Precisely. By definition, there is a difference between (0.\dot9) and (1). The problem is that some people cannot accept the obvious because they are confused by the complicated. Specifically, what confuses them is the fact that there is no decimal number that can express this difference. This is based on the erroneous notion that the set of decimal numbers is the set of all numbers. That’s not true.
I’m sure the difference between the two numbers can be expressed with a hyperreal number. But then, there are people who do not think that hyperreal numbers are numbers . . .
Either way, you don’t have to find a numerical representation of a quantity in order to know that such a quantity exists.
Thank you for that edifying discourse, E. I was intrigued, and feel I should now risk the following hypothesis; all the great mathematicians were great because they were gay.
You heard em, Andy. Either make some significant changes to your sexual orientations, or continue to flunk this thread. Makes no difference to me.
I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That’s where his infA originated (as opposed to infB). And his reasoning was simply that trying to say “2 * infinity” is like saying “2 * length”. For it to make any sense, you have to be more specific.
I thought it was an interesting argument that there are more numbers than decimals; however, anytime you use a ratio, decimals are implied. Can you contradict this?
Silhouette and I have provided a few compelling arguments already that you haven’t addressed as to WHY 2*infinity is a contradiction.
You keep stating it, but you avoided our proofs through contradiction.
Maybe you think there’s a backdoor that circumvents our “proofs”, that’s fine, but I haven’t seen it yet.
I personally used the “cheat” argument and the “2 gods” argument and the “quantum flux” argument.
I can let silhouette speak for his own arguments, the “straw/box” argument of his is really good.
There is no decimal representation of (\frac{1}{3}), for example. (0.333\dots) isn’t it. That’s a different number.
If you’re asking for a rigorous proof, I don’t have one. And I don’t think it’s necessary. You can’t demand of me to prove more than it’s necessary to prove in order to show that (0.999\dotso) does not equal (1). I sometimes do so, but it’s only because it’s fun, not because it’s necessary.
On the other hand, since you are the one claiming that the set of decimal numbers is the set of all numbers, it’s up to you to prove it.
You provided no compelling arguments, no proofs. You merely defined the word “infinity” in your own way and used the logical implications of such a concept to “prove” that I am wrong even though that’s not the way I define the word “infinity”. (In fact, that’s not the standard definition of the word.)
The “straw/box” argument merely shows that he does not understand the simple fact that there are no inherently correct definitions of words.
You stated outright that you don’t believe that 1=0.999…
So what are we arguing about?
You don’t believe in convergence theory and convergence theory is necessary to quantify infinity.
You know? There’s a big difference between infinity and infinite sequences.
1/3 * 3 equals nine in base 10; 10/3 equals 3*3 which equals 9 to get back to ten.
So… really… we have an operator trick.
Now, if as you state, infinity doesn’t converge, then there are no orders of infinity that can be used, because as non convergence infinity ceases to be applied as a unit of measurement which is subject to operators such as additions and powers.
It’s by definition that there’s always another (9) in (0.\dot9) and therefore never ever any room for anything at all between (1) and (0.\dot9) no matter how hard you go.
“Infinite means there is always more” - sure: just more 9s in this case though. The recurring “dot” is applying to the 9 it’s on top of, not anything else.
The result of doing this is that there’s “infinitely no space left”: an infinite lack of space for even one finite infinitesimal - never mind an infinite number of them!
The idea that dividing one by 3 and then multiplying the result by 3 leaves an infinite gap between what you started with and finished with… - this should be a pretty good clue that you’re inverting the sense of what infinity is if you think that. There’s no infinite gap when dividing 3 by 3 and then multiplying the result by 3 - be consistent.
The idea that the closest possible decimal quantity to (\frac{1}3) is (0.\dot3), therefore there’s an infinite gap between them should sound some alarm bells too.
This is perfectly circular by James here.
To have the conditions to do something (use maths with infinities) you need to have those conditions (a defined infinity).
I can clearly state that in order for black to be white, you first have to define a black white.
The conclusion that the in-finite can be de-fined mathematically is assumed in the premise that a standard in-finite can be de-fined.
Again with the treatment of the quality of having no quantity as quantity.
“For non-specified ending to make any sense, you just have to be specific with the non-specified ending”…
I am saying that defining (giving ends) to infinity (endlessness) is a contradiction in terms.
This is correct though.
I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.
“0.000” defines the start and “1” defines the end. With all ends accounted for, the “…” attempting to imply infinity is a contradiction in terms.
