and teachers lost their ability to actually understand them, so that it increasingly became about memorizing operations, I began to loose interest.

I'm honestly grateful. There is an internal logic to math that has an incredible pull that can engulf a mathematicion more perditionously than WoW will a nerd. Without ever knowing what math is, or why it pulls. Questions...

ONLY PHILOSOPHY CAN ANSWER

Pedro I Rengel
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### Re: Is 1 = 0.999... ? Really?

Magnus you’re playing disingenuous games with infinity here:

1.) you use it as a solid object when it suits you
2.) you use it as a procedure when it suits you

0.0 ...123 is a PROCEDURE, yet, I’ve heard you several times in this thread (for different arguments) define infinity as only an object!

You need to make up your mind! It’s impossible to argue someone switching back and forth, especially, if they don’t acknowledge it!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Pedro I Rengel wrote:I don't like that this thread is making me want to sniff paint.

Enough excuses! With philosophy it's "it's all pointless," with chess it's "I don't have time to practice," with math it's "I sniffed glue."

I was frankly gifted at math in school. As the topics got more complex though and teachers lost their ability to actually understand them, so that it increasingly became about memorizing operations, I began to loose interest.

I'm honestly grateful. There is an internal logic to math that has an incredible pull that can engulf a mathematicion more perditionously than WoW will a nerd. Without ever knowing what math is, or why it pulls. Questions...

ONLY PHILOSOPHY CAN ANSWER

Let's be clear on one thing:

Philosophy is a radically new thing, which began in Greece. The philosophical texts we have from that time are more sporadic reflections of it than philosophy itself.

Before Greece NOTHING LIKE IT EXISTED.

It is unlike and stands appart from all other things. It is incompatible with any other form of thinking.

Deleuze's great crime was spiritualizing philosophy. Ain't nothing spiritual about it. Baudrillard knew this, and found a way out.

He duped, co-opted Guattari. Guattari was 0 spiritual by hos own path. That's what drew him in about Marx, his rigorous pseudo-science. He was all Das Kapital, 0 Communist Manifesto. But that's how they do it don't it? They give ypu the rigour and then draw you into the spiritual...

There is nothing natural, or inevitable, or universal, about philosophy.

I knew Capable was a serious philosopher when he dedicated an aphorism to his suspicion of adjectives. He did so not inspired by, but parallel to Nietzsche. That's a philosopher.

There is nothing philosophy is for, and philosophy is for nothing.

But we will still conquer the fucking world for it.

Pedro I Rengel
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:The whole issue for non-mathematicians is that they don't look the same because they're represented with different digits. Turns out it's true despite appearances - is that so very hard to accept?

A similar argument can be made against you (and your ilk.) Namely, the reason you don't believe that infinities come in different sizes is because when you take $$1 + 1 + 1 + \cdots$$ and add $$1$$ to it, you get confused by the result because it looks like what you started with -- which is $$1 + 1 + 1 + \cdots$$. Failing to understand that you're dealing with one and the same symbol representing two different things, you conclude -- erroneously so -- that all infinities are equal. Deceived by the appearances, the accuser is revealed as being guilty of their own accusations (: Accuse the other before they accuse you. Is that a motto of yours?

it's a quantity that is always smaller than it is

$$0.\dot01$$ is not a quantity smaller than "it is". It's not a quantity smaller than itself.

An infinitely small gap must be even smaller than that, if it has any size at all, in order to satisfy being small enough.

We're talking about $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. Every partial product is greater than zero, thus, the result of the infinite product is greater than zero.

But we're not talking merely partial products

We're talking about every partial product of $$0.\dot01$$. Not merely a subset. And each one of the infinitely many partial products is greater than $$0$$. That's how you know the result of the sum is greater than $$0$$.

I'm waiting for substance here beyond "there must be something" based on that contradiction.

It'e enough to prove that the gap exists. It's unnecessary to know its exact value.

