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...
15
50%
Other
5
17%
 
Total votes : 30

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

Postby Ecmandu » Sun Jan 12, 2020 6:25 pm

obsrvr524 wrote:
gib wrote:1) When we talk about a set being infinite, we mean it has an infinite number of members, not that the value of each member is closer or further from infinity. So we don't say the size of set {1, 2, 3} is less than the size of set {2, 4, 6}. They're both 3 members in size.

2) The rate at which members of a set approach infinity is different from the size of the set. You can say the rate with which set B approaches infinity is greater than the rate with which set A approaches infinity, but that doesn't mean set B has more members than A.

Well okay but just take any one value of the sets as a count for a new set. The new set for B will always be longer than the new set for A. So as x goes to infinity, the new set for B will be infinitely larger in count or longer in "length" than the new set for A, even though they are both infinite.


You’re still ignoring me.

Do you want the answer to this or not ?

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

Postby obsrvr524 » Sun Jan 12, 2020 6:27 pm

Ecmandu wrote:My argument addressed this better Than gibs approach, yet you ignore it:

The “highest” order of infinity is:

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

I ignore it because I can't make any sense out of it.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 12, 2020 6:32 pm

obsrvr524 wrote:
Ecmandu wrote:My argument addressed this better Than gibs approach, yet you ignore it:

The “highest” order of infinity is:

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

I ignore it because I can't make any sense out of it.


The sense of it is that you can do concept replacements of numbers, and order the concepts in a list.

I’m not listing an actual number here, I’m using that abstract representation of the number with words, to place 1:1 correspondence for the “highest order of infinity”

I call it my cheat.

If mathematicians balk at my abstractions, they can only look at how others balk at theirs.

This is fair game what my proof is.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 12, 2020 6:44 pm

Gib wrote:Like I haven't pointed out the flaw numerous times before.


You haven't.

If you stop pretending that you did, you will stop being frustrated (which means you will stop posting funny gifs that do nothing but reduce the quality of the discussion.)

I know that frustration is a fashionable thing but that does not mean it's a positive thing.

And BTW, adding the red apple but removing another apple doesn't contradict #2, it just means something was left out.


It does because premise #2 also states that the line didn't change in any other way (which means that no green apple was removed from it.)

Let's say with infinite sets, the size of the set is undefined, then you can't increase the size by adding one because, well, how do you increase undefined?


First of all, the size of infinite sets is not undefined.

To say that a size is undefined is to say that it's unspecified i.e. that it's unknown.

But to say that a set is infinite is to say that the number of its elements is greater than every finite number. By definition, every infinite set is larger than every finite set. So it's not like we know absolutely nothing about the size of infinite sets.

But even if the number of things is undefined (i.e. unknown), by adding one to it you necessarily increase it.

Any number + 1 < Result

That's what the word "add" means.

If you don't like that, then let's say the size of the set is infinite. Then:

\(\infty\) + 1 = \(\infty\)

The size doesn't change.


That's not an argument.
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Re: Is 1 = 0.999... ? Really?

Postby gib » Sun Jan 12, 2020 7:58 pm

obsrvr524 wrote:Gib,

Magnus already addressed that counter argument with:

To say that you "ADD" something is to say that you made it larger, longer, taller, or whatever fits.

What you are saying is that you cannot add to an infinite set.


Magnus can define things however he wants. Doesn't make it coherent.

Look, I can define subtraction as "making a ham and cheese sandwich". So then 4 - 2 = ... what? What does that mean now that I've defined subtraction as making a ham and cheese sandwich? It's incoherent, right? Just the same, I find the notion of making an infinite set larger, longer, etc. incoherent. I've been trying to help Magnus anchor his points by asking him to define what it means for an infinite set to be "bigger" (or whatever) than another infinite set, but the last response he gave me was: an infinite set is bigger than another if the number of units that constitutes it is more. Useful definition for finite sets but just as problematic for infinite sets.

