Is 1 = 0.999... ? Really?

That’s because the word “infinity” has multiple definitions. One of them is “endlessness” which is not a quantity (because it need not refer to lack of quantitative end.) The other is “endless quantity” which is a quantity that is greater than every finite number.

If you can say that something is more than (or less than) something else, then you can say that that something (and that something else) is a quantity.

Not really. It makes sense either way.

Endlessness and red aren’t numbers. (\infty) is. This is evident in the fact that it represents a quantity greater than every finite number.

I know you’re ignoring me Magnus,

But here’s the deal.

If you believe calculus,

Every “finite sum”, has an infinite amount of infinite sequences converging towards it if you believe in convergent theories.

The reason I call them “finite sums” in quotes is because they are all “equal” to infinite series.

That means that any counting number you pick, is “equal to”, an infinite series, which means that are all infinite.

I don’t buy this argument, but if you believe in calculus, you do!

Ok, Magnus, but you know what I say to people who come up with their own definitions: you’re on your own.

Error: assertion denied!

Error: assertion denied! Error overload! Exploding!

That’s true.

That’s true but is irrelevant and the sole point of it is to show that you know what a limit is.

That’s true.

That does not follow. Indeed, the opposite is what follows. Because (0.\dot01) never attains (0), this means there is a gap between (0.\dot01) and (0).

When you say that the gap cannot be defined to exist, what you mean is that it cannot be represented using one of the numbers they taught you in school.

If they don’t teach it at school, it doesn’t exist.

Tell that to Google.

Silhouette is going to have a seizure!

What’s interesting to me about this thread is that you, Silhouette and I all fall on different spectrums…

You: orders of infinity exist, 0.999… does not equal 1

Silhouette: no orders of infinity exist, 0.999… does equal 1

Me: no orders of infinity exist, 0.999… does not equal 1

Gib hasn’t commented on his orientation yet.

Needless to say, it makes for very interesting discussion.

No matter how you try to explain the difference between red and green, you cannot get a dog to see color.

Actually, they don’t.

Mathematicians do say that the result of division by zero is undefined but they do not say that it is undefinable. Undefined (\neq) undefinable.

(\frac1\infty) is valid (i.e. there is absolutely nothing wrong about it) even if we’re working with the insufficiently defined concept of infinity that Observer is talking about.

The only problem is that such a symbol of infinity does not necessarily represent the same infinite quantity wherever it appears in the equation, making it possible to say such things as (\infty + 1 = \infty) and (\frac1\infty \times \frac{1}{10} = \frac1\infty) without being wrong. The bigger problem is that it leads to erroneous conclusions such as the one we can see in many Wikipedia “proofs” of the equality between (0) and (0.\dot9). But this isn’t a problem with the definition as much as it is a problem with people not following its implications.

By changing the definition of the symbol of infinity from “an infinite quantity that is not necessarily the same as the one represented by the same symbol elsewhere in the equation” to “an infinite quantity that is the same as the one represented by the same symbol elsewhere in the equation”, expressions such as (\infty + 1 = \infty) can no longer be said to be true.

We know very well what the word “infinity” means. You don’t.

Magnus:

Infinity + 1 either equals infinity or it does not.

You are using the logic of finite math:

Infinity + 1 = infinity + 1

With non finite math,

Infinity + 1 still equals infinity

Do they have proof that 0 * ∞ = 0?

I’m not so sure that it does. How could that be proven?

And if it doesn’t, they can only say that a/0 is undefined, not undefinable.

“Infinity” doesn’t exist. An infinite thing can exist (and does I’m sure) but infinity is defined as the point where parallel lines meet. Since parallel lines do not meet, infinity cannot exist.

James’ infA exists as a specific infinite series. That is not the same as infinity.

Well… you’re going to have to reign your “friend” in, because he’s using axioms like 1/infinity!

Not really. It depends on what (\infty + 1 = \infty) means.

Magnus,

We live in infinity. Like physics states, you cannot add matter or subtract matter from existence, all you can do it move it around.

This means that you cannot use infinity proper as something that can be affected by operators.

It’s a display.
Whilst it’s possible to make an objective mistake in how information internally relates to other information, you cannot make an objective mistake in how you subjectively decide to display that information.
But you CAN mislead, and come to mistaken conclusions as a result of this misleading.

It’s painfully clear that the blues sum up to the yellow and the reds sum up to the purple - those two separate interactions are objectively fine, and the utility of your subjective choice to display the respective sums underneath what they sum is nice and convenient. That part’s fine too.

Where you’re misleading is in allowing this subjective display positionings of the blues and the yellow to affect the objective position of the purples, which do not overlap or have anything to do with the operation of summing the blues to get the yellow. Likewise for the reds, you could subjectively display them anywhere you like and they’d still objectively sum to the purples. It’s only subjectively convenient to put them next to the blues so the visual grouping of the greens is clearer, but even if you displayed them anywhere else that you liked: the first element of the greens would still be the first element of the blues, the second element of the greens would still be the first element of the reds and so on.