I still love learning and can't wait for anyone to show me something I've not learned or thought of before

There are people who talk the talk and then there are people who walk the walk.

Whether empirical or conceptual is a matter of Epistemology either way

Nothing to do with epistemology.

The question is whether the output as a whole is distinguishable from the previous input states. I can watch an apple and a banana be put into the same basket and ask someone else who didn't watch the process to describe the product, and they'd be able to accurately describe it as an apple added to a banana. No information is lost upon the completion of the operation.
If you asked the same of someone adding an apple to an infinite line of apples, someone else who didn't know the input process would only recognise the result as an infinite line of apples, and could only guess at any operation previously performed. You yourself can only know the result as "an infinite line of apples plus 1 apple" as a result of your memory of a past state - as opposed to the current state.

That's why I'm saying that you (and Gib) are conflating empirical matters with conceptual ones.

The question is not an empirical one. We are not asking "Given two infinitely long pipes, how can we determine whether they are equal or not?" We are asking "If there's an infinite number of green apples in front of us and we add a red apple at the start of the line, and nothing else changes, is the number of apples greater than it perviously was or is it the same?"

We don't care about the poor guy who knows nothing about the situation. Why do you insist on listening to what a clueless guy has to say? Of course, since he's clueless, he'll tell you the line didn't change. But that doesn't make it true.

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

Philosophy is a radically new thing, which began in Greece.

"with the exception of the principles of mathematics which i will develop, the philosophical nonsense i learned and adopted from the egyptians will no doubt be attributed to me by me fellow countrymen, and yet i will not let this prevent me from expanding it further into realms of nonsense so immense as to be incomprehensible by modern men lest they be educated by plato, another greek space cadet who is about to go viral." - pythagoras, 'the lost chronicles'

okay that was funny.

i don't know how many different ways i can say this my dude, but the stuff that the word 'philo sophia' means and represents did not begin with the greeks nor does it belong to them. the word does, but not the stuff. all this eurocentric bias in the western world about the 'fathers of philosophy' is nothing more than a fetishsized trend that exists because it has no competition... because what philosophical speculation by other people that did exist in writing didn't survive the centuries like the greek shit did. and you don't need to be a historian to realize this. all you need is the most basic of anthropological insight to deduce that any ancestor after the neanderthal and with a 'modern' brain, was capable of asking stupid questions just like the greeks.

you'll understand eventually. in the next thousand years or so the human species will undergo neurological changes that will, literally, prevent them from asking those same stupid questions. we're starting to see this new, improved human being appear even today. we call them nihilists, which is code for 'wtf are you talking about? are you high?'

here's a suggestion for ya. it could very well be the case that philosophical questions which haven't been answered for two thousand years might not be a real questions. just a guess, though.
promethean75
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### Re: Is 1 = 0.999... ? Really?

A similar argument can be made against you (and your ilk.) Namely, the reason you don't believe that infinities come in different sizes is because when you take 1+1+1+⋯ and add 1 to it, you get confused by the result because it looks like what you started with -- which is 1+1+1+⋯. Failing to understand that you're dealing with one and the same symbol representing two different things, you conclude -- erroneously so -- that all infinities are equal.
I don't know why you say that you agree with
$$\infty+1=\infty$$
when you really only agree with
$$\infty_0+1=\infty_1$$

Who knows what $$\infty_0$$ amounts to?
Last edited by phyllo on Wed Jan 15, 2020 9:15 pm, edited 1 time in total.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Enough excuses! With philosophy it's "it's all pointless," with chess it's "I don't have time to practice," with math it's "I sniffed glue."

and that's just the excuses i come up with for not doing things i'm not good at. you ain't even seen the excuses i come up with for not doing shit i am good at.
promethean75
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:I have two sets, A and B, which each contain one instance of each integer from 0 upwards, in ascending order $$\{0,1,2,3,...\}$$.

Okay, so we have two identical infinite sets $$A = \{0, 1, 2, 3, \dotso\}$$ and $$B = \{0, 1, 2, 3, \dotso\}$$.

