Infinity is not a placeholder. Placeholder is a base concept. Infinity is an operator.
Well, you’re sorta going off with your own definitions here, but fine. I still don’t get how (\frac{\infty}{100}) gets us a “smaller” infinity than (\infty). I’m not even sure how one divides infinity. Let’s get back to basics. Infinity just means: no end. The symbol (\infty) means the property of having no end, or no finality, or no limit, etc. If (\frac{\infty}{100}) is still infinity (albeit a smaller infinity according to you), that just means there still is no end, no finality, no limit, etc. What does it mean to say it’s a “smaller” no end, no finality, no limit, etc.? It seems pretty binary–either something has an end or it doesn’t.
I think you might be objectifying the concept of infinity–i.e. you’re thinking of it as an object, and therefore subject to dividing up into smaller parts.
I don’t deny that there can be different orders of infinity (which is what I think James is getting at), but you don’t get there just by adding 1 to infinity. It’s more complicated than that. Take the example of the two parallel lines Ecmandu brought up. He says that since there is an infinite number of points in the first line, adding the second line, which also has an infinite number of points, doubles the number of points. Whether the arithmetic works like that or not (I don’t think it does), that’s not an example of a higher order of infinity. A higher order of infinity is more like a plane compared to a line–something you arrive at by compounding an infinity of infinities. The lines are only infinities of points, not infinities of infinities. But a plane is an infinity of infinities because it is an infinity of lines which in turn is an infinity of points.
The idea is like this: infinity, if you want to imagine it as something that you can somehow reach, is a “transcendent” object. To get there, you have to transcend all finite things (in the case of numbers, all numbers). It’s impossible, just like transcending space and time is impossible for physical beings like us, trapped within space and time. No matter how high you count, you’re no closer to infinity, just as no matter how far through space you travel, you’re no closer to being outside space. But if you want to suppose you somehow could reach infinity, you can imagine skipping the journey of counting (or traveling through space) and magically arriving there. In that case, you must objectify infinity–meaning you must now think of it as an object–i.e. a finite thing–this is your new unit, your new building block, your new fundamental particle in a higher universe–it is your new point. It’s like when you transcend all points in the line, you get the line itself. You can then treat the line as the new unit and start over adding lines together. Now counting consists of counting these lines, these infinities, and the new infinity to strive towards is the plane, the new transcendent object in this higher universe.
Going back to counting points, as in the case of counting up the points in the second line, is to go back below the first order infinity. You may think of it as going back to a different universe (i.e. a different line), but this is not the same as a different order of infinity. Adding the first infinity to the first point in the second infinity is not valid. It does not equal (\infty) + 1. It’s adding apples and oranges. The infinity and the point are not only completely incommensurate objects, but they, in a sense, don’t even exist relative to each other (that’s why infinity is “transcendent”–it is “beyond” the universe of points–the point relative to the infinity could be thought of as the infinitesimal). You can add up objects in a box, but you cannot add the box to the objects (for example, 2 apples + 3 apples = 5 apples; but what about 2 apples + the box the apples came in? What does that equal?). But you can add several boxes together. The arithmetic works only in the same universe, not across universes.
It means we have to be careful when mixing up infinities in our arithmetic. We can’t just assume the same mathematical rules will apply.
That’s true. I’d even say it’s smaller than all negative finite numbers.
These aren’t infinities. Infinitesimals are infinitely small, but an infinity means an infinite number of things. 0.999… isn’t infinite either–it’s a finite number (maybe even 1)–but it has an infinite decimal expansion (according to this specific notation).
I get stuck on what an infinite number of organisms means. Not being able to adequately define it, I’m at a loss to say whether you can add more organisms to it to get something greater than infinity. Part of the problem is that it’s impossible (I can’t imagine it in any case) to have an infinite quantity of anything in the real world. Space might seem like an exception but space is more like a lack of something rather than a something. What would an infinite number of organisms look like? What would happen if you had it? I’d have to figure this out before I can say yes or no to the question of: would you have twice as many organisms if you added another planet of infinite organism.
Yes, this is what I was getting at above with infinities being the new units in a transcendent universe. But you get two trains, not two times the number of carts. You get two planets of infinite organisms, not two times the number of organisms. These are the units, the basic indivisible building blocks–trains, planets, lines, infinities–and you make an invalid move when you jump between contexts, first treating them as units, next treating them as infinite sets of smaller units. If you talk about adding the two trains together, stick to the trains as a whole, never talk about the carts.
Of course you do. Come on, Anderson. This “something” is finite. The normal rules of quantities and arithmetics will always apply to finite things.
This is where I struggle. I’m hard pressed to imagine how one divides infinity. Where do you begin to divide an infinite train? OK, you can divide the train wherever you’re standing–unhinge the two carts that happen to be in front of you. That’s dividing by 2, but what about higher numbers? How would you divide an infinite train into 3 parts? After you unhinge the carts in front of you, where do you do your next unhinging?