There's a common misconception of what "addition" means in mathematics. You see it when people argue that 1 + 1 sometimes equals 1, as in when two raindrops join to make one large raindrop. My pet peeve with this is they're defining "addition" as a process or something going through a change. But when mathematicians talk about addition, they mean something more static, like if you took a photo of the two raindrops, how many raindrops would there be all together in the photo. Addition in mathematics is a way of talking about equivalent meanings, not the result of a process. 2 + 2 means 4. That is, to say there are two objects and there are two other objects is just to say there are four objects. We're not talking about what happens to those objects, whether they "come togther", or they meld to become one giant object, or when they come together they collide and break into a million pieces, etc. It's about how quantities can be expressed in different combinations and permutations.

Magnus's definition makes sense in terms of a process. If you add an apple to a line of apples, the line becomes longer (at least in the finite case). But that's "add" as in "you take the apple and place it at the front of the line such that it becomes part of the line." To that I say, of course you can add to an infinite set. You can drop an apple into a barrel of infinity apples. But as a mathematical concept, it's incoherent. If you "add" one apple to an infinity of apples, in the mathematical sense, you just get infinity again. What it means for that resultant infinity to be "larger" than the original infinity is what I'm grappling with and what Magnus hasn't been able to help me with. So far, it remains incoherent.

I did like your definition of infinity as a property of a set, rather than a quantity. I think that's right. It's the property of being endless. But as such, it's pretty binary. A set is either endless or it's not. I don't know what "less" endless means.

obsrvr524 wrote:Well okay but just take any one value of the sets as a count for a new set. The new set for B will always be longer than the new set for A. So as x goes to infinity, the new set for B will be infinitely larger in count or longer in "length" than the new set for A, even though they are both infinite.


Not sure whay you mean? You mean something like this:

A = {1, 2, 3,...} ==> A' = {{x}, {xx}, {xxx}...}
B = {2, 4, 6,...} ==> B' = {{xx}, {xxxx}, {xxxxxx}...}

They're both still infinite, and the size of each subset doesn't determine the size of the parent set.

The size of {{x}, {x}, {x}} is the same as {{xx}, {xx}, {xx}}. They're both 3.

On the other hand, if you want to do this:

A = {1, 2, 3,...} ==> A' = {x, ... x, x, ... x, x, x,...} = {x, x, x, x, x, x,...}
B = {2, 4, 6,...} ==> B' = {x,x, ... x, x, x, x, ... x, x, x, x, x, x...} = {x,x, x, x, x, x, x, x, x, x, x, x...}

Both sets are still infinite, but set B appears to be growing faster than set A. This assumes that the time it takes to replace each number in each set with a bunch of x's is the same, but yeah, the rates would be different. But that doesn't tell you anything about the size of each set.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 12, 2020 8:03 pm

Magnus argues that infinity + 1 equals a greater value.

Where did the 1 come from?

An infinite set:

0(because the 1 is gone now) 2,3,4,5,6,7,8,9...

Equals

1,0,0,0,0,0,0,0,0...

In 1:1 correspondence

Otherwise the 1 would have to be tacked in the end of an infinite sequence and would never be expressed :

0,1,2,3,4,5,6,7.... 1!

That’s absurd!
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Re: Is 1 = 0.999... ? Really?

Postby gib » Sun Jan 12, 2020 8:14 pm

Magnus Anderson wrote:If you stop pretending that you did, you will stop being frustrated (which means you will stop posting funny gifs that do nothing but reduce the quality of the discussion.) Yeah, 'cause this is a super high quality discussion already.

I know that frustration is a fashionable thing but that does not mean it's a positive thing.


I'm actually not that frustrated. The giffies are for comic relief.

Magnus Anderson wrote:To say that a size is undefined is to say that it's unspecified i.e. that it's unknown.


Or that it doesn't make sense.

Magnus Anderson wrote:But to say that a set is infinite is to say that the number of its elements is greater than every finite number. By definition, every infinite set is larger than every finite set. So it's not like we know absolutely nothing about the size of infinite sets.


I'll concede to this. But I still don't know how it makes sense to say an infinite set is larger than another infinite set. The whole reason why every infinite set is larger than any finite set is that you can't get any larger than infinity.

Magnus Anderson wrote:
\(\infty\) + 1 = \(\infty\)

The size doesn't change.


That's not an argument.


Oh, and just reasserting "...by adding one to it you necessarily increase it" over and over is?