This is what happens when you try to define infinity by giving it ends: you make it finite (the opposite of what it is).
If there could be an infinity of something, e.g. 0 digits entirely enclosed by finite bounds, then the “finite” bounds would be indefinitely separated such that they’d never come to exist - just like the end of infinity never comes to exist by both definition and derivation.
(0.000…1) is a numerical representation of an invalid number. Just goes to show that just because you can represent something numerically, doesn’t mean it’s a number. Fortunately my logic went in the other direction, that to be a number, you have to be able to represent it numerically. Let’s avoid “affirming the consequent” here, eh?
So as I was saying, numbers are necessary in any valid format. The invalid (0.000…1) therefore doesn’t count.
I’d accept any valid numerical representation, decimal or otherwise.
(\sum_{n=1}^\infty\frac9{10^n}) or (0.\dot9) are fine because here the infinity isn’t finite nor the finite infinity. The finites and the infinites are separate: the finites are defined in a very specific way such that the infinity can then say “now go do that defined thing an infinite number of times in the one way that infinity can be infinite”.
(\frac1\infty) is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated. Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It’s an instruction to operate on defined finites in an ongoing way - it’s never “the end” of having been operated on - because obviously “the end” i.e. finitude is the opposite of endlessness. (2\times\infty) is invalid for the same reason.
Just be consistent! That’s all you need. Math wouldn’t be math if it wasn’t consistent.
Again, this violates consistency.
Why is it okay to divide 9 by 3 to get 3 and then multiply that by 3 to get back to 9, but not ok to divide 10 by 3 and/or then multiply the result back by 2 to get (10) or (9.\dot9) by using the exact same method?
You’ll no doubt accept that 10/4 is 0.25 but somehow as soon as the remainders don’t resolve to 0, it’s not valid?
You probably didn’t even stop to think that your logic here would invalidate the irrational and transcendental numbers, right? I bet you think they’re okay, but the exact same thing isn’t okay when it goes against your point.
You can call black “white” and it would still be what it is and and not its opposite.
I keep saying it can be interesting to see what happens if you treat infinites as finites, but that doesn’t make it true - only useful.
Only count numbers as numbers and you’ll be up to speed.
But then this says it all really.
I’m trying to get you to sufficiently delve into the details but if rigor isn’t necessary to you, there’s not been any point in me trying.
Infinity is the quality of having no quantity is something I’ve said over and over, which is exactly how it’s both defined and derived. This isn’t “my own way”, it is “the” way.
These kinds of resorts of yours are pretty low - to just deny that all this work people have gone to for your sake was never even done. It’s at the very least rude, if not outright ignorant.
I’m sorry, I don’t accept the definitions of the words you’ve used to say there are no inherently correct definitions of words.
You see how self-annihilating this kind of argument of yours can get?
If you can’t legitimately define the terms for your argument to be valid, then throw out words altogether, right?
Try imagining the possibility that you’re wrong about something and think about how you’d react and resolve it. If the answer is to double down indefinitely, then if neither of us ties up this resolved issue, this debate will go on endlessly. Defining a finite resolution to such an infinite scenario would be invalid - is invalidity what you’re aiming for?
Now, suppose that you want to quantify (i.e. represent using some sort of symbol) the number of whole apples that is in front of you.
And suppose that you’re only familiar with the set of natural numbers.
Tell me, how would you represent what you see in front of yourself?
Note that we’re not interested in approximate values, only in exact values. How would you answer the question “What’s the exact number of whole apples in front of you?”
No natural number can describe the quantity of whole apples that is in front of you. The set of natural numbers allows you to represent one apple, two apples, three apples and so on, but what’s in front of you doesn’t quite fit any such description.
What does this mean? Does this mean that the number of whole apples in front of you is equal to one (unfortunately, no zero among natural numbers) or does it mean that your vocabulary is limited?
It would be really strange to say that the number of whole apples in front of you is one for you would be denying your senses (or at the very least, you would be contradicting yourself.)
That would be a very backward kind of reasoning, one that I’ve found to be frequently used in sophistical arguments.
What the above situation indicates is not that your senses are deceiving you, nor that your prior assumptions are incorrect, but quite simply that your vocabulary is limited i.e. you have no symbols with which you can represent the quantity that is in front of you.
That’s why you have to extend your vocubalury with new kinds of numbers, such as rational numbers.