I state however that B can be constructed from A by first inserting 0 to start the list, and consequently populating the rest of B by adding 1 to each element in A.

In which case we have $$B' = \{0, 0_{+1}, 1_{+1}, 2_{+1}, 3_{+1}, \dotso\}$$ which has more elements than $$B$$.

This construction of B from A is entirely possible, but for each element in the list, there is one-to-one correspondence between the identical sets.

And that's not the case.

The implication that B has an extra element because 0 was added to it separately to the construction of the rest is inconsequential, because for every element in A there is still an identical element in B - literally forever and with no gaps.

Not true. There is no one-to-one correspondence between the two sets. What you're doing here is called magic trick.

$$A = \{0, 1, 2, 3, \dotso\}$$
$$B = \{0, 1, 2, 3, \dotso\}$$
$$B' = A \cup \{0_{new}\} = \{0_{new}, 0, 1, 2, 3, \dotso\}$$

You never get to any un-matching element in B from its very start and onward forever

You do.

It was only the alternative construction of B from A that gave the "appearance" that correspondence was skewed from being one-to-one. Superficial.

Actually, it is what you're doing (i.e. listening to Cantor) that is giving the appearance of one-to-one correspondence.
Last edited by Magnus Anderson on Wed Jan 15, 2020 9:28 pm, edited 1 time in total.

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

phyllo wrote:I don't know why you say that you agree with
$$\infty+1=\infty$$

That's true but that does not mean that infinities are equal.

It's true in the sense that if you have an infinite number and you add 1 to it, you get an infinite number as a result (not necessarily the same infinite number, but still an infinite number i.e. not a finite number.)

On the other hand, if $$\infty$$ represents one and the same infinite quantity on both sides (and wherever else it appears), the expression is false.

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

Lol @ eurocentric bias. Alright there Greta.

Don't break your back bending over backwards like that. For them there explanatuons.

Pedro I Rengel
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### Re: Is 1 = 0.999... ? Really?

promethean75 wrote:
Enough excuses! With philosophy it's "it's all pointless," with chess it's "I don't have time to practice," with math it's "I sniffed glue."

and that's just the excuses i come up with for not doing things i'm not good at. you ain't even seen the excuses i come up with for not doing shit i am good at.

Ha ha
yeah Im still waiting to hear those.
The strong do what they can, the weak accept what they must.
- Thucydides

BTL

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

Magnus Anderson wrote:
Silhouette wrote:The whole issue for non-mathematicians is that they don't look the same because they're represented with different digits. Turns out it's true despite appearances - is that so very hard to accept?

A similar argument can be made against you (and your ilk.) Namely, the reason you don't believe that infinities come in different sizes is because when you take $$1 + 1 + 1 + \cdots$$ and add $$1$$ to it, you get confused by the result because it looks like what you started with -- which is $$1 + 1 + 1 + \cdots$$. Failing to understand that you're dealing with one and the same symbol representing two different things, you conclude -- erroneously so -- that all infinities are equal. Deceived by the appearances, the accuser is revealed as being guilty of their own accusations (: Accuse the other before they accuse you. Is that a motto of yours?

"And my ilk" here comes the segregation by dissociation language for easier demonisation and premature dismissal...

I don't have any mottos, no. It's ironic that you were falsely accusing me of assertion rather than explanation before I pointed out your hypocrisy, and now you're accusing me of "accusing the other before they accuse you"
But it's been projection of your insecurities onto others from the very beginning - very human of you.

I've repeatedly denied the equality of infinities - and you conclude that I conclude that infinities are equal.

Your symptoms of irrationality are really shining through now, if they were merely prevalent before - they're brazen now.

You're not only getting deceived by appearances but even hallucinations now!

Magnus Anderson wrote:
Silhouette wrote:it's a quantity that is always smaller than it is

$$0.\dot01$$ is not a quantity smaller than "it is". It's not a quantity smaller than itself.