Yes, you can do this. But again, this is in the context where the infinity is just the unit. And while you can have half a unit, it breaks down when you cross that line and translate the “unit” into an infinite quantity of things. Remember, infinity means “no end”–so when you go from dividing infinity as a “unit” to dividing infinity as “no end to the number of things”, it ceases to be clear what that means, or what the results are. I’m not sure what to say about dividing an infinite population of organisms into two halfs. I’m pretty sure each half must also have no end to the number of organisms within them, but does this mean there are less organisms in each half than in the whole? Does it mean there is the same number? I don’t know. (And I believe the consensus among mathematicians is that there is no answer to these questions because “no end” is not defined as a number.) It’s pretty clear that when you treat infinity as a unit, half a unit is less than a whole unit, but the quantity here is the number of units, not the number of things making up the units. If you want to talk about the quantity of things that make up the unit, you have to jump a line that transcends the universe under consideration, and the rules of the game change.
But how can you tell? Did I leave a foot print at my prior position? Remember, we’re talking about a universe with nothing in it except me and the space around me. Does motion through such a universe even make sense? More to the point, are you really closer to the edge of the universe, or are you the same distance away? Is (\infty) - 2 < (\infty)? Or is (\infty) - 2 = (\infty)?
I’ll reply to all of your points, gib, but before I do that, I want to tackle this one:
Yes, we can say that an infinite train is a unit, and like with all other units, we can do basic arithmetic with it. So, one infinite train + one infinite train = two infinite trains. But also, since units can be divided, thanks to rational numbers, you can take one infinite train and divide it into, say, 60 inches long segments. What you’ll get is (\frac{\infty}{60}) 60-inch segments. You can also divide it into carts, if you want to. So yes, if you have no problem with rational numbers, it makes perfect sense to talk about infinite trains in terms of their parts.
Here’s the remainder of my response to you:
The rules of arithmetic apply to everything that can be quantified. Everything that can be expressed as a number of something is subject to the rules of arithmetic. It’s pretty clear that infinity can be quantified. One infinite train + one infinite train = two infinite trains. Infinite train / 100 = 2 x Infinite train / 200. It’s obvious. Now, the question is: what makes you think that infinity cannot be quantified? It’s not enough to state it. It’s not enough to say that arithmetic does not apply to rational numbers. You must have a rationale behind it. And it must be a good one. You can’t just say something like “Well, rational numbers aren’t natural numbers, and arithmetic applies only to natural numbers, so arithmetic cannot be applied to rational numbers.” You can’t just say something like that and expect others to endorse your position.
You say “this something is finite”. Well, an infinite train, being a single thing, is also finite. It’s one infinite train. And one is a finite number.
Here’s an example. There’s an infinite queue of people standing in front of you. How do you divide it? There are many ways to do so. But let’s say you want to divide it into two equal halves. What would you do? Again, many ways to go about it, but let’s pick the simplest approach: you take every other person out of that queue and you put them into another queue. That’s how you divide infinity. It’s like taking the set of all natural numbers and then dividing it into a set of even natural numbers and a set of odd natural numbers. You can’t say the resulting sets are equal to the set we started with because you created these resulting sets by subtracting a smaller infinity of numbers from the initial set.
Wherever you want. Of course, since infinite trains have no end, you cannot start with the end. But you can start anywhere else. It does not matter.
Removing two carts is not dividing by 2, it’s substracting 2. Dividing by 2 would be removing every second cart. Dividing by 3 would be removing every third cart. And so on.
Two units are more than one unit regardless of the number of smaller units they are made of. The only requirement is that these units are equal in size. And there is absolutely nothing strange about making claims such as “There are two planets each populated by an equally infinite number of organisms”. “Equally” here means that for every organism on planet A there is a corresponding organism on planet B.
I am pretty sure that James is not merely saying that there are different orders of infinity. Rather, he’s quite literally saying that one added to infinity is a greater infinity than the one we started with. If you want, I can quote him for you.
You can’t “get there”. There is no “end” you can get to. What you can do is complete an infinite number of actions within a finite period of time. Aristotle called this sort of infinity “completed infinity” or “actual infinity”. It’s a concept many have problem with because it confuses people.
But they do 
Which is why I spoke of infinity in the broadest sense of the word.
This isn’t a discussion about what’s possible in the real world. This is a discussion about the meaning of the words that we use. And even it is impossible to have an infinite quantity of anything in the real world (which I disagree with) it shouldn’t be too hard to imagine what a planet populated by an infinite number of organisms would look like.
That’s irrelevant. It has little to do with the subject.
Because of the math that immediately preceded the statement.
This is a strange requirement.
On one hand you’re demanding exact equality in train length, not merely approximate, and on the other hand you’re demanding a visual evaluation, which is always going to be approximate.
Visual evaluation will never be perfect in the same way that you never get to the “end” of infinite exactness in practical terms (such as sight) because infinite time is a practical concern that cannot be realistically achieved.