With rational numbers you can say there is exactly (\frac{1}{2}) whole apples in front of you.
The set of natural numbers contains no numbers smaller than (1). That does not mean that numbers smaller than (1) do not exist, right? It simply means that the set of natural numbers is a limited vocabulary, and hence, not the set of all numbers that can be imagined.
Suppose now that there is an infinite number of apples in front of you and that you’re only familiar with real numbers. You’d be facing a similar situation. The set of real numbers contains no such number, and yet, the quantity that is in front of you requires it. What would you do? You’d do the same thing: you’d extend your vocabulary.
And what if you had this same infinity of apples in two different places? How would you represent that? “Infinity” wouldn’t be enough because you’d want to represent the obvious difference between the two endless quantities. For this isn’t the same infinite quantity you saw before. It’s clearly twice the size. You can say the quantity is infinite but only in the general sense of the word. But if you wanted to be specific, you’d have to say something like (2\times\infty) or (2\times\omega) or (2\times \text{infA}). You’d have to pick a standard against which infinite quantities would be measured.
The same applies to the difference between (0.\dot9) and (1). You’d have to extend your existing vocabulary by introducing infinitesimals.
The lesson to be taken is that JUST BECAUSE YOU HAVE NO SYMBOLS WITH WHICH YOU CAN REPRESENT SOME KIND OF REALITY, IT DOES NOT MEAN THAT THAT KIND OF REALITY DOES NOT EXIST.
Ecmandu and Silhouette argue that it’s impossible to add two infinite quantities together and get a bigger infinity quantity because they work with their own concept of infinity according which the word “infinite” means “the property of having no ends of any kind” where “end of any kind” has a very, very, very, very broad meaning. Ultimately, the only thing they are showing is that THEIR concept of infinity cannot be added to, subtracted from, multiplied, divided, etc. None of that has any relevance to the subject at hand.
So, Magnus , you’re saying that all infinities can be bound, you’re saying that infinity can be bound in general? You’re saying that the endless ends every time? (Contradiction)?
It’s by definition that there’s always another (9) in (0.\dot9) and that means that there’s always a gap between (1) and (0.\dot9). This is obvious as hell, the fact that you have the guts to deny it is worrisome.
That’s correct but it’s a strawman (a product of your imagination.) Noone said, and noone’s logic implied, that dividing one by 3 and then multiplying the result by 3 leaves a gap between what we started with and finished with.
I assume what you mean by “infinite gap” is compatible with “infinitely small gap”. If so, that’s true: there is an infinitely small gap between the two values. If you think that this should sound some alarm bells, you should explain why.
The point is that if you want to determine whether (2\times\infty = \infty) is necessarily true or false, you have to put the two infinite quantities in relation to each other. Otherwise, the answer is indeterminate: it might be true but it also might be false. The same applies to expressions such as (2\times\text{finite number} = \text{finite number}). If you want to know whether such an equation is necessarily true or necessarily false, you have to define the relation between the two finite numbers e.g. if you say that the second finite number is twice the size the first number, then the equation is necessarily true. Otherwise, the answer is indeterminate.
Your argument is that no relation can be defined between two infinite quantities. Which is a ridiculous claim.
And that means exactly what? How do you give “ends” to an infinite set? Don’t talk about sequences, that’s irrelevant. Take the set of natural numbers. The set is infinite. How do you give “ends” to it? You can give it an “end” by making it finite, but by doing so, it’d no longer be infinite. (And noone is doing this, anyways.)
Your problem is with statements such as “There are two planets populated by an infinite number of organisms existing at two different points in space.” You have this weird idea that such statements are contradictory because they speak of quantities of things that are spatially bounded i.e. things that do not exist everywhere but only within a limited portion of space. You have a problem with the idea that there can be an infinite number of things within a finite portion of the universe. You think this is a contradiction because it does not abide by your concept of “true” infinity, the concept of infinity as “truly endless”. The fact is that noone cares about your concept of infinity. It’s not the one that most people work with. And it’s certainly not the one I am using. You can’t say I am contradicting myself if I am not saying what you think I’m saying (and I’m not saying what you think I’m saying because I’m not working with your concept of infinity.)
That would be you channeling the spirit of Ecmandu and also the spirit of Carleas and many other people who make the same exact silly claim because they did not bother to understand what the symbol represents. Basically, you’re too confident about how well you understand other people’s positions.
There’s no (1) at the end of the infinite sequence of (0)s. How many times must it be said?