Asserting \(\infty\) + 1 = \(\infty\) is just the answer to your question. It's the reason why adding an apple to the infinite line of apples doesn't make it bigger. I could go deeper, but I have a feel no depth would ever convince you (you could look up my replies to obsrvr for a hint).

And besides, I'm not doing that. You convince me that what applies to finite sets applies to infinite sets.
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Re: Is 1 = 0.999... ? Really?

Postby Silhouette » Sun Jan 12, 2020 8:35 pm

obsrvr524 wrote:I don't see how that makes any sense at all. It's very obvious that the symbol \(\infty\) certainly is defined and well known (for centuries).

Additionally infinity itself is defined and well known. The only problem that I see is that it is not defined sufficiently for maths operations to be sensibly used on it.

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 \(\frac1\infty\)

Saying infinity itself is defined and well known from a list of "what it is not" is like saying we know very well what's in a hole because we've defined its edges. The definitions you quoted:

"1" is just rewording the noun form to the adjective form without mention of any of the semantics of the word that we've merely changed the syntax of.
"2" brings up "ends" in the context of time, space and quantity, and then saying "not that": saying what something "isn't" isn't saying what something "is".
"3" brings up non-specific "great numbers" and "amounts" and confirms this non-specificity with the word "indefinitely", and alters the syntax further to the adverb form "infinitely".
"4" is a practical application of "7" for optics and photography that suggests some implied theoretical limit that phenomena only ever tend towards in practice, but never actually get to to "know" it directly.
"5" brings up "finite variations" in terms of dimensions/quantities and then says "not (affected by) that".
"6" brings up "finite numbers" and then says "not that" because it's "bigger" in some non-specified way.
"7" is the generalised case of things like "4" that we can only "assume" due to implications of what it looks like would probably happen if we could ever get there, without ever actually getting there to "know" it directly.

Do we know what infinity is now? Well, we know what it isn't....?
So we've successfully known that we don't really know what \(\infty\) is (for centuries).

Magnus Anderson wrote:
Silhouette wrote:It's absolutely relevant since the core of your argument is to regard the infinite as finite.

That's not true. What I'm doing is I'm treating infinite quantities as if they are quantities.

That's your road in to treat infinites as finites, yeah, because "quanta" can be dealt with in specific ways "as though" they have properties that can be defined even if you excused yourself for doing so with the apology "but don't worry, it's still infinite..."
Infinite sums have properties that can be defined and you can deal with those finite properties in specific ways to give the illusion that the whole thing that you're dealing with can be defined and dealt with in specific ways. The infinite quantity you can't deal with though, and it's so important to be precise and separate the two or you're going to get conflations and misunderstandings such as yours. Take your infinite product of tenths: there is literally no end to it, its limit is zero as the only number that it's tending towards, because anything else can always be divided into ten again. You don't get to any "1" or "end", you can't because there is no end - and therefore no gap can be defined to exist at all. It just "looks like" maybe it should using your intuitions about finites.

It's not a coincidence that quantity derives from a question (how much?), not an answer. Any answer is either going to be (de)finite or "I don't know" in(de)finite. For "I don't know", you can point in a direction that you could either get to the end of or you can't. Getting to the end means the quantity was finite, just very large, and not being able to get to the end turns the answer into "I know that I can't know". Requiring that something go on forever, such as in an infinite sum, you're stating you know that you can't get to the end and know the entirety of what you're dealing with. This is why infinites don't have size, only finites do. Infinites might be able to have "implied" bijection but you never reach the end to confirm the implied size equality or inequality. It's indefinite: you can't define it - that's what infinite means. Attempts can lead to contradictory outcomes and might tempt you to just look for the outcome you want and stop your reasoning there, like you keep doing. That's why it's a mistake to deal specifically with infinites - because you will encounter mistakes, implied by the contradiction in terms of doing so in the first place ("specifically" and "infinites").

Magnus Anderson wrote:
Silhouette wrote:The definite symbol \(\infty\) is not definable - it represents the exact opposite of definable even though the word/symbol/signifier takes the same form as things that are definable.

Maybe you should start with the word "definable". What does it mean? What does it mean to say that something is not definable?