There’s also physical complications of visual light having a minimum wavelength, and if scales go below this wavelength you won’t be able to see anything - even if instruments that can detect below this wavelength translate back to visual light wavelengths, the illusion of zooming deeper will never be exact due to the same restrictions. Even if the visual data is translated into another type of data that doesn’t suffer from such restrictions, we’ve only figured out how to detect the effects of particles rather than the particles themselves directly - and there’s always the theoretical smallest possible length “the Planck length” that would need to be transcended to keep going towards the unreachable “exact” infinity. So as I started this paragraph, the practical considerations of getting to infinity is itself a contradiction in terms.
In light of these issues with exact train lengths, we can at least perform a simple sense check to see if the trains will at least look approximately the same length - even if that both is and isn’t what you were asking for with your mixture of “visual” and “exact” evaluations. After about 4 wagons, the naked eye is going to find it hard to see the difference that 0.01 inches makes, at which point we will see the first 4 wagons of Train A being being 90+9+0.9+.09 = 99.99 inches long, and Train B being 93.75+5.859375+0.3662109375+0.02288818359375 = 99.99847412109375 inches long (5D.C+5.DC+0.5DC+0.05DC = 63.FF9C inches long in hex, where 100 in decimal is written as “64” in hexadecimal). They both “look” like they’re converging towards 100 decimal inches in length at least, with additional wagons not going to cause the train lengths to exceed 100. The remaining lengths will just appear as non-distinct “train” that will top off the remainder to look like 100 for both trains at all possible magnifications.
But enough about the visual test you’re proposing - the only “exact” methods are going to have to transcend the practical into the theoretical, for which you have to accept the math with its concepts such as “one-to-one correspondence” in infinite series:
$$s=\sum_{x=0}^\infty \frac9{10^x}\to \frac{s}{10}=\sum_{x=1}^\infty \frac9{10^x}$$
$$s-\frac{s}{10}=\sum_{x=0}^0 \frac9{10^x}=9=s(1-\frac1{10})$$
$$s=\frac9{0.9}=10=\sum_{x=0}^\infty \frac9{10^x}=9.\dot 9$$
Finally we are in agreement.
(\infty\ = \infty) makes no sense.
Neither does Magnus’ “2 + 2 = Finite number = 4 + 4” for the same reason: because finite and infinite are non-specific qualifiers (qualities), not quantities.
({2+2, 4+4} \in {finite}) doesn’t mean (2+2 = 4+4) are equal any more than ({\infty, \infty+1} \in {infinite}) means (\infty\ = \infty+1)
The same goes for Socrates and Aristotle not being one in the same person just because they’re both philosophers: “(\in)” (\neq) “(=)”
All of this is fine if (\infty) represents a finite number, which of course it doesn’t. It’s the literal opposite, in fact.
The transfer principle is supposed to hold, but what really is (\frac{\infty}{90}) or (\frac{\infty}{100})? 90 and 100 are quantities but (\infty) is not. Endlessness divided by some quantity is still endlessness, even if instead of dividing it into 90 sections, you divide it into sections of 90 like in your example. So “a” is infinite 90 inch segments and “b” is infinite 100 inch segments - both of the set “infinite” but not equalling each other or anything at all because they’re not quantities.
If you don’t respect this fundamental categorial difference, you get absurdities like the total length being shorter than its parts like with (a < \infty \hspace{0.1cm} \text{ninety inches segments}) and (b < \infty \hspace{0.1cm} \text{hundred inches segments}). It’s like hiding infinity in a set of equations to make 1=2 in those silly little proofs, it’s just a mistake.
Cantor’s diagonal argument is misleading.
There is a hidden mechanic within the structure of the argument itself that only makes it seem persuasive on a superficial level.
“One-to-one correspondence” requires a 1:1 ratio between “the number of digits in each listed series” and “the number of possible combinations of those digits in whatever numeral system is being used” (so that every diagonal or whatever possible combination of digits could be found).
In decimal, for every number of digits “n” in each series, you are going to need 10^n series to find all possible combinations of those digits - so obviously there is no 1:1 correspondence here.
There is an “n:10^n” correspondence.
Even for binary series, there’s an “n:2^n” correspondence, which still is a long way off 1:1 - moreso the more digits in each listed series.
For example, a list of 1-digit-long series would require 2 combinations (1:2), 2 digits would require 4 combinations (also 1:2), 3 digits would require 8 combinations (3:8 correspondence) and so on.
For a 1:1 correspondence, you would need a numeral system of base “b” such that “n = b^n” for however many digits “n” are in each listed series.
This would require a seemingly abstract numeral system of base (n^{1/n}), yet this is actually satisfied by the unary integer base of “base 1”.
This is obvious if you think about it.