Sure, it’s an “invalid” number, whatever that means. And if you’re saying it’s not a representation of a quantity that exists, why is it so? Because you say so?
It’s a number, it has been shown to be a number, it’s just that you don’t want to accept it.
Why is that an invalid format? Because you do not understand it?
You’d accept anything familiar to you and reject anything that is not. Sort of like rejecting rational numbers because only natural numbers are familiar to you.
As if (0.000\dotso1) is a finite infinity i.e. a contradiction in terms.
Basically, you think it’s a contradiction in terms. That’s all you said. Of course, without any argument whatsoever.
No, infinity is not an operation. That’s the spirit of Ecmandu taking you over. Don’t let it take you over.
Yes, it’s invalid because you think it’s a contradiction in terms because you don’t understand what it stands for.
That’s exactly thet point of this thread. (0.\dot9 = 1) betrays consistency.
(9\div3\times3 = 9) and (10\div3\times3 = 10) are true. What’s not true is that (10 = 9.\dot9). Again, a strawman.
Because it’s obvious that (10\div3 \neq 3.\dot3). That’s how you can know that the division algorithm that we use does not give us correct results when dividing certain numbers.
The word “infinite” simply means “endless”. I have no idea what “the quality of having no quantity” means. “No quantity” sounds more like “zero” than “infinity”. But “zero” is also considered a quantity. I am not really sure there’s anyone beside you (other than Ecmandu) who defines infinity that way.
Is it a contradiction in terms to say that an infinite number of things exist within a limited portion of the universe? Depends on what you mean by “infinite”. In general, when we say “an infinite number of things” we mean endless in terms of quantity i.e. we want to say that the number of things is endless. The fact that they are located within a limited portion of the universe (which means they have spatial ends) does not change the fact that their number is infinite (that they have no end when it comes to their quantity.) So there is no contradiction whatsoever. Similarly, saying that there is an infinite quantity of things in one place and an infinity quantity of things in another place is not contradictory. It’s only contradictory if you’re working with a different notion of the infinite. But such a notion is irrelevant.
Well, I’ve been doing a bit more than just asserting. I gave some counter-examples as a means of proof by contradiction:
Not exactly proof that (\frac{\infty}{50}) = (\frac{\infty}{60}) = (\frac{\infty}{70}), but definitely proof that (\frac{\infty}{50}) > (\frac{\infty}{60}) > (\frac{\infty}{70}) can’t be true.
I also tried to explain why you have no right to switch between contexts when talking about infinity:
^ But given your lack of response to this, I think it went over your head.
^ That, and I did argue earlier in this thread that infinity is not a number, and therefore arithmetic doesn’t necessarily apply to it. I know you can take the symbol (\infty) and plug into mathematical equations and do algebra with it, which is probably what you’re getting at when you say you can do arithmetic with it, but (\infty) in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it’s place and the same rules would apply: (\alpha)/60, (\phi)/60, /60. (\infty) in this case doesn’t mean “the highest number possible”, it means “some undetermined quantity”… but it has to be a quantity, otherwise it doesn’t apply. The only sense I agreed with you that you can do arithmetic on (\infty) is if you treat (\infty) as a unit (where the quantity would be 1… like 1cm ← cm is the unit, 1 is the quantity). This is why I said you shouldn’t switched contexts willy-nilly–you’re switching from saying “1 of something symbolised by (\infty)” to “and infinity of something else (like train carts)”. When you do that, the math suddenly doesn’t work. (\infty) in this case is not simply defined as “an infinite number of things”, it’s defined as “some undetermined thing” (like a variable). But if you want to say you can do arithmetic on infinity (as an infinite number of things), I deny your right to do so.
That’s pretty damn basic. I’d even say definitional… as in, I never thought you would have asked for proof given that that’s essentially how we define “shorter”. If Max is shorter than Harry, that means the tip of Max’s head is below that of Harry’s. If neither has a tip at the top of their head (because they’re infinitely tall or whatever), what does it mean to say one shorter than the other?
Look, this dumbing-down approach of “you’re just denying yourself/lacking the language for it” is sadly wasted on me - I know all too well how it is to deal with people whose vocabulary is all too conservative and restricted.
In fact, I love thinking outside the box - unreasonably so. I love the opportunity to do so.
I also love the rigorous rationalisation of exactly how you get outside the box, and the appreciation of every tiny step with all the exact minutae involved to get there, including the illegitimate ones.