Subjective display choices have absolutely no influence on the objective fact that element 1 of the greens equals element 1 of the purples with perfect correspondence. The second green and the second purple are also equal, and so on with perfect objective bijection. Your display choices have nothing to do with this.

Again. You’re getting carried away by superficials. Stop it.
You’re trying to conflate subjective clarity with objective positioning.

Again, it’s perfectly relevant: limits are at the core of “infinite series” whether sums or products, differentiation, integration - for any and all the many mathematical applications of divergence and convergence.

There is no “product itself” of a divergent infinite product - you never get there. The only thing you can do is identify some limit that it’s tending towards because of the very fact that it never gets there.

Again, your non-mathematical thinking and lack of knowledge on the topic betrays you. Stop pretending you have expertise even when you admit you don’t.

(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. That’s why adding one results in boundlessness both before and after you do it, and therefore (\infty) applies either way, with any finite adjustment contradictory and meaningless. 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’s like a drop into an infinite ocean. In a finite ocean, even a drop would raise the sea level by some miniscule amount depending on the size of the ocean. In an infinite ocean, the drop would be 1 finite drop all the way until it went into the ocean, which would be sizeless both before and after, with the finitude of the drop completely absorbed into the infinitude of the ocean - its defined finitude annihiliated. The equation of adding 1 to infinity is a statement about the result of such an addition, which remains as infinitude. Before the equation, either the apple or drop is still finitely one. After that, it isn’t anything, and the change is literally 0. This doesn’t mean 1=0, this is only the kind of contradiction between finites that you get when dealing with undefinables/infinities, which is exactly why you can’t treat infinite quantities like finite quantities that you can equate and operate on consistently.

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.
The correct approach is to understand that treating infinites alongside finites as a contradiction, which is proven by the fact that it leads to contradictions between finites. Relating infinites to finites as though they were compatible is a prior error to the contradiction of the kind 1=0, which is merely a symptom of your initial error. Semantically this makes perfect sense as well, since there’s only one way infinites can be infinite, no matter what tinkering you do to them with finites.

Yeah the phrase is my own invention - and a damn good one too! But I guess since the standard definition is “already so good”, saying literally nothing about what infinity “is” and only what it “isn’t” then I guess that makes my idiosyncratic summation completely invalid :wink: Your “logic” always makes me laugh :laughing:

I have no doubt that the irony of this comment will be lost on you, considering it is an assertion that lacks argument, but whatever - it’s been clear what I’m dealing with for a while now.

It’s also hypocritical:

Great reasoning.

Great reasoning.

“Not quite.” seems to be up there with “you only tell me what to think, you don’t explain” as your most frequent response.
Oh the hypocrisy…
But let me guess your excuse for your double standards on explanation: it’s “unnecessary to do so” and “One doesn’t have to prove more than it’s necessary”, or maybe you “don’t have to go any further than this” or “I don’t have to respond to anyone unless I wish to do so.”
So many of these gems just falling out your mouth, I could start up a jewellery shop!

Just repeat your debunked claim. Provide no explanation.
I’ve been saying all along that infinity having no destination is exactly why (0.\dot9 = 1)

See, things like this just confirm your irrationality.
No hesitation by you to jump to obviously false extremes with the presumed intention to rile me up :slight_smile:

Oh, and “great reasoning”.

Oh boy :laughing: It’s right here buddy:

Man, the things you try and get away with! Saying I’m hallucinating that you finally admitted an obvious truth that I can easily find and quote back to you :laughing:

So there’s only one way that infinite sets can be infinite but there’s more than one way that infinite sets can be infinite in size.

Contradictions abound!

And there I was saying “truth is a relative term in that it relates two things and their likeness.”

And here you are saying “apple” on its own has no truth-value, as if that was remotely close to countering what I said.

I told you to put the straw man down.

My frustration is piqued by the consistency of things like the above, where you insist you’ve not misunderstood me when “exactly how (you) interpreted what (I) wrote” is in direct contradiction to something I said - e.g. just above.
My patience, however, whilst uncharacteristically thin, holds regardless.

I know what your claim is. You think there’s no issue at all in treating the indefinite as a definite symbol.
You proceed to operate upon that symbol as though it represented a definite quantity, and thus claim validity in your conclusion and you were dealing with an indefinite all along.

You also claim that I think symbols have to look like what they represent, which is something I’ve now repeatedly stressed I don’t mean.

Signifiers sharing the property of “being defined” with what they signify does not mean sharing the property of “looking like each other”. This obsession you have with appearances is just more evidence that you’re getting carried away by appearances. Whether or not something is “defined” is a product of its essence, as bounded, not its appearance.

As explained above, just no.