Again, the point goes straight over your head...

It has to be a quantity smaller than itself to be valid, which is invalid, making it invalid either way. It's a contradiction in notation either way - it's logically not a possible quantity.

So even this contradictory "quantity" is too big to fill the gap between $$1$$ and $$0.\dot9$$ - any possible concept is too big. It literally defies its own existence as soon as any existence is proposed. Zero possible existence resolves to non-existence and therefore no gap. It's very simple, but simple logic seems a little beyond your skills.

Magnus Anderson wrote:
Silhouette wrote:An infinitely small gap must be even smaller than that, if it has any size at all, in order to satisfy being small enough.

We're talking about $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. Every partial product is greater than zero, thus, the result of the infinite product is greater than zero.

Every partial product is greater than zero, yes - because it's only partial! Thus the same logic carried over to any "non-partial-product" of some kind of "infinite product" is invalid. Again - poor logic skills on your part.

Magnus Anderson wrote:We're talking about every partial product of $$0.\dot01$$. Not merely a subset. And each one of the infinitely many partial products is greater than $$0$$. That's how you know the result of the sum is greater than $$0$$.

Partial products of this contradictory and fictitious "quantity" of $$0.\dot01$$ are the completed set. They just don't include any contradictory and fictitious "non-partial-product" because you never get to one for an infinite product.

You're trying to pretend you know your way around these mathematical terms and it's just annoying because you so plainly don't! Stop embarrassing yourself.

Magnus Anderson wrote:
Silhouette wrote:I'm waiting for substance here beyond "there must be something" based on that contradiction.

It'e enough to prove that the gap exists. It's unnecessary to know its exact value.

The fact that it's impossible to narrow down any possible value should ring alarm bells that this "proof that the gap exists" is flawed.

I'm showing you where it's objectively flawed. It's not like you have any stake in this trivial topic that's already been answered - why have you become so emotionally invested in it? You owe it nothing, it's not part of who you are - it's just an amusing riddle to occupy amateurs for a bit until they realise that the proven answer teaches them a lesson about judging books by their cover...

Magnus Anderson wrote:
Silhouette wrote:I still love learning and can't wait for anyone to show me something I've not learned or thought of before

There are people who talk the talk and then there are people who walk the walk.

I'm still waiting on Ecmandu to teach me something about his Convergence Theory and how he's using the basic concept of carrying differently to come up with such strange series of equations. I told him very clearly I want to learn about this.

So I'm walking the walk - though I already learned this topic. I'm getting nothing out of this if I don't help to teach you anything. Maybe you're angry at me coz I'm so much smarter than you and you're rejecting my lessons out of spite? I don't know, but it's petty whatever your reasons.

Magnus Anderson wrote:
Silhouette wrote:Whether empirical or conceptual is a matter of Epistemology either way

Nothing to do with epistemology.

It would be so convenient for you if matters of knowledge didn't apply to you for this thread, huh?

Magnus Anderson wrote:
Silhouette wrote:The question is whether the output as a whole is distinguishable from the previous input states. I can watch an apple and a banana be put into the same basket and ask someone else who didn't watch the process to describe the product, and they'd be able to accurately describe it as an apple added to a banana. No information is lost upon the completion of the operation.
If you asked the same of someone adding an apple to an infinite line of apples, someone else who didn't know the input process would only recognise the result as an infinite line of apples, and could only guess at any operation previously performed. You yourself can only know the result as "an infinite line of apples plus 1 apple" as a result of your memory of a past state - as opposed to the current state.

That's why I'm saying that you (and Gib) are conflating empirical matters with conceptual ones.

The question is not an empirical one. We are not asking "Given two infinitely long pipes, how can we determine whether they are equal or not?" We are asking "If there's an infinite number of green apples in front of us and we add a red apple at the start of the line, and nothing else changes, is the number of apples greater than it perviously was or is it the same?"