I've been doing this for a while now, but of course you haven't noticed "and therefore I didn't".
"Definable" derives from the ability to give ends/bounds to something. Entirely definable means you can physically or at least mentally encapsulate the thing or concept in every possible way.
Not definable means you can't do this (no shit). In the purest and most consistent sense, this means absolutely everything - the entirety of existence.
You can combine the two into a set of finites that can be operated on infinitely, but you cannot conflate these opposites or even imply that a mixture of the two is only one or only the other if you want to be sufficiently precise.

Without at least minimal definition, you can't operate on "absolutely everything" with anything else because there is nothing else - therefore if you're dealing with a "quantity", there is at least some minimal definition to it that goes against its infinity. So if you're dealing with something that involves infinity, it necessarily has finititude involved as well, and with finitude being the necessary ingredient to deal with something, it's the finitude that you're dealing with - not the infinitude.

This is why with your dealings with quantities are with finites and not infinites, and if infinitude is involved, it is separate from your quantitative dealings until you've defined exactly what finites you want to "go on forever" and how finitely you want to do this.

Magnus Anderson wrote:If the purpose of symbols is to merely represent something, and not to look like that something, then there is absolutely nothing questionable about the act of symbolization. (Cryptography must be a very questionable practice.)

If there's an infinite line of green apples in front of you and you say "Look, there's an infinite line of green apples in front me!" the statement isn't false by virtue of not looking like what it represents. The word "true" does not mean "a symbol that looks exactly like that which it is trying to represent". It merely means "a symbol that can be used to represent that which it is trying to represent".

You're doing that thing again where you're explaining to me something I've just explained to you as though you're teaching me something.

I confirmed I'm not saying that words are required to look like what they represent and I confirmed that this is the whole point of having a word in the first place, so what's with the straw man that I'm saying a sentence is false because the words don't look like what they represent?
When something is close to what it represents, then it can be said to be "true to" what it represents such as with a piece of art or a scientific formula or theorem - truth is a relative term in that it relates two things and their likeness.

Words that are intended to not be the thing they represent are therefore symbolisation intrinsically lacks in truth - on purpose, due to the utility of deviating from truth. This is therefore absolutely questionable in terms of truth, and that's the whole point of doing it. The word "word" does pretty well in being true to what it represents, and the word/symbol of infinity/\(\infty\) is at the opposite end of this relative scale. It is maximally questionable.

Magnus Anderson wrote:
Silhouette wrote:the specific instance of a finite symbol representing an infinite quantity is absolutely central to the thread and your misunderstanding of "infinity" as finite

You didn't explain why.

Sure I didn't #-o To repeat myself yet again, if you fail to treat quantities as infinite, you can end up thinking there's a gap between \(1\) and \(0.\dot9\) because for all finites there is a gap, but with infinites, the gap never arrives. Ever.
I love how you're acting like I've never said this kinda thing several times - perhaps you're going to act like this explanation isn't an explanation too!

Magnus Anderson wrote:Here's the problem. You're saying that one can only pretend that the symbol \(\infty\) can be used to represent infinity. But that's not true. I don't have to pretend. The symbol \(\infty\) CAN be used to represent infinity without any sort of pretense.

So I was right when I said that you're one of those people who think that the symbol must look like the symbolized in order to be able to say that the symbol represents the symbolized. According to you, if the symbol does not look like the symbolized, you can't say the symbol represents the symbolized, but you can pretend that it does. Useful contradictions and all. Beside being wrong, what you're doing here is justifying contradictions in the name of utility.

I just said that symbol \(\infty\) CAN be used to represent infinity, but there is maximal pretense in doing so as I explained above. You CAN do it, but doing so pretends there's some kind of finite symbol that sums up the opposite of finitude, giving the appearance that you can deal with it like any other symbol that denotes finitude, when really you'd be dealing with the exact opposite of finitude that cannot be dealt with.

One is always dealing with the signified when one deals with signifiers in their place. All finite signified things can be dealt with to some degree, which is consistent with the finitude of their signifers. Signified infinity cannot be dealt with to any degree, which is inconsistent with the finitude of its signifier \(\infty\) 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.

Magnus Anderson wrote:The symbol \(\infty\) does not represent undefinability. It represents infinity. Infinity and undefinability are two different concepts.