If the unary digit to be used is 1, then the list of series maps with perfect correspondence as follows:
s1 = {1…}
s2 = {11…}
s3 = {111…}
s4 = {1111…}
s5 = {11111…}
and so on…
It doesn’t matter if you add in any whitespace to these series to make each series look the same length as blanks or anything else do not count as a 2nd digit in unary that affects the quantity in any way.
Any diagonal or line of any kind (that doesn’t double back on itself) taken through these series will not reveal any new combination that hasn’t already been listed.
Cantor’s diagonal argument is refuted on the grounds of the numeral systems he used.
All possible quantities can be denoted in unary just as well as in binary, decimal or any other numeral system so there is no issue here.
QED
Silhouette,
We approach philosophy of math the same way.
I actually proved that all you have to do to allow the diagonals is to add another dimension, say 1 and dimension 1.1 (which covers all the diagonals)
So then I developed another proof called “dimensional flooding”… but dimensional flooding only works in a well ordered set (sequential)
As we all know, the rationals are enumerated by what’s called “a scattered set”
There is no reason to disbelieve that a scattered set can enumerate all of the reals (not just rationals)
You went into a bit of greater detail than me to explain that infinity is not a quantity.
I try to keep thing as simple as I can by explaining infinity as an operator: what does plus divided by plus equal? It’s absurd!
If people can understand that infinity is not a quantity but rather an operator, they’ll stop making this mistake.
bro i’ve been tellin andy to respect the fundamental category difference for years and he ain’t tryin to hear it. every time i tell him, he just kinda stares off and begins pacing back and forth. i dunno, maybe he sees something we don’t. all i know is that he can’t be bothered when he’s pacing like that. he doesn’t even answer his phone.
But (\infty) isn’t a rational number. Both the numerator and the denominator have to be rational numbers (come to think of it, any real number will do).
On the other hand, you can think of (\infty) as a unit, in which case it’s more like 1 (i.e. one infinite train), and divide that into 60 parts, but each part isn’t a 60 inch long cart, but just 60 infinitely long train segments.
Is this what you meant when you said “infinity” has two meanings? 1) As a set of an infinite number of things, and 2) as a whole unit (like the whole population of infinite organisms)? I grant that as a unit, you can divide (\infty) into as many pieces as you want, and each piece is smaller than the whole. But it’s smaller in terms of the unit, not the infinite number of things making up the unit. 1/2 a unit is smaller than a whole unit. But each piece, no matter how small, still contains an infinite number of things. I don’t know what it means to say one piece has a smaller infinity of things than the whole.
Yes, I think we agree on this. You can treat infinity as a unit, and when you do that, you can run wild with arithmetic. But what I’m saying is you can’t just switch between context willy-nilly. You can’t treat infinity as a unit, and in the same statement switch to treating it as a never ending set of things. For example, you can say that an infinitely long train can be divided into two parts, but in this context, you’re treating the train as a single unit, not an infinite set of carts. You’re abstracting the idea of an infinite set of carts and thinking of it as a “block”. You’re imagining it as a finite thing. So you can imagine this block being split into two or more parts, and each part would be visualized as smaller than the whole. But now you’re stuck in this context. You can’t just switch and say: well since each part is smaller than the whole, each infinite set of things in each part is smaller than the infinite set of things in the whole. That’s meaningless. It makes no sense to say there are two infinite sets yet there are fewer things in one set than the other. The only reason you can say the parts are smaller than the whole in the first context is because you made the whole into a unit (a finite block) in which case you can divide it and expect the normal rules of arithmetic to apply. If you switch back to the other context, you shouldn’t have been able to divide the infinite set. That’s cheating.
Ok, that’s a good way of talking about division of infinities. Where I get stuck is imagining how the infinitely long queue gets “shorter”. To say each resultant queue is half the size of the original queue is to say it’s only half as long, or “shorter”. I just don’t know how to make sense out of that. For something to be “shorter” in length is to imply it has a beginning and an end. You’d have to imagine putting it next to something else and observing that it ends before the other thing ends. But we both agreed that the two resultant queues are still infinite. To me, that means I can’t imagine putting those queues next to the original one and seeing that they’re shorter. They would still seem to be the same length.
I meant spitting the train into two halfs. You’d pick two carts and disconnect them from each other, not from the rest of the train. But yes, taking every second cart is a better example.
Yes.
The same is true of the set of natural numbers and the set of even numbers:
1 → 2
2 → 4
3 → 6
…
For every natural number, there is a corresponding even number (and it is half the even number). Yet you say the natural numbers is twice as big as the even numbers.
That’s fine. I’m talking about orders of infinity (or what mathematicians mean by that term).
Not sure where you pulled this point from, but I did say this: “To get there, you have to transcend all finite things…” You’re right that we can’t “get there”, but we do all the time in our thought experiments. When you talk about a planet populated by an infinite number of organisms, you imagined having the infinity of organisms. How did your imagination do that? It didn’t count organisms starting from 1 until it got to infinity. It transcended the journey and instantly “arrived” at infinity. You didn’t visualize an infinite number of organisms, but you did posit their existence. ← This is what I mean. But I also added that there are rules that must be observed when we do this sort of thing.