We don't care about the poor guy who knows nothing about the situation. Why do you insist on listening to what a clueless guy has to say? Of course, since he's clueless, he'll tell you the line didn't change. But that doesn't make it true.

You're resorting to nonsense about conceptual matters being separate from empirical matters because deep down you recognise the only "traction" you have left on this issue is to insist that your argument doesn't "need" to be testable.

The reason we care about the poor guy who wasn't there when you added or did whatever finite operation to infinity, is not to simply "take his word as truth", it's because whatever his perspective is it highlights something fundamental about information theory - and we're dealing with information here. You don't want to care about him because unfortunately for you, information theory is inconvenient for you and undermines your point. Your resort to deny any and all contradictory evidence to your argument is just "confirmation bias". I know it's difficult to admit you're wrong, but at least you'll escape this debate with respect if you learn how to overcome your emotional shortcomings that are currently denying you of any integrity and authenticity.

Magnus Anderson wrote:
Silhouette wrote:I have two sets, A and B, which each contain one instance of each integer from 0 upwards, in ascending order $$\{0,1,2,3,...\}$$.

Okay, so we have two identical infinite sets $$A = \{0, 1, 2, 3, \dotso\}$$ and $$B = \{0, 1, 2, 3, \dotso\}$$.

Correct! Progress?

Magnus Anderson wrote:
Silhouette wrote:I state however that B can be constructed from A by first inserting 0 to start the list, and consequently populating the rest of B by adding 1 to each element in A.

In which case we have $$B' = \{0, 0_{+1}, 1_{+1}, 2_{+1}, 3_{+1}, \dotso\}$$ which has more elements than $$B$$.

If you want to display it like that, sure. It's the same as B (and A) either way.

Magnus Anderson wrote:
Silhouette wrote:This construction of B from A is entirely possible, but for each element in the list, there is one-to-one correspondence between the identical sets.

And that's not the case.

I mean, it literally necessarily is the case by definition - you even accepted the fact that they are identical sets insofar as they have one-to-one correspondence.

As soon as I construct it in a different way to give the illusion that there is no bijection (or that there is "injection" in this case) the identical sets actually objectively lose their bijective correpondence?
From the very start the first terms correspond perfectly with one another. As do the second terms and so on forever.... where is the extra element in B? At the end of the infinite set?? We covered this already, Magnus

Magnus Anderson wrote:
Silhouette wrote:The implication that B has an extra element because 0 was added to it separately to the construction of the rest is inconsequential, because for every element in A there is still an identical element in B - literally forever and with no gaps.

Not true. There is no one-to-one correspondence between the two sets. What you're doing here is called magic trick.

$$A = \{0, 1, 2, 3, \dotso\}$$
$$B = \{0, 1, 2, 3, \dotso\}$$
$$B' = A \cup \{0_{new}\} = \{0_{new}, 0, 1, 2, 3, \dotso\}$$

Why does B' now have an extra 0?!

You're accusing me of a magic trick??? I think you're getting confused with yourself again. This is you literally conjuring something from nothing.

I mean, maybe you meant $$B' = \{0_{new}, 0_{+1}, 1_{+1}, 2_{+1}, 3_{+1}, \dotso\}$$ ?
Either way this simplifies as $$B' = \{0, 1, 2, 3, \dotso\}$$, which is identical to both B and A as you accepted from the very start...

Magnus Anderson wrote:
Silhouette wrote:You never get to any un-matching element in B from its very start and onward forever

You do.

Really? Where exactly? This is more of your mysticism around the "conceptual", which is "beyond epistemological concerns and doesn't require testability", right? Keep that to the religious subforum, buddy - this is math. You need comprehensive proofs, which are not incidentally way beyond your ability.

Magnus Anderson wrote:
Silhouette wrote:It was only the alternative construction of B from A that gave the "appearance" that correspondence was skewed from being one-to-one. Superficial.

