In mathematics, “a = b” means that the two symbols “a” and “b” have one and the same meaning i.e. that everything that can be represented by the symbol “a” can also be represented by the symbol “b” and vice versa.
“2 + 2 = 4” is true because everything that can be represented by “2 + 2” can also be represented by “4” and vice versa.
However, when we say something like “2 + 2 = Finite number” we are working with a different concept of equality. In such a case, “a = b” means that everything that can be represented by one of the two symbols can also be represented by the other symbol but not necessarily the opposite. This is evident in the above expression. Namely, everything that can be represented by “2 + 2” can be represented by “Finite number” whereas the opposite does not hold true i.e. there are things that can be represented by “Finite number” that cannot be represented by “2 + 2” e.g. any finite number other than 4, such as -250, 0, 1/2, 5, 100, etc. This isn’t problematic per se unless it leads to fallacious arguments such as the following:
2 + 2 = Finite number
4 + 4 = Finite number
Therefore, 2 + 2 = 4 + 4
The concept of equality we’re working with does not permit this kind of deduction.
Put another way:
A = The set of all things that can be represented by “2 + 2”
B = The set of all things that can be represented by “4 + 4”
C = The set of all things that can be represented by “Finite number”
A is a subset of C
B is a subset of C
Therefore, A is a subset or a superset of B
It should be perfectly obvious now why the conclusion does not follow. Just because A and B are part of something it does not mean that A contains B or that B contains A or that they are the same part.
People can neglect this because they can forget (and in some cases not even notice) that they are working with a different concept of equality. You can’t abide by the rules that apply to one concept of equality if you’re working with another one (unless the same rules apply, which is not the case here.)
In the same exact way, one can make the following argument:
(1+1+…+1) + 1 = Infinity
(1+1+…+1) + 2 = Infinity
Therefore, (1+1+…+1) + 1 = (1+1+…+1) + 2
The premises are correct but the conclusion does not follow – and it’s wrong.
I also agree that, by definition, to add some quantity (finite or infinite) to some other quantity (finite or infinite) means to increase that other quantity. This means that, if you add 1 to infinity, you get a larger number. “Infinity + 1 = Infinity” is only true in the sense that both “Infinity + 1” and “Infinity” are infinite numbers (i.e. they both belong to the same class of numbers.) It’s not true in the sense that the two numbers are of the same size.
Given that it’s obvious that 0.999… =/= 1, the question is, why do Wikipedia proofs look so convincing? Because they do, at least to me, and figuring out where’s the flaw isn’t so trivial.
Take this proof, for example:
What’s wrong about it? Where’s the mistake?
After much pondering (and reading, of course) I’ve come to the conclusion that the mistake lies in the fact that the following two numbers aren’t the same.
They are equivalent in the sense that they belong to the same class of infinite numbers that can be represented using the symbol “0.999…” But they are not equivalent in the mathematical sense of the word. I am inclined to think that they are not equivalent because the number of digits after the dot in the second number is smaller than in the first one. And this is due to the fact that we got that number by taking 0.999…, multiplying it by 10 and then subtracting 9 from it. But when you take 0.999… and multiply it by 10, one of the digits from the right side moves to the left side. Sort of like how when you take 0.999 and multiply it by 10, all of the digits move one position to the left, with one leaving the right side and entering the left side. It becomes 9.99 which has two, rather than three, 9s after the decimal point.
But since 0.999… isn’t even a number, 10 of them added together cant be a number either. So it isn’t valid to say that 10x = 9.999… and assume there are the same number of 9’s in the string. You would have to add a zero to the end of the string. How do you do that?
The rule of shifting a digit isn’t the math. It is merely a convenient shortcut to normal maths operations.
They had another one that James pointed out using 1/3 = 0.333… and that 3x0.333… = 0.999… James pointed out that you can’t add expressions that end with “…” until you define the limits of what “…” means in your specific example.
The bottom line was that 1/3 => 0.333… It leads to the infinite string 0.333… It is not equal to because the string is insufficiently defined.
Does (0.111\dotso = \frac{1}{9})? How do you prove it?
[s]But the step colored in red is the most interesting to me.
How can 90 times something equal 9 plus 9 times that same something?
How can ( 90\times5 = 9 + 9\times5 )? It’s clearly not true. 450 does not equal 54.[/s]
Obviously, I’m forgetting that there are two units involved: the mentioned something (90 of which are on the left side and 9 on the right) and an spoken unit of which there are 9 on the right side.
In hexadecimal, 10/3 is 5.(5)
In vigesimal, 10/3 is 6.(6)
Multiply 5.(5) by 3 in hex to get F.(F)
Multiply 6.(6) by 3 in vig to get J.(J)
10 in any standard positional numeral system divided by 3 and multiplied back by 3 gets you an answer of the form “x.(x)” where x is the highest single value digit in the numeral system - as well as “10”
So “cross-numeral-system”-ly, 9.(9) is the same size as F.(F) is the same size as J.(J) for decimal, hexadecimal and vigesimal respectively - because they all equal 10.
