Is 1 = 0.999... ? Really?

Did you really just ignore this whole post ???

Besides, how do you “reach infinity faster”?!

viewtopic.php?p=2755866#p2755866

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.

Couple things:

  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.

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

Using this argument, we can determine that if you split it, you’re left with:

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

Etc…

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

Etc…

No matter how you divide from the “highest order” the sets will all be in 1:1 correspondence.

Done.

Proof.

QED.

You guys lost.

Now continue your meaningless discussions.

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.

The pattern of always being longer never changes.

You’re still ignoring me.

Do you want the answer to this or not ?

viewtopic.php?p=2755884&sid=dfdee997b272ae54ab79e6c6362c2cb0#p2755884

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.

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.

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.)

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.

That’s not an argument.

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.

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.

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!

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

Or that it doesn’t make sense.

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.

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.

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).

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”).

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.

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.

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!

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.

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.

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

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.

:laughing: :laughing: :laughing: :laughing: :laughing:

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. :laughing:

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

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

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)?

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

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 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.

Sure. I’m going to do everything while you do nothing.

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.

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.)

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.

That’s not true.

Not quite. They mean different things.

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”.

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

You know nothing.

You’re hallucinating.

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

Actually, it’s defined as endlessness.

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.

Really, see my reply to obsrvr.

Oh heck, I’ll do it:

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.

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

Ha! You wish!

You can start any time.

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

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

Not really. It makes sense either way.

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

I know you’re ignoring me Magnus,

But here’s the deal.

If you believe calculus,

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

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

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

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

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

Error: assertion denied!

Error: assertion denied! Error overload! Exploding!

That’s true.

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

That’s true.

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

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

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