Oh really! Well the appearance of their spelling looks different so you must be right!
Like I demonstrated to obsrvr, trying to define infinity runs into the problem that you can only either tautologously change the syntax of the word, a synonym, or define finite things and say "not that", which doesn't actually say anything about what infinity "is". It just says things about what it is not. This is all you can do, because infinity is undefinable. Turns out the two words have the same derivation as finitude - what a coincidence. The prefix of "de-" just means "from" so indefinite just says "not from boundedness" instead of infinite saying "not bounded". Big difference!

Mathematicians literally say that things like division by zero is undefinable. Non-mathematicians like to play around with the answer of "infinity" as though you could get to such an answer and say it equals something. Mathematicians will point out that it tends towards infinity, that its limit is infinity, but they don't make the mistake of saying that infinity is a destination that you can get to - it's just a placeholder for some progression to be "divergent".

I've just explained why they're not different concepts and you've simply stated that they're two different concepts.
I love how you keep accusing me of simply "telling you" when I consistently explain everything, and you simply "tell me" things all the time.

Magnus Anderson wrote:When I use a symbol to represent something (e.g. an infinite line of green applies), I am not describing what that symbol means, I am simply using a symbol to represent that something.

So you're telling me that thing I've already told you again. Thanks again.

Magnus Anderson wrote:There is absoultely nothing that cannot be represented using a symbol. All it takes is to pick a symbol (you can literally pick anything) and say "This symbol represents this thing".

The problem is that you do not understand what it means to say that a symbol is representing something. You have this naive idea that to represent something means to find a symbol that looks exactly like that something. That's not true.

In other words, you think you're an expert semiotician while in reality you know nothing about semiotics

As above I already know you CAN use a symbol to denote infinity - people do it all the time: \(\infty\) See?
But people can also do things that don't make sense like write the sentence "This sentence is false". Just because you can do something, doesn't mean it makes sense.

I know exactly the difference between signifiers and signifieds - it's something I've been explaining to you.
I explictly said that a symbol doesn't have to look exactly like what it represents, and that not looking like what it represents is the whole point.
PUT. THE STRAWMAN. DOWN.

What you're getting confused by is that finite symbols denoting different finite things isn't a big problem, but denoting infinitude with finitude makes it "look like" you can deal with infinitude as though it was finitude. You cannot.
This doesn't mean I'm saying all words need to look exactly like what they represent - that's retarded. Especially since I'm saying the exact opposite for all finite things. As you finally admitted, there's only one way that infinity is infinite, and this one instance makes all the difference since infinity literally says it cannot be finite or represented as finite in any way.

Magnus Anderson wrote:
Silhouette wrote:I don't want any credit nor hope for any personal gain from the success of people learning why something basic is true.

That merely means you're one of those people who do not think but merely follow whatever is popular at the time.

You are deferring to the authority precisely because you have no idea why \(0.\dot9 = 1\).

:lol: :lol: :lol: :lol: :lol:

How the fuck did you jump from "impartiality and objectivity" to "partiality to contemporary popularity"?!!!!!!!!!!!!!!!!!!!!
The same way you jump from "infinites" to its opposite "finites" I guess. Just tell me you're not doing that, and I'll magically believe you though...

I'm saying it doesn't matter if you accept my correct and logical explanations or someone else's who is demonstrably even better at doing what I'm doing - we'll be explaining and concluding the same thing.
It's irrational for you to reject my correct and logical explanations, so I'm suggesting that if that irrationality is because of "me" maybe you could get over it if it was someone else who said the same thing, who you're even less justified in saying they don't know what they're talking about.
You've put me in this bizarre situation where, given your irrationality in rejecting my rationality, it's rational for me to suggest irrationality for you because even with your failure to be persuaded by rationality, the end result of the different means of irrationality is the same as it would be if you didn't fail to accept my rationality, since it's the same rationality, just by professional mathematicians instead of me. :lol:
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 12, 2020 8:38 pm

gib wrote:Yeah, 'cause this is a super high quality discussion already.


Regardless of its present quality, if you start posting funny gifs, you're gonna make it worse than it was (and significantly so.)

I'm actually not that frustrated. The giffies are for comic relief.


Yes, you need comic relief, because you are frustrated (:

I'll concede to this. But I still don't know how it makes sense to say an infinite set is larger than another infinite set. The whole reason why every infinite set is larger than any finite set is that you can't get any larger than infinity.