What does that mean? In what sense of infinity does it make sense to say infinity is less than 1? (with the exception of -infinity).
Would look like? Dude, an infinite number of anything is impossible to visualize. I can grasp the concept, and I can grant it, but when it comes to following it up with its implications, the details matter. I’m simply saying that what you get when you add an infinite number of something to an infinite number of something depends on how infinite quantities work, and it’s especially troublesome if we’re talking about physically existing things. What is the nature of a physically existing set of infinite things? It’s not like adding finite things together. You add 10 organisms to 10 more organisms, you get 20 organisms. But what happens when you add an infinite number of organisms to an infinite number of organisms? I don’t know how quantities like that play out in the physical world. Would it just be twice as many organisms? Would the one infinity simply blend with the other such that you still just have infinity?
I understand it seem intuitive that an infinite set of things added to another infinite set of things gives you twice as many things, but I take this is an assumption, and I pause at it. I ask: is this assumption warranted? And I don’t have an answer. Until I get an answer, I don’t run with it.
But it does. It’s exactly the same problem. If the edge of the universe is an infinite number of steps away, and I take two steps towards it, am I now (\infty) - 2 steps away from the edge of the universe? Or am I still (\infty) steps away? Are these the same things? You’re saying, no, they are not the same. (\infty) - 2 < (\infty). I’m saying (\infty) - 2 = (\infty). How many steps away one is from the edge of the universe is just another example to be added to the list, a list which so far consists of carts in a train, organisms on a planet, points in a line, etc.
Let’s take a look at that math.
What you’re doing here is correcting a minor mistake, namely, you’re saying that the second train is 100 times longer than what I said. I am not really sure that’s a counter-argument. Beside that, you’re asserting that (99.999… = 100) and (FF.FFF… = 100) without explaining why. It’s an assertion, not an argument.
Endlessness divided by some quantity is still endlessness, sure, but that does not mean it’s of the same size.
When you take a finite number and divide it by other finite number you get a finite number (instead of an infinite one.) Similarly, when you take an infinite number and divide it by other number you get an infinite number (instead of a finite one.) In both cases, however, you are not guaranteed to get a number of equal size. (8) divided by (2) is not (8) it’s (4). It’s a finite number, just like (8), but it’s not the same number. Similarly, (\infty) divded by (2) is not the same (\infty) it’s (\frac{\infty}{2}). It’s an infinite number, just like (\infty), but it’s not the same number.
The idea that infinite quantites cannot be equal to each other is simply not true.
What does it mean to say that two infinite quantities are equal to each other? Suppose you have two infinite queues of people. What does it mean to say that they are equal? It means that for every man in the first queue there is a man in the second queue you can pair it with. If this holds true, the two ininite quantities are equal. But it need not be the case. You can have larger and smaller queues. This sort of thing isn’t difficult to understand, the only thing required is a will to understand it. If you have no such will, and all you’re interested in is justifying some position you adopted without thinking on your own, then, well, you won’t understand it.
The total length of these infinitely long sticks is not shorter than the length of their parts. Quite the opposite. Not sure where you got that idea from.
You can look at it both ways. You can say it’s 60 infinitely long train segments and you can represent it like (60\times\frac{\infty}{60}). Or you can say it’s an infinite number of 60-inch segments and represent it like (\frac{\infty}{60}) 60-inch-segments. This is the commutativity law of multiplication. Are you saying that multiplication ceases to be commutative when working with infinity?
The unit in question is an infinite quantity of things. So each piece being smaller than the whole (which is the unit) implies that each piece is a smaller infinity in relation to the whole.
You can because they are the same thing. You are merely changing how you’re calling it.
It’s a unit that is made out of an infinite number of carts. The train does not cease to be infinite when you treat it like a unit.
If there is a finite number of people in your town, and you abduct a bunch of them, you’ll know there are fewer people left, even though you never knew the total number of them. Clearly, it’s not true that you need to know the size of a set in order to be able to tell that it’s smaller than another. This holds true for infinite sets as much as it does for finite sets. If there is an infinite queue of people in front of you, and you remove one of them, the queue would be smaller by the fact that you removed one person from it.
That’s not really true.
You’re taking my words too literally.
You get (2\times\infty) organisms.
Sure. What about people who say things like (2 + 2 = 4) is an assumption that they pause at until they get an answer to the question “Is it warranted?” Basically, what you’re saying here is you’re not sure that (\infty + \infty = 2\times\infty). But I am sure, not merely because I didn’t make a pause like you did, but because I derived it using logic.
You are exactly (\infty - 2) steps away from the imagined edge, assuming that the edge is located (\infty) steps away from your starting point.
I had the impression that you went a little beyond simple example dealing with the concept of infinity.
