Good point. I think the very question of the OP is problematic because there is a difference between *everything imaginable *and everything possible (no matter how unlikely).
One can imagine an impossibility. Therefore the answer to the OP must be ‘no’. But I wonder if the OP might actually want to know: “given infinity, is any and every possibility inevitable?”
(FTR: I am not trying to be a pain in the arse here…really trying to wrap my head around this. That said…)
Seems to me a trick is being pulled here. You are asking me to insert something between and infinite series of “9’s” and one.
Can’t I do the same to you the other way?
Looks like the hang up is on iterations as opposed to an infinite string that “just is”.
So, if I ask you to calculate the odds of an infinite number of coin flips coming up heads I imagine the number would look something like 0.000000(forever). While it is never quite actually zero can you insert a smaller number in there?
That was in fact my question, the insertion of “imagine” is just problematic and I should have worded it how you did. I do think this thread has pretty well convinced me to change my opinion on the answer though. I’m at least leaning towards a no.
I’ve found another way of thinking about it. Of all the possibilities even just limited to our life here on earth in hypothetical other incarnations, the extreme majority of them would involve people behaving in wildly unreasonable ways.The world I inhabit is logically structured as far as I can tell, and the odds of me happening to be in one of the “logical” realities would be so extremely small that I should assume that it’s not a coincidence. If that makes any sense.
I’m missing something too because I don’t see how you can do it if .9999… = 1, as has been hashed out here many times before.
I certainly don’t have the chops to address the turn the thread has taken. but the answer to the OP’s question is no. Infinity doesn’t demand that everything possible occurs. Infinite diversity doesn’t even demand it. Consider an infinite set consisting of all integers except 7. It’s infinite in size and infinite in diversity. But you won’t find 7 anywhere in there. And yet 7 is a theoretically possible quantity. It could have been in that set. But it just plain isn’t.
Just because the universe/multiverse is/might be infinite, that doesn’t mean that everything that could happen does happen – it just means that there’s no end to the things that do happen.
–Cliffy
Ok…I can follow that something doesn’t have to happen. Like the flipping coin. It doesn’t have to ever land on tails and could be heads forever.
That said can we say it is really, Really, REALLY likely that in an infinite universe there is a Cliffy doppelganger out there somewhere? Maybe a bunch? Maybe an infinite number of them?
The thing about infinity is it is really colossally huge. Graham’s Number is peanuts in comparison (and that number is so big our universe is too small to write it out…literally).
If you keep rolling the dice for an arrangement of atoms that equals Cliffy you are likely to succeed with an infinite number of tries. Indeed succeed quite often. It may not have to happen but the probability is pretty good I would think.
There is no number you can name that is greater than .99999… but less than 1. Therefore, there is no distinguishable difference between .99999 and 1. Therefore, .9999…=1.
Now, imagine you flip a coin an infinite number of times. What are the odds that the coin will always come up heads? Zero. Does that mean it is impossible for the coin to always flip heads?
No.
Because think about this. Suppose you start flipping the coin and get HTHTHTTTTHTHTTHTHTHTHTTTHTHTHTHTH, followed by an infinite sequence of heads and tails in a random pattern. What are the odds that the coin will generate this sequence? Zero. The odds for any sequence in an infinite series of flips is zero. You are exactly as likely to get any sequence of random numbers as any other, and the odds of getting any sequence is zero. But, you are guaranteed to get a sequence with probability of 1. Therefore, the sum of all the probabilities of getting a particular sequence must sum to 1. But the probability of any particular sequence is zero.
This is the same problem. If I ask you to choose a real number between 0 and 1 at random, what are the odds that you will choose a particular real number? Zero, because there are an infinite number of real numbers between 0 and 1. I can prove there are an infinite number, because for any two real numbers between 0 and 1 that you pick, I can pick a real number that is between them. I can do this no matter what numbers you pick, and I will never fail. So, given an infinite number of reals, does that mean you can’t pick a real number between 0 and 1?
So, 0+0+0+0+0… = 1?