Actually, it is what you're doing (i.e. listening to Cantor) that is giving the appearance of one-to-one correspondence.

Listening to the guy who first invoked the notion of 1-to-1 correspondence in the first place, when dealing in matters of 1-to-1 correspondence and how it works - according to its inventor... this sin is just like going to school, right? Unforgivable!
The first element in one set corresponds with the first element in the other set and so on, perfectly, forever.... that's not an illusion

I think you're done, don't you?

Take a break. Relax. Mull it all over - it'll be good for you.

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

Silhouette wrote:You're resorting to nonsense about conceptual matters being separate from empirical matters because deep down you recognise the only "traction" you have left on this issue is to insist that your argument doesn't "need" to be testable.

What makes you think that only empirical claims are testable? Mathematics is not an empirical science, and yet, it's perfectly testable. "2 + 2 = 4" is not an empirical claim, and yet, it's perfectly testable.

I never said nor implied that this thread is about untestable claims. I merely said that it's not about the kind of questions that you (and Gib) have been obsessing over ever since you joined this discussion.

This thread has nothing to do with science, epistemology and information theory. It has to do with mathematics and logic.

Why does B' now have an extra 0?!

Because that's what you're doing. You're creating a new set, $$B'$$, by adding $$0$$ to it (the extra $$0$$ you speak of) and then by adding all of the elements of the set $$A$$ while relabelling them in order to make the resulting set $$B'$$ look identical to $$B$$.

$$B = \{0, 1, 2, 3, \dotso\}$$
$$B' = A \cup \{0_{new}\} = \{0_{new}, 0, 1, 2, 3, \dotso\}$$
$$B' = \{0, 1, 2, 3, 4, \dotso\}$$

In the final step, you're renaming $$0_{new}$$ to $$0$$, $$0$$ to $$1$$, $$1$$ to $$2$$, $$2$$ to $$3$$ and so on. That's how you create the illusion of one-to-one correspondence between the two sets.

Either way this simplifies as $$B' = \{0, 1, 2, 3, \dotso\}$$, which is identical to both B and A as you accepted from the very start...

It does not simplify. When you write it like $$B' = \{0_{new}, 0, 1, 2, 3, \dotso\}$$, it's pretty obvious that $$B'$$ is greater than $$B$$ because it has all of the elements that $$B$$ has plus the new zero you added. But when you "simplify" it, you make it look as if they are identical. That's the trick.

So where's the extra element in $$B'$$? Well, $$0_{new}$$ is the extra element. Don't conceal it and you'll see it.

I think you're done, don't you?

That would be you.

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

Ecmandu wrote:So Magnus...

Here’s my argument:

The “highest order of infinity”, is my cheat:

1.) rational number
2.) uncounted infinity
3.) different rational number
4.) different uncounted infinity
Etc...

I’m not listing the numbers here, just the concepts.

When you divide the interpolated orders of numbers, you get:

1.) rational number
2.) 0
3.) rational number
4.) 0

Etc...

Also!

1.) 0
2.) uncounted infinity
3.) 0
4.) uncounted infinity

Etc...

No matter how many times you divide it, they are still in correspondence. Mathematicians use abstractions to make their proofs. This is a symbolic abstraction (I’m not listing numbers themselves). Symbolic abstractions are fair game in mathematics!

My other argument against you is the dimensionality argument.

You claim planes are infinitely larger than lines.

That’s not true.

A 2D image can not only appear to be moving (3D), but can also render 3D images. Unary can do the same for 2D images. I can link some of these “optical illusions” if you like.

What this means is that a plane is not bigger than a line.

Those are my two arguments against orders of infinity.

Once you “cry uncle” submit, I can start debating Silhouette on 0.999... /= 1

I forgot my other argument:

If you add a 1 at the front, you have bumped the entire infinity up 1 space, which is exactly the same as adding a 1 at the end (both of which means the series wasn’t really infinite to begin with)

I’m sure I have more, but these three should suffice.
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