Let’s take a look at what you said:
Yep, 90 inches long + 9 + 0.9 etc. for Train A and 93.75 inches long + 1/16 of that + 1/16 of that etc. for Train B.
So:
(90\sum_{x=0}^\infty 1/{10}^x = 100)
(93.75\sum_{x=0}^\infty 1/{16}^x = 100)
isn’t 100 times what you said, it’s exactly what you said.
A minor mistake of yours, but you’re right that it’s not a counter-argument.
Yes, I did explain my argument and not merely “assert” that (99.999… = 100) and (FF.FFF… = 100).
My first couple of posts on this thread 6 days ago on page 44 demonstrate the principle in the simplest way I could think of:
(1\div3=0.\dot3)
(0.\dot3\times3=0.\dot9)
As basic as explanations can get, right?
I’m sure you don’t need me to explain the basic steps of division and multiplication that result in these values, do you?
Mathematics is consistent so dividing some quantity by a value and multiplying the result by the same value gives you the same quantity you started with.
There’s no issue for (9\div3=3) and (3\times3=9) so why is there for 10 instead of 9?
The same goes for 100: (100\div3=33.\dot3) and (33.\dot3\times3=99.\dot9)
I even restate the very same principle directly to you on my 3rd post on this thread from 3 days ago on page 45 for hexadecimal and vigesimal:
(10\div3=5.\dot5) in hex and (10\div3=6.\dot6) in vig.
(5.\dot5\times3=F.\dot{F}) in hex and (6.\dot6\times3=J.\dot{J}) in vig.
The same basic steps of division and multiplication explain what you’re saying I’m merely asserting here too.
And there’s absolutely no difference in the basic division and multiplication by 3 in any of the above calculations than there is for any value in any standard positional numeral system, whether decimal, hexadecimal, vigesimal etc.
I even formalised this explanation just earlier for why (10 = 9.\dot9) in decimal, and this can easily be altered to do the same in hex or vig etc.:
In line 1, (s=\sum_{x=0}^\infty \frac9{10^x}) lets the infinite the sum of (9+0.9+0.09+…) be denoted by “(s)”.
(\frac{s}{10}=\sum_{x=1}^\infty \frac9{10^x}) restates the same infinite sum but starting from (0.9) instead of (9).
The reason this is (s/10) is because:
(\frac{9/{10^0}+9/{10^1}+9/{10^2}+…}{10}=\frac{9/1+9/{10}+9/{100}+…}{10}=9/{10^1}+9/{10^2}+9/{10^3}+…=\sum_{x=1}^\infty \frac9{10^x})
The next line simply subtracts (s/10) from (s)
This is (\sum_{x=0}^0 \frac9{10^x}) because the only difference is the first term in (s) is missing from (s/10), which is (9).
I close line 2 by factoring out (s) from (s-\frac{s}{10}) to get (s(1-\frac1{10}))
To make the beginning of line 3 clearer, (9=s(1-\frac1{10})) is obviously the same as (s({0.9})=9)
Divide both sides by (0.9) to get (s=\frac{9}{0.9}=10)
So we get from establishing (s) as (\sum_{x=0}^\infty \frac9{10^x}), which unpacked is (9.\dot9), to (s=10)
Therefore (9.\dot9=10)
A little less basic, but it explains why (99.999… = 100) if we simply alter (s) to start from (x=-1) instead of (0) and (s/10) starts from (x=0) instead of (1).
Don’t assert so certainly that I’m simply asserting and not explaining when I have explained myself just fine.
If there’s anything you don’t understand or accept then single it out and ask - don’t just assert like you’re accusing me of doing.
Endlessness literally means size is undefined - size is for quantity and undefined quantity is a quality (the quality of being the opposite to quantity).
The quality of “being endless” versus the quality of being “even more endless”, or “not quite as endless” is just meaningless.
What you mean when you try to communicate a little more/less endless is obvious, but it still makes no sense:
if something goes on forever, there isn’t anything that goes on “more forever” than that, or “slightly less forever” than that - it’s forever either way.
Something either goes on forever (infinite) or it doesn’t to some quantifiable degree (finite).
Obviously I understand your simple arguments - like I’ve already said a few times, and you admit: you’re treating finite quantities no differently from infinite qualities.
As I said before, it’s a category error.
You can restate your error an infinite number of times, but quantity (\neq) quality.
Obviously 8/2 is not 8, it’s 4: because these are finites.
But infinite endlessness divided by 2 is still endlessness. It’s undefined. It doesn’t mean all undefined things are the equal just because they’re elements of the same set (as I also covered just earlier: “(\in)” (\neq) “(=)”). But their lack of equality doesn’t mean they’re unequal either - this would commit the false dilemma fallacy.
Equality is simply invalid for the opposite of a quantity. All endlessness is undefined in the same way (type) but each instance is not therefore the same (token) as another. Infinites:qualities:types. Finites:quantities:tokens. Infinites are not finites and treating them the same results can look precise but is ultimately meaninglessness.