Here’s the thing. Infinity is only a concept. A real working concept. There are not infinite parallel universes, because that would imply infinite mass and light. If any leaked through in any way once, it would do so infinitely because any possibility is not only probably but infinite, and our universe would be a big ball of light and/or collapsing in on itself.
That does not eliminate finite parallel universes.
As for non-parallel universes, according to the laws of physics and specifically relativity, we will never confirm their existence, so they might as well not exist. A physics convention will have to pass new laws.
The funny thing is that people are very prone to generate all sorts of uncanny abstractions like infinity that do not exist in observable reality and then wrack their brains over those imaginary things using cognitive devices and logic that are derived from and operate in purely finite domains. Don’t you think it’s plain wrong?
Yes, you just need an infinite number of zeros.
Of course, the real answer is no, you can’t add 0+0+0+0… and get 1. But if you divide 1 by infinity, you get zero. Which is why infinity isn’t a number and you can’t do arithmetic operations like division with it, because if you do then arithmetic breaks down and 1=2.
So, there are an infinite number of reals between 0 and 1, but you can still count to 1, you just can’t count to 1 by counting the real numbers between 0 and 1, starting at the smallest real number greater than 0. There is no smallest real number greater than zero, any such number you propose I can always name a smaller one.
So, there are an infinite number of possible ways to flip a coin an infinite number of times, and each way has a probability zero of being chosen. If you want to argue that means it’s impossible to flip a coin an infinite number of times, then I won’t argue with you.
No, I think it’s fun. It seems to me that there are two possibilities, infinite existence of some kind or eventually there will be nothingness. Even if my puny human brain cannot understand either and it certainly has no practical effect on anything, I still like trying.
I had a calculus instructor who said that the infinite .9999 sequence was equal to 1 because you couldn’t come up with a number to add to infinite .9999 to make it equal to 1. Thus both numbers could be interchanged for the other.
Is this provable or by definition?
Neither, you can’t divide by infinity because infinity isn’t a number.
The point about what infinity means is a bit clearer when we talk about numbers between 0 and 1, than if we talk about numbers starting and 1 and then continuing on forever. You can start counting 1, 2, 3, 4, 5, 6, and then imagine keeping on going forever. But you can’t start counting the numbers between 0 and 1, because even though there’s a number you could call the first number, 0, there’s no number you could call the second number. So you can’t even start counting them.
“Dividing by Infinity” and “Dividing by Zero” have so many ambiguous results it is better to discuss “Limits.”
The Limit of [1 divided by X], as X approaches Infinty, equals Zero ≠ (is not the same as) 1∕∞ = 0.
Just as in arithmetic, the expression 1∕0 is “undefined,” because to allow division by zero, even if we call the result “Infinity,” allows any conclusion from any starting premise (the shorthand being, you can “prove” 1 = 2.)
The probability of any given string of an infinite iteration of heads and tails in the coin-flip scenario is zero, yet the probabilty that there will, in fact, be an infinite string of heads and tails “after” the experiment has terminated is 1, or dead certain.
There is no contradiction, even though the “sum” of an infinity of Zeros (the probability of any given string) eqals One.
The problem is that there are two ways to view it.
If you view it as the sum of an infinite series, then there are all the partial sums, counting upward, as you increase the index integer.
If, on the other hand, you view it as a completed operation – a “completed infinity” – then, yes, definitely, there is no “error function” or undefined “last term” the way there is in a real Taylor Series.
And, with a number like “1” you get to do this. We can construct “1” exactly. There isn’t any remainder.
With a number like e or pi, you can’t. You can’t construct them. The Taylor Series will always have an undefined last term.
So, I’m on your side regarding numbers like 1, or 1/3. They are constructible.
I think I have to disagree… Because you can’t “flip a coin an infinite number of times.” You can’t “jump right to the end” the way you can with the decimal expansion of 1 or 1/3. There isn’t any end!
Otherwise, you’ve just argued that you can add zero plus zero plus zero plus zero plus zero…and it sums to one.
(Disclaimer: I do not like “constructive math!” I think it’s a goofy cop-out. But it is useful in some ways, especially when someone brings “infinity” to dinner.)
(Infinity is a terrible dinner guest. It eats everything…forever. And rarely washes its hands.)