Not quite. The whole reason why every infinite set is larger than every finite set is that infinity is by definition a number larger than every finite number (but not, as some claim, the largest number possible.)

So if you agree that infinity is a number larger than every finite number, do you still think that infinity is not a number (or that it makes no sense to say that an infinite set has size)?

Oh, and just reasserting "...by adding one to it you necessarily increase it" over and over is?


Reminding you of the definition of the word "add" (or the operation of addition.)

Asserting \(\infty\) + 1 = \(\infty\) is just the answer to your question. It's the reason why adding an apple to the infinite line of apples doesn't make it bigger.


There is only one sense in which \(\infty\) + 1 = \(\infty\) is true. It's true in the sense that any infinite number plus one gives you an infinite number. However, it's not true in the sense that any infinite number plus one gives you an infinite number of the same size.

I could go deeper, but I have a feel no depth would ever convince you (you could look up my replies to obsrvr for a hint).


I feel the same about you (and Silhouette and Ecmandu.) But I don't make a fuss about it. If I don't want to argue, I simply stop arguing.

All in all, you're wrong (and horribly so) and if you don't want to argue anymore, that's perfectly fine by me.

And besides, I'm not doing that. You convince me that what applies to finite sets applies to infinite sets.


Sure. I'm going to do everything while you do nothing.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 12, 2020 9:35 pm

Silhouette wrote:Words that are intended to not be the thing they represent are therefore symbolisation intrinsically lacks in truth - on purpose, due to the utility of deviating from truth. This is therefore absolutely questionable in terms of truth, and that's the whole point of doing it. The word "word" does pretty well in being true to what it represents, and the word/symbol of infinity/\(\infty\) is at the opposite end of this relative scale. It is maximally questionable.


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.

I just said that symbol \(\infty\) CAN be used to represent infinity, but there is maximal pretense in doing so as I explained above.


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.)

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.

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.

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.

Well the appearance of their spelling looks different so you must be right!


Not quite. They mean different things.

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".

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.

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


You know nothing.

As you finally admitted


You're hallucinating.

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

Postby gib » Sun Jan 12, 2020 9:42 pm

Magnus Anderson wrote:Not quite. The whole reason why every infinite set is larger than every finite set is that infinity is by definition a number larger than every finite number (but not, as some claim, the largest number possible.)


Actually, it's defined as endlessness.

Magnus Anderson wrote:So if you agree that infinity is a number larger than every finite number, I don't. do you still think that infinity is not a number (or that it makes no sense to say that an infinite set has size)?


Absolutely, infinity is not a number! I don't have this as well thought out as possible, but the reason infinity is larger than any finite quantity is because it's beyond quantity--meaning it's not a quantity but a direction is insinuated. That is to say, "beyond" implies that you approach it by going in the positive direction, the direction we call "greater". But still, it goes beyond the numbers and therefore isn't a number itself.

Magnus Anderson wrote:Reminding you of the definition of the word "add" (or the operation of addition.)


Really, see my reply to obsrvr.

Oh heck, I'll do it:

gib wrote:There's a common misconception of what "addition" means in mathematics. You see it when people argue that 1 + 1 sometimes equals 1, as in when two raindrops join to make one large raindrop. My pet peeve with this is they're defining "addition" as a process or something going through a change. But when mathematicians talk about addition, they mean something more static, like if you took a photo of the two raindrops, how many raindrops would there be all together in the photo. Addition in mathematics is a way of talking about equivalent meanings, not the result of a process. 2 + 2 means 4. That is, to say there are two objects and there are two other objects is just to say there are four objects. We're not talking about what happens to those objects, whether they "come togther", or they meld to become one giant object, or when they come together they collide and break into a million pieces, etc. It's about how quantities can be expressed in different combinations and permutations.

Magnus's definition makes sense in terms of a process. If you add an apple to a line of apples, the line becomes longer (at least in the finite case). But that's "add" as in "you take the apple and place it at the front of the line such that it becomes part of the line." To that I say, of course you can add to an infinite set. You can drop an apple into a barrel of infinity apples. But as a mathematical concept, it's incoherent. If you "add" one apple to an infinity of apples, in the mathematical sense, you just get infinity again. What it means for that resultant infinity to be "larger" than the original infinity is what I'm grappling with and what Magnus hasn't been able to help me with. So far, it remains incoherent.