This is not to say that “9.9 = F.F = J.J” or “9.99 = F.FF = J.JJ” and so on for any finite number of dec/hex/vig places, whether across numeral systems or in the same numeral system when such notations are possible.
This is where the illusion lies.
J.JJJ is closer to 10 than F.FFF, than 9.999, and so on for more and more dec/hex/vig places - so you might be fooled that the vig system will always allow bigger quantities than the hex, and more still than the dec.
But the differences between J.JJJJ, F.FFFF, and 9.9999 are smaller than the differences between J.JJJ, F.FFF and 9.999.
This difference gets smaller and smaller as you add more digits - and for however many finite number of times you do this.
Perhaps you might attempt the argument that this difference tends towards the epsilon number ε, but just the same as dividing by 3 and multiplying by 3 doesn’t change the quantity you started with, but can reveal a different way to represent the same quantity, as soon as you add in the quality of endlessness to the number of times you add more digits, there’s literally no difference. The difference really is 0, and not even ε. Perhaps you could even argue that ε tends towards 0, which is why it literally gets there as soon as you add in the quality of endlessness. Perhaps you could even say that the quantity ε really is 0 once the quality of endlessness replaces quantity (its opposite).
It’s the same illusion that I’ve already covered that the act of adding the quality of endlessness (the unquantifiable) to quantities is what changes your intuitions of finite quantities such that 10 = “x.(x)” in the manner I just explained.
10 is the representation in terms where the property of endlessness is redundant (i.e. adding finite or infinite 0s after a decimal point changes nothing) and,
“x.(x)” is the representation in terms where the property of endlessness is necessary (i.e. adding finite "x"s after a decimal point changes the quantity, and none of these quantities will be the same as if infinite "x"s were added).
This is the root, which I’ve already explained, of why some people can’t get their head around such topics as these: they treat the quality of not having a quantity as a quantity.
As I keep saying, the mathematical point of the whole exercise is to see if there’s any utility in treating the untrue as true i.e. Experientialism.
So this is already solved, but I can go through more individual cases if anyone still needs it - though they’ll all come back to these same points that I’m reiterating.
For example, the following is just another erroneous instance of “using one’s intuitions about finites for infinites”:
You can’t add a zero to the “end of the string” when there is no end of the string.
The string is endless - that’s what “…” means, though I’m using the neater parentheses notation to refer to recurring digits.
There is no “gap” that emerges “at the end” of a literally endless string when you multiply endless digits by 10.
The appearance of the shifting digits is just the superficial result of performing the math, rather than the math itself - exactly right.
But the fallacy here is to say that “the superficial appearance” isn’t “the real”, therefore a result that takes the form of this superficial appearance isn’t valid.
It’s a red herring.
The real mathematical operations are still there! Distracting attention to the superficial and only disproving the superficial does nothing to disprove the validity of the real mathematical operations that are being distracted from.
Magnus, your objection of red underlining in your second-to-last post does the same thing.
Your most recent post is harder to make sense of, because of the crossing out.
Obviously “90×5=9+9×5” isn’t true, because 5 =/= 0.(1)
0.(1) isn’t any old number “n”, the equation “90 x n = 9 + 9 x n” only works when n = 0.(1)
The 0.(1) is the same throughout each equation, which is only obvious when you don’t treat the endlessness of 1s as a finite number of 1s “with an end”. There is literally no end of the 1s, which is the only way it can equal 1/9.
Yeah Andy I wuz gonna say just hook up with sil and he’ll sort you out. Him or that ‘wtf’ guy who doesn’t post anymore. You know that guy? I’ve never said a word to him. Guy scares me to def.
Yeah stick with sil. You’re already smarter than me, and when sil gets done with ya you’ll be way way smarter than me. And that’s something to aspire to
Comparing 0.999… to 0.FFF… is like comparing two rocket ships both heading towards the end of space (or an infinite distance away). One approaches it twice as fast as the other, but neither will ever get there. Therefore, you cannot say: when the slower one reaches infinity, the faster one will reach a further infinity.
Silhouette adds that the ships are also decelerating. The slower one will never surpass the faster one but the gap between them is always shrinking such that given an eternity, the slower one will finally catch up to the faster one such that they will reach infinity (if that were possible) at the same time.
What about two infinitely long trains that are not even moving, merely standing in place, occupying space? Both are made out of an infinite number of wagons, but one’s wagons are longer than the other. Which of the two trains is longer?
Train A starts with a wagon that is 90 inches long. Every subsequent wagon is 10 times shorter. So the next wagon would be 9 inches long, the next one 0.9 inches long, the one after it 0.09 inches long and so on. Again, the train is made out of an infinite number of wagons, so there is no “last wagon”. The chain of wagons is without an end, it’s infinite.
Train B starts with a wagon that is 93.75 inches long. A bit longer than that of train A. Every subsequent wagon is 16 times shorter. So its subsequent wagons are 0.05859375 inches long, 0.00366210937 inches long and so on. This one is infinite as well.
Are you going to tell me that they are equally long?
If you were to align them by one of their ends and zoom-in on their other ends indefinitely, you’d never see them being equally long. Why would you then say they are equally long?
