Magnus Anderson wrote:Silhouette wrote:It's just idiotic to use that as an excuse to shift over the separate purple set that doesn't even include the yellow set that's being used to superficially shift it.

I am not using the yellow rectangle to shift the purple one. Rather, the purple one is where it is because of the position of its equivalent rectangle (the red one.)

The purple rectangle represents the red one and the yellow one represents the blue one. The blue one and the red one are independent from each other -- they have no terms in common.

You have yet to show me my mistake.

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.

Magnus Anderson wrote:Silhouette wrote:Either way, \(\lim_{n\to\infty}\) of this infinite product is \(0\).

Which is completely irrelevant. Just because the limit of an infinite product is \(0\) does not mean the infinite product itself is \(0\).

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.

Magnus Anderson wrote:Silhouette wrote:One apple at the "starting bound" of a "boundless" line of green apples doesn't make it "more boundless".

Not sure what it means to say that an infinite line of green apples is "more boundless". I certainly didn't say such a thing. I said that the number of apples is greater than before.

\(1+\infty\) 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.

Magnus Anderson wrote:Silhouette wrote:The quality of having no quantity is not a quantity.

"The quality of having no quantity" is your own idiosyncrasy that has nothing to do with the standard definition of the word "infinite".

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

Your "logic" always makes me laugh

Magnus Anderson wrote:Silhouette wrote:Add it literally anywhere in the line, as you say - no difference. It would make a difference to a finite line, for sure. Adding a new first element to an infinite set just gives you an infinite set with a new finite bound - the finite "1st element" changed, shifting all successors down by 1 place infinitely.... - no size change occurs. It would occur for a finite set, sure, but you can't have a "longer" infinite endlessness even if you change a finite constraint to how it starts.

Here we go again. One assertion after another, no arguments whatsoever.

What do you think you can achieve by merely restating your beliefs?

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:

Magnus Anderson wrote:Silhouette wrote:It's therefore not a problem to denote finite signified things with a finite symbol. It is a problem to denote infinity with a finite symbol.

That's not true.

Great reasoning.

Magnus Anderson wrote:Silhouette wrote:Well the appearance of their spelling looks different so you must be right!

Not quite. They mean different things.

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!

Magnus Anderson wrote:Silhouette wrote:they don't make the mistake of saying that infinity is a destination that you can get to

I make no such mistake. In fact, it is people who claim that \(0.\dot9 = 1\) that make that mistake

over and over again.

Just repeat your debunked claim. Provide no explanation.

I've been saying all along that infinity having no destination is exactly why \(0.\dot9 = 1\)

Magnus Anderson wrote:Silhouette wrote:I know exactly the difference between signifiers and signifieds - it's something I've been explaining to you.

You know nothing.

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

Oh, and "great reasoning".

Magnus Anderson wrote:Silhouette wrote:As you finally admitted

You're hallucinating.

Oh boy

It's right

here buddy:

Magnus Anderson wrote:A set cannot be more or less infinite. It cannot be partially infinite. It's either infinite or it is not.

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

Magnus Anderson wrote:Silhouette wrote:there's only one way that infinity is infinite

There's only one way that infinite sets can be infinite does not mean that infinite sets don't come in sizes.

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!

Magnus Anderson wrote:Symbols (such as words) have no truth-value on their own. The word "apple" is neither true nor false on its own. It's merely a symbol with certain meaning attached to it. (The meaning of a symbol being the set of all things that can be represented by that symbol.) It is only when you use that word to represent some portion of reality that it acquires truth-value. For example, when you use the word "apple" to represent what's inside some box. Such an association can be represented with a statement such as "There is an apple inside the box". That's either true or false. Either what's inside the box can be represented with the word "apple" or it cannot be. But the word "apple" on its own has no truth-value.

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.

Magnus Anderson wrote:You're implying that I misunderstood you. I didn't. That's exactly how I interpreted what you wrote. You said that the symbol \(\infty\) can be used to represent infinity but that there is maximal pretense in doing so. My claim is that there is no pretense whatsoever. The fact that you think that there's a pretense involved is what indicates to me that you do in fact think that symbols have to look like what they represent. (I'm not really sure you understand what I mean by this. Your impatience is on the rise, so I can expect you to misunderstand me more and more.)

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.

Magnus Anderson wrote:Silhouette wrote:You CAN do it, but doing so pretends there's some kind of finite symbol that sums up the opposite of finitude

There you go. You just said that the symbol of infinity must "sum up", i.e. must look like, infinity.

As explained above, just no.

Magnus Anderson wrote:Note that I'm not saying that you think that symbols must look exactly like (i.e. 100% like) what they represent.

I'm saying that you think that symbols must look like what they represent to a certain degree (the exact number is irrelevant.)

I'm saying that they don't have to. I'm saying that they can look completely different from what they represent (as in cryptography.)

The statement "There is an infinite line of green apples in front of you" looks nothing like what it represents, but if the thing in front of you is an infinite line of green apples, then the statement is true.

The meaning of a symbol is something that exists independently from the symbol.

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.

Magnus Anderson wrote:Silhouette wrote:Mathematicians literally say that things like division by zero is undefinable.

Yes, they do, and what that means is that "division by zero" is an undefined expression i.e. there is no meaning assigned to it.

Unfortunately for you, the word "infinite" does have a meaning, so it's not undefined. The word "infinite" means "without end".

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!

Magnus Anderson wrote:Silhouette wrote:Take your infinite product of tenths: there is literally no end to it

That's true.

Progress.

Magnus Anderson wrote:Silhouette wrote:You don't get to any "1" or "end", you can't because there is no end

That's true.

Keep practicing.

Magnus Anderson wrote:Silhouette wrote:its limit is zero as the only number that it's tending towards

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

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

Magnus Anderson wrote:Silhouette wrote:and therefore no gap can be defined to exist at all.

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.

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.

Magnus Anderson wrote:Silhouette wrote:Mathematicians literally say that things like division by zero is undefinable.

Max wrote:Yes, they do, and what that means is that "division by zero" is an undefined expression i.e. there is no meaning assigned to it.

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.

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".

Magnus Anderson wrote:Silhouette wrote:It's true that it's not defined sufficiently for maths operations to be sensibly used on it, yes. Hence the invalidity of things like \(\frac1\infty\).

\(\frac1\infty\) 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.

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.

Magnus Anderson wrote:Silhouette wrote:So we've successfully known that we don't really know what \(\infty\) is (for centuries).

We know very well what the word "infinity" means. You don't.

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"...