I did like your definition of infinity as a property of a set, rather than a quantity. I think that's right. It's the property of being endless. But as such, it's pretty binary. A set is either endless or it's not. I don't know what "less" endless means.


When you say \(\infty\) + 1 > \(\infty\), to me that's the same as saying endlessness + 1 > endlessness, or perhaps red + 1 > red.

^ All things that aren't numbers so don't make sense to plug them into math.

Magnus Anderson wrote:I feel the same about you (and Silhouette and Ecmandu.) But I don't make a fuss about it. Who's making a fuss? If I don't want to argue, I simply stop arguing.


Who said I wanted to stop arguing? I said I want to stop trying to convince you step #3 is your flaw.

Magnus Anderson wrote:All in all, you're wrong (and horribly so) and if you don't want to argue anymore, that's perfectly fine by me.


Ha! You wish!

Magnus Anderson wrote:Sure. I'm going to do everything while you do nothing.


You can start any time.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 12, 2020 9:56 pm

gib wrote:Actually, it's defined as endlessness.


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.

Absolutely, infinity is not a number! I don't have this as well thought out as possible, but the reason infinity is larger than any finite quantity is because it's beyond quantity--meaning it's not a quantity but a direction is insinuated. That is to say, "beyond" implies that you approach it by going in the positive direction, the direction we call "greater". But still, it goes beyond the numbers and therefore isn't a number itself.


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.

Magnus's definition makes sense in terms of a process.


Not really. It makes sense either way.

When you say \(\infty\) + 1 > \(\infty\), to me that's the same as saying endlessness + 1 > endlessness, or perhaps red + 1 > red.

^ All things that aren't numbers so don't make sense to plug them into math.


Endlessness and red aren't numbers. \(\infty\) is. This is evident in the fact that it represents a quantity greater than every finite number.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 12, 2020 10:01 pm

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

Postby gib » Sun Jan 12, 2020 10:38 pm

Magnus Anderson wrote: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.


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

Magnus Anderson wrote: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.


Error: assertion denied!

Magnus Anderson wrote:Endlessness and red aren't numbers. ∞ is. This is evident in the fact that it represents a quantity greater than every finite number.


Error: assertion denied! Error overload! Exploding!

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

Postby Magnus Anderson » Sun Jan 12, 2020 10:55 pm

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


That's true.

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.

You don't get to any "1" or "end", you can't because there is no end


That's true.

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

Postby Magnus Anderson » Sun Jan 12, 2020 10:59 pm

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


Tell that to Google.

Google wrote:infinity
/ɪnˈfɪnɪti/
noun
1.
the state or quality of being infinite.

2.
MATHEMATICS
a number greater than any assignable quantity or countable number (symbol ∞).
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 12, 2020 11:02 pm

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


That's true.

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.

You don't get to any "1" or "end", you can't because there is no end


That's true.

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.


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

Postby obsrvr524 » Sun Jan 12, 2020 11:27 pm

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

Postby Magnus Anderson » Mon Jan 13, 2020 2:00 am

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.

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.

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

Postby Ecmandu » Mon Jan 13, 2020 2:41 am

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

Postby obsrvr524 » Mon Jan 13, 2020 2:52 am

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.

Magnus Anderson wrote:Infinity + 1 either equals infinity or it does not.

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

Postby Ecmandu » Mon Jan 13, 2020 3:41 am

obsrvr524 wrote: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.

Magnus Anderson wrote:Infinity + 1 either equals infinity or it does not.

"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!
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Mon Jan 13, 2020 1:59 pm

Ecmandu wrote:Infinity + 1 either equals infinity or it does not.


Not really. It depends on what \(\infty + 1 = \infty\) means.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Mon Jan 13, 2020 5:54 pm

Magnus Anderson wrote:
Ecmandu wrote:Infinity + 1 either equals infinity or it does not.


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

Postby Silhouette » Mon Jan 13, 2020 6:35 pm

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.


Image

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 :lol:

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 :lol: 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 :lol:

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"...
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