I know what you think you’re explaining and exactly how you’re getting there - but with the apparent lack of telepathy at my disposal, I don’t seem to be able to get you to understand that I understand every single thing you’re doing and saying, whilst also rejecting it.
You tell me: how do I communicate the fact that I don’t need apple analogies to understand what you’re saying, whilst still rejecting it?
Tell me how, because you’re just wasting time, keystrokes and computer bits for both of us.
By the way, if you had “infinite apples in front of you”, you’d be an apple and there would be nothing but apple everywhere.
You can’t do that “twice”. Everywhere is already taken.
If you had an infinite line of apples extending out “in front of you”, you’d not be able to see beyond a small number of them, you’d never be able to get to (\frac{1}2) way along to even divide it into 2, nor would you ever be able to divide all them into any finite number of apples. There would be a finite limit on apples going upwards, downwards, left, right, even backwards in this situation - just not forwards: the same as the natural numbers. Thus the line of apples is distinguished from an entire universe of apples by virtue of its finite constraints up, down, left, right, back, and in all other ways other than forwards. So any “size” of such a line of apples would be dictated by the size of the finite constraints, not the infinity that stretched before you. The infinity has only 1 type (quality), and no tokens (quantities): I am being very very very specific about the kind of ends that are going on here - don’t give me your shit about how I’m being broad.
You can imagine “apples in front of you” in any way you like, and the same principle applies - you cannot get to the “end” of that line and add any more.
But pay attention to the following:
You could even set up a line of apples directly underneath the first, and there would still be infinite apples. The fact that you started the line with 2 apples on top of each other doesn’t give the infinite line “more” or “less” apples. You changed a finite constraint. The infinity is just as unforgivingly endless in its monolithic single extension either way. There was no end before, and there’s still no end - quantity is just as meaningless either way.
You regard the change in the finite number of apples that start the infinity as a change in the infinity.
This is just wrong, I’m sorry.
It’s the finitude that governs the size. Any infinity has only one way of being infinite, and it defies all possible quantification, or operation using quantities.
You can “imagine” what it would be like “hey, let’s say it was the infinity was affected by changes to finite constraints” and go on from there, and you have your arguments.
Fine.
Do that.
It’s not legitimate, but let’s see what happens.
That’s not true. You have to understand that I am not using the word “infinite” the way that you do. In fact, apart from a handful of people, noone is.
That would be true if what I meant by “infinite number of apples” is “there is nothing in space other than apples”. But that’s not what I meant. The word “infinite” simply means “endless”. It does not mean “endless in every single way one can think of”. You can have an endless number of things within a portion of the universe that is not endless in certain regards. Of course, in order to accomodate all of the infinitely many things, the portion itself must have enough room and that means it must have infinite room, so the portion must be infinite in at least one direction but there is no requirement to be infinite in all directions.
Again, the word “infinite” does not mean “the property of being the only type of thing that exists”.
That’s irrelevant.
I’m afraid you’re confusing conceptual matters with empirical ones. Here in this thread, we’re discussing concepts. We’re not talking about whether it’s possible, nor how is it possible, to determine whether things that can be represented by our concepts exist. So if you’re claiming that there’s no infinite quantities in the universe, or that it’s impossible to establish whether infinite quantities exist, you’re wrong, to be sure, but most importantly, what you’re saying is irrelevant.
You can divide the infinite line of apples into two smaller equally-sized infinite lines of apples by removing every second apple from it and placing it elsewhere.
Of course you can.
The set of natural numbers does not have such limits. That’s because it’s a set.
The infinite line of apples does because it’s a physical object, not merely a set, and the fact that it’s infinite in only one direction (and not all) is completely irrelevant to the subject at hand. This is because we’re talking about THE QUANTITY OF APPLES, not about THE PHYSICAL LINE OF APPLES. The quantity of apples can only be infinite in ONE direction, and if it is infinite in this one direction, then it IS infinite.
If a hungry man walked past the infinite line of apples, grabbed one and ate it, that act alone would tell us that the infinite line of apples has one apple less than before.
You don’t have to go the “end” of the line in order to add more apples. You can add them elsewhere (just like with finite sets.)
You’re trying to apply to infinite quantities what applies only to finite quantities (which makes you guilty of your own accusations.)
/60. (\infty) in this case doesn’t mean “the highest number possible”, it means “some undetermined quantity”… but it has to be a 