Think of it this way: an infinitely long piece of string “attached” to an infinitely long piece of string connects what ends together exactly? An infinitely long piece of string has no ends because it’s infinite (the word infinite literally derives as “no ends”). Without ends to connect together the notion of adding two infinitely long pieces of string doesn’t even make sense. It’s a contradiction in terms.
I have the will to reject contradictions in terms.
I have the will to understand all these arguments you’re asserting, and the ability to understand what you’re saying just fine.
They’re not difficult to understand, you’re absolutely right about that, they’re just founded on invalidity.
I can point out the invalidity, but cannot force you to understand or accept it.
I’ve already covered the notion of “One-to-one correspondence” myself several times, which is what you’re now explaining to me that I don’t understand or don’t want to understand.
I understand that if two infinite sets have corresponding pairs they’re deemed “the same size” - I’m just pointing out that endlessness having a size in the first place is meaningless.
Infinity is literally sizelessness - that’s what endlessness amounts to. Size is a product of the difference between “ends” e.g. the size of a string is the difference between one of its ends and the other.
An infinite string has no ends by defintion and derivation, so size is not something it can have.
I even just wrote a post disproving Cantor’s diagonal argument, which is founded on the concept of One-to-one correspondence and sizes of infinity - you really think I don’t get this stuff?
I got it from your equations (a < \infty \hspace{0.1cm} \text{ninety inches segments}) and (b < \infty \hspace{0.1cm} \text{hundred inches segments}).
You defined “a” as the length of an infinitely long stick,
You stated this as “less than” an infinite number of the 90 inch segments that it was divided into.
Therefore the equation (a < \infty \hspace{0.1cm} \text{ninety inches segments}) means the “length of an infinitely long stick” is “less than” the “infinite number of 90 inch segements that it was divided into”.
That’s where I got the idea from.
The same goes for b.
Perhaps you meant to communicate something else by those two inequalities, but my interpretation of them that I just explained is perfectly logical.
Well, I am familiar with this proof. It’s one of those Wikipedia proofs that James mentions. The problem with it is that it assumes that (1\div3=0.\dot3). How do you prove that?
This proof is very similar to that Wikipedia proof that I mentioned several posts ago and it suffers from the same problem.
The problematic equation:
$$ \frac{9.\dot9}{10} = 0.\dot9 $$
The (0.\dot9) that you get when you divide (9.\dot9) by (10) is not the same as the one that is found in (9.\dot9) before division. This is because the number of terms in the two infinite sums is not equal. Your proof ignores this (normally, since you do not accept that infinities come in different sizes, or if you do accept it, you do not accept it fully.) That’s why your proof is an example of sophism.
Let’s rewrite the above equation using summation notation:
$$ \frac{9 + \sum_{i=1}^{\infty} \frac{9}{10^i}}{10} = \sum_{i=1}^{\infty+1} \frac{9}{10^i} $$
Notice that the first sum goes from (1) to (\infty) whereas the second one goes from (1) to (\infty + 1). Properly speaking, the second (0.\dot9) is greater than the first (0.\dot9) but in your proof this is not obvious because you’re not tracking how many terms each of the two infinite sums has.
This isn’t so obvious, and if one is not capable of accepting more obvious insights, it might not be rational to expect them to accept less obvious ones. But I’ll try to make it clear anyways. Just in case.
Suppose you have three infinite sets. Let them be made of whatever you want (apples, oranges, people, wagons, etc.) We can represent them like so:
1 1 1 …
1 1 1 …
1 1 1 …
There are many ways to describe this quantity. You can, for example, describe it as three inifnite sets of ones.
You can also describe it as an infinite set of groups of three ones.
But there is one particular description we’re interested in. We want to express this quantity as a sum of three ones and three other equally-sized things.
The simplest way to do so is as a group of three ones and three non-finite groups of ones. This is true but is problematic because you can’t put it in a transitive relation with previously mentioned descriptions.
This means that we need to refer to a specific infinity and not infinity in general.
So let’s say that (\infty) does not refer to infinity in general but very specific infinity, namely, the size of horizontal blue sets of ones in this image. Three horizontal blue equally-sized sets whose size is represented by (\infty) can be described as (3\times\infty).
The question is: does (3\times\infty = 3 + 3\times\infty).
Of course not. You can show this by subtracting (3\times\infty) from both sides. What do you get? You get (0 = 3).
Let’s now visually compare the two representations, namely, (3\times\infty) and (3 + 3\times\text{non-finite sets of ones}).
(3\times\infty) is in red, the other expression is in blue. Notice how each one of the blue non-finite sets of ones has fewer elements than each one of the red infinite sets of ones? Each one of the blue infinite sets is one element smaller than each one of the red infinite sets.