This thread makes me very crotchety, or perhaps I just am a very crotchety person and this thread makes me very aware of it. I find I find something to disagree with in almost every post, either in wording or substance (probably post #47 was the only one I was basically happy with, and even there, I had reservations about the first line).
Thoughts:
A) Whether “infinity” is a “number” or not is a completely irrelevant terminological question; the word “number” is a label which we can assign as we like it or not. I am fond of saying infinity is a number in certain number systems (e.g., affinely or projectively extended arithmetic, which perfectly well validate all the “1/infinity = 0”, “2 + infinity = infinity”, etc.,-style arithmetic claims people are naturally led to make), just as 3, 0, 1/2, -5, and 6 + 2i are; the concept of arithmetic with infinity admits family resemblances to other arithmetic systems, of the sort that justify similar language to me. However, it doesn’t matter whether you want to use this language or not; what matters is just the properties of infinite processes, regardless of what labels one wants to use in describing them.
B) There is no contradiction in assigning a value to 1/0 in some arithmetic language-game, so long as appropriate care is taken. See, again, the aforementioned system of projectively extended arithmetic.
C) Probability is just a way to describe proportions. It’s just the study of semipositive quantities totalling to 1. There’s no intrinsic bridge principle linking probability and possibility; one could perfectly well study probability as having nothing to do with and nothing to say about possibility. Yes, if one imposes a uniform measure on 2^N, or [0, 1], or such things, each particular singleton takes up less than 1/n of the whole, for arbitrarily high natural numbers n; this does not compel us to say anything about what is possible behavior for a physical or idealized coin.
D) Indeed, even should one want to play a language-game in which a link between probability and possibility is imposed, it is very hard to identify “has probability zero” with “is impossible”, for reasons already given. If one insists on doing so, one can bend oneself into saying something along such lines (e.g., “‘Definable’ events of probability zero are impossible; our definition of a ‘fair coin’ and the use of alethic language thereof should include the condition that, when flipped infinitely, such a coin must produce a sequence of heads and tails which is not a priori definable”) in a mathematically fruitful way, but there are significant (to the point of monstrous) subtleties to be carefully appreciated. This is the one point on which I find I have least crotchetiness, in that others have made the same argument just as well already.
E) For what it’s worth, most mathematicians who use the label “constructivists” are perfectly happy to discuss the sense in which “0.99999…” stands for 1, irrational numbers, e, pi, and all the rest of it. [Brouwer more than anyone else brought mathematicians’ attention to the idea of distinguishing between “completed” and “potential” infinities, yet Brouwer had no qualms with consideration of pi; one can perfectly easily construct an algorithm which describes the value of pi (or sqrt(2), or 0.9999…) to arbitrary precision]. I know no one in here is taking a contrary position, but I just want to point out that the aforementioned “constructivist” is, if accurately described, subscribing to a particularly narrow viewpoint not representative of most who use that name (that one should only study the arithmetic system of rationals, and no other possible numerical rule-games, apparently).
F) Furthermore, constructivism isn’t, typically, a “cop-out”; a constructivist is typically perfectly well capable of appreciating a deduction in classical, non-constructivist logic, as a deduction within a particular style of formal system or from particular premises, appropriate for particular narrow purposes. A constructivist is someone who is also willing to study other formal systems, with the thought that these are appropriate for other purposes. A willingness to investigate which principles are “constructively” valid is no more a cop-out than a willingness to study nonabelian groups or non-Euclidean geometry or any other instance of passage from the study of some familiar class of structures to the study of some other, more general class of structures.
G) Also, I need to buy groceries. It’s a damn shame the local Safeway is shut down for construction; now I have to go to the Andronico’s, where the prices are typically a bit higher, and which also closes two hours later. This has become a real problem, what with my habit of late-night shopping for…grumble, grumble
(And now I’m grumbling about all the things in my own post whose wording I take belated exception to. Life is pain.)
Here’s a supposed proof that .9999… equals 1.
Let x = .9999…
10x = 9.9999…
10x - x = 9.9999… - .9999…
9x = 9
x = 1
If x = .9999… and also x = 1, then .9999… = 1