Cryptography is entirely irrelevant here, because even encoded information takes definite form, just like decoded information and all symbols regardless of cryptography.
But since you mention cryptography, try encrypting a number that’s infinite into a finite number, using only numbers.
What, you can’t? That’s correct.
You need further symbols than just numbers to explain that what you’re really dealing with (an infinite number) cannot be expressed using just numbers.
You need to go on to explain that you’re expressing what can’t be finitely expressed in a finite form anyway - but let’s just hope there’s a mathematician on the other end and not Magnus, to be sure nothing gets lost in translation, and that the receiver doesn’t go on to treat a number that couldn’t be expressed with just finite numbers as though it could be expressed and operated on and dealt with as though it could be expressed with just finite numbers.

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.

And yes, thanks for confirming I was right about something.
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:
Allowing some definite answer “n”, multiply the expression (n=\frac0{0}) by the fraction’s denominator to get (n\times0=0) and we quickly see that any quantity “n” multiplied by zero yields zero. “n” can’t be a definite answer: proof by contradiction. This is what undefined means.

We’ll make a mathematician out of you yet!

Progress.

Keep practicing.

As above, limits are at the core of infinite series.

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).

The undefinability of any “gap” means there’s no grounds to distinguish (0.\dot9) from (1), nor (0.\dot01) from (0).

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.
“Undefined” (\to) “a gap exists” is erroneous. But it is valid to say that any such gap can never be defined therefore it’s undefinable and cannot be said to exist at all.
Any reversal of something like “it can’t be said to not exist either” does not mean “It can be said to exist” - this would commit the formal fallacy of “affirming a disjunct”.

Fortunately for me (0.\dot01) isn’t a valid quantity in the first place as it’s bounding either end of an endlessness, which is a contradiction. So it’s unsound to use it in any of the further reasoning that you’d need to define any gap never vanishing.
Combine that with “A gap can be said to exist” being a formal fallacy and bingo: “no gap can be said to ever exist” is valid and the equality cannot be disproven.

Many things exist that you don’t learn at school - you should learn about some. Numbers that logically cannot exist are not one of them.
Go back to school - maybe at least then you’ll learn how to learn, and we could actually talk.

So when mathematicians say “undefined” do they mean “well, for now at least”?

A mathematical result being undefined now doesn’t mean it might be later on. It means it cannot be defined period. This is no different from “undefinable”.

The problem of “(\frac1\infty) not necessarily representing the same infinite quantity wherever it appears” is its undefinability. Every single instance of infinity is undefined in itself, never mind the fact that it’s also undefined across all the different equations that are also misleadingly said to “equal” infinity! Differentiating all the different instances of infinity does nothing to define each individual instance, since each individual instance is undefined in itself.
This doesn’t mean “well we can just treat it as definable for all the infinite ways in which it’s undefinable”. That’s as stupid as your claim that Wikipedia proves the equality between (0) and (0.\dot9). No doubt a typo, but stupid either way.

This is the core of what you’re misunderstanding - that even one “infinity” for which we’ve constructed the finites around it in a very specific way, the infinity in the construction is no more or less specific than any divergent result you get from it. It’s undefined “going in” and undefined “going out” - and no matter how precise you are with the finites that you’re operating on infinitely many times, the undefined element of “infinitely many times” is as undefined as any divergence you get.

Only convergence can tend to very precise and specific finite values - even if they’re only the limit, to which we never actually ever arrive. It’s still impossible to define any difference between zero and any “abstract mysteriousness” that you try to assert as “existing”. Whatever magic you try to conjure, it’s always smaller than that, and smaller than that indefinitely such that it cannot ever be said to exist even in your imagination.

I’ve already validly deconstructed obsrvrs’ “definitions” of infinity.
Again - more unexplained assertions by you to accompany all your complaints that you don’t see the explanation in my explanations therefore it’s just an assertion.

The fallacy of “proof by assertion” is keeping such close company with your fallacy of “argument from incredulity”…

“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 “they” is you too. Maths is objective so it’s your proof just as much as it is anyone else’s.

And as I just said to Magnus, this doesn’t mean temporarily undefined. It means it’ll always be undefinable.

Indeed infinity does not exist. InfA doesn’t either, because it contains within it the “…” that implies “infinitely more times”, which as we’ve just learned is an undefined number of times, which could only lead to an undefined result if it could ever reach any result at all. It has a divergent limit that cannot be definitely specified one way or another - and calling it “infA” doesn’t solve this.

I think that Magnus is exactly right about that and that your claim is exactly wrong.

After we debunk orders of infinity, I’m coming back after silhouette.

That’s nice.

I mean, I just disproved it (again) in the post right at the end of the previous page:

But as long as you just simply “think” that’s exactly wrong, does that make you exactly right?

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.

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.)

Not really.

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”.

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.

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.

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).

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.

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.

(\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).

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.

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.

Magnus,

Your graph is nice, but it’s actually not different from simply stating 0.333… * 3 = 0.999…

I want to stay on orders of infinity with you before really laying into silhouette