First of all, a train infinite in length wouldn’t have a ‘first’ wagon, andy. Omg . And don’t try telling me the first wagon exists ex nihilo and pulled itself up by it’s own axle straps. And second, train wagons only 90 inches long would be a bad investment and terribly inefficient. The whole point of the railroad system is to haul large payloads over vast distances. What the hell can a train of 90 inch wagons haul, dude?
I hate to be the bearer of bad news, but unfortunately your doubt is a result of your miscalculation.
The second carriage of Train B is 5.859375 inches long, not 0.05859375 inches long.
16 times shorter than 93.75 inches is not 0.9375/16.
A simple mistake, no harm done - on the plus side, I found the MathJax post that you recently bumped to be extremely useful:
$$90\sum_{x=0}^\infty 1/{10}^x = 100$$
$$93.75\sum_{x=0}^\infty 1/{16}^x = 100$$
Each train in your example is the same length, yes.
Or rather: Zeno was wrong with his paradoxes, right?
Because you say so?
Even though at no viewing distance do they look equally long, they are nonetheless equally long?
If the two trains are truly equally long (and not merely approximately equally long) there should be a viewing distance (or zoom level) at which, and beyond which, the two trains look exactly equally long. That is, however, not the case.
The miscalculation is an insignificant mistake. The point remains. At no viewing distance (or zoom level) do these trains ever look equally long.
I would have accepted “undefined” as well. I certainly would not have accepted that train B is longer because its carts are longer. Imagine two infinitely long sticks. They’re both infinitely long. For all intents and purposes, equal. Now imagine chopping up one stick into 90 inch long segments and the other into 93.75 inch long segments (what an odd number to choose). Is the one with 93.75 inch long segments suddenly longer?
It seems to me that infinite things don’t have a “length” per se. So you cannot say that two of them have the “same length”. I think that is the problem being fueled. Saying that something if infinite isn’t sufficient to make claims concerning comparisons to other infinite things.
Saying that A is tall and also that B is tall, doesn’t mean that A and B are equal. I don’t think it takes a maths wiz to figure that out.
Just a couple of posts earlier, I addressed this issue. Basically, you’re right (and certainly so) in the sense that both sticks have a length that is infinite (rather than finite) but you’re not necessarily right in the sense that they are equally long. Your argument is akin to the following fallacious argument:
2 + 2 = Finite number
4 + 4 = Finite number
Therefore, 2 + 2 = 4 + 4
The result of 2 + 2 is indeed a finite number. And so is the result of 4 + 4. But given that when we say something like “2 + 2 = Finite number” we’re operating with an equivalence relation that is not transitive, it does not follow that “2 + 2 = 4 + 4”.
An equivalence relation is transitive if a=b and c=b implies a=c. That does not apply to the above. If both Socrates and Aristotle are philosophers, it does not follow that Socrates and Aristotle are one and the same person.
But let’s say that what you mean by above is that the two sticks are BY DECLARATION equal and not that they are equal because they are infinite.
(93.75 = 10\times0.F) where (0.F) is a hexadecimal number. Remember that I made a claim that (0.FFF…) is larger than (0.999…)? The one you tried to dispute? I substituted your example with rockets with my example with trains where one train is (99.999…) inches long and the other is (FF.FFF…) inches long. But instead of multiplying 0.F by FF to get the real length of the longest wagon, I mistakenly multiplied it by 10. Hence the strange number. Phew. That was one hell of a justification, particularly because it’s not really important.
Okay. So you have two equally-infinitely long sticks. Let’s write that down as (a = \infty\ inches) and (b = \infty\ inches). You chop one stick into 90 inch long segments. What do you get? You get (a = \frac{\infty}{90}\hspace{0.1cm}\text{ninety inch segments}). You take that other stick and you chop it nicely into 100 inch long segments. What do you get? You get (b = \frac{\infty}{100} \text{hundred inch segments}). They are still equal. Note that (a < \infty \hspace{0.1cm} \text{ninety inches segments}) and (b < \infty \hspace{0.1cm} \text{hundred inches segments}).
The question comes down to whether you believe Cantor or not. Are there orders of infinite’s where some infinities are larger than other infinities (Cantor) or not.
I side against cantor … I think all infinities are equal.
See… and I always hated James for doing this, your adding an infinity past another infinity to make that argument.
It’s one thing to add 0.3… to 0.4…to get 0.7…
It’s an entirely different thing to add point zero four after point zero three, because point zero three never ends. This is what James did with infinitesimals.
I’m neither a mathematician nor an expert on what James thought but I think that Cantor said that the set of natural numbers and the set of rational number are of the same size. I disagree with that. I believe James disagrees too.
I am sure no Cantor is required in order to understand that infinities come in different sizes. All you need is basic logic. A single planet populated by an infinite number of organisms is less than two planets each populated by an infinite number of organisms.
That’s exactly what cantor argued. Cantor is famous for arguing orders of infinity.
I disagree with cantor. I am positive that one planet with an infinite number of people has exactly the same amount as 2 planets with an infinite number of people.
Infinity is not subject to the same rules as temporal/spacial logic.