The correct equation is:
$$ 3\times\infty = 3 + 3\times(\infty - 1) $$
Notice also that:
$$
3\times\infty = 3 + 3\times\infty - 3
\
3\times\infty = 3\times\infty
$$
One more point of disagreement. Let me put forward my own opinion. Endlessness implies no end but no end does not imply no size.
Two sets (whether finite or infinite) are equal if and only if for every member of the first set there’s a member in the second set that you can pair it with. If this does not hold true then they are not equal in size and one set is either larger or smaller than the other.
If I say “Here’s an infinite set A and an infinite set B which stands in relation to set A in such a way that for every member of the set B there are five members of the set A” then I’ve defined the size of the two sets.
The confusion might stem from your excessively literal interpretation of words. I’m not really sure that saying “infinite set A is bigger than infinite set B” means “infinite set A is more endless than infinite set B”.
You said it but you had nothing to back it up (so far.)
Yes, in the same way that finitude divided by 2 is still finitude.
You just said that endlessness divided by 2 is endlessness. That means it’s defined. Unless you changed your mind?
The result of an operation is undefined if there is no result to it. 1/0 is undefined because there is no result to it.
Just a few moments ago you said that the (0.\dot9) in (9.\dot9) is equal to the (0.\dot9) that you get when you divide (9.\dot9) by (10).
$$
s = 9.\dot9\
\frac{s}{10} = 0.\dot9\
s - \frac{s}{10} = 9
$$
Basically, what you’re saying is “Look, infinities can’t be compared, any two infinities are neither equal nor unequal, their difference is undefined, but let me show you that (9.\dot9\ = 10) by subtracting (0.\dot9) from (9.\dot9) and getting a nice little (9).”
Yeah, that’s why you claim that (9.\dot9 - \frac{9.\dot9}{10} = 9) because you can’t subtract two infinite sums – they are the opposite of quantity.
You can’t take an infinitely long stick and place another one (finite or infinite) at its end because there is no end. It’s a contradiction in terms to speak of an infinitely long stick connected to an end of another infinitely long stick. That’s where we agree. Where we disagree is that adding two infinite quantities means precisely this. It does not. I can add one apple and one apple to get two apples even if they are worlds apart. Noone cares where they are in space.
Then you should do so.
If they are the same size this means they have size in the first place. What’s meaningless is saying that they have no size but that they nonetheless do.
Not really.
Size has little to do with ends.
Well, if you really want my opinion . . . but I think it’s unnecessary.
I defined “a” as the length of an infinitely long stick in terms of inches not 90 inch segments.
$$
a = \infty \hspace{0.1cm} \text{1-inch segments} =
\frac{\infty}{90}\hspace{0.1cm}\text{90-inch segments} < \infty \hspace{0.1cm} \text{90-inch segments}
$$
Magnus!
You didn’t quote his best argument!
“What does it mean to say there is a greater or lesser endlessness?”
It means that 2 infinite lines are more than 1.
I don’t want to say what it means to argue otherwise.
I suppose what you’re asking is what does it mean for something to be more or less endless. Well, I personally don’t know, but then again, I also don’t know what it means for something to be more or less finite. The problematic part is that you think this question is somehow related to the question posed in the OP. I don’t see how. If your argument is “Because more/less endless is undefined, it follows that greater/lesser infinity is also undefined” then you’re wrong. Consider that more/less finite is undefined, and yet, we all know that finite sets have size. Whether finitude/infinitude is binary (either finite/infinite or not finite/infinite) or infinitary (more or less finite/infinite) has no impact on the claim that infinite sets come in different sizes. When you say that some infinite set A is bigger than some infinite set B, you are NOT saying that infinite set A is more endless than infinite set B in the same way that when you say that some finite set A is bigger than some finite set B you are NOT saying that finite set A is more finite than finite set B.
One could simply say that the countable infinities are the lesser ones and the uncountable ones are the greater ones
So those with a one to one correspondence would be the lesser ones and those without it would be the greater ones
All infinite sets are endless by default but some just have more members than others
I am pretty sure that Cantor in general (not merely his diagonal argument) is irrelevant to the subject at hand even though it might sound strange to some considering that Cantor was the one who introduced the idea that infinites come in different sizes.
Consider that Cantor said that the set of natural numbers ({1, 2, 3, \dotso}) and the set of even positive integers ({2, 4, 8, \dotso}) are of the same size. This is right but not completely right. It is right in the sense that the two sets can be equal in size. It is not completely right because the two sets are not necessarily equal in size.
No set (finite or infinite) has inherent size. All sets have size in relation to other sets.
Did Cantor tell us that for every even positive integer there can be five natural numbers? In such a case, the set of positive even integers would be larger than the set of natural numbers.
We need a standard. We need a set against which all other sets will be measured.
If we say that the set of even positive numbers refers to the subset of the set of natural numbers, then the set of even positive numbers is half the size of the set of natural numbers. You can’t say its size is equal to or greater than the size of the set of natural numbers. That would be sophistry.