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April 5th, 2017, 02:59 PM   #41
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Originally Posted by v8archie View Post
I can't say that I agree that your way of looking at it is the only correct way.
Is this for me or the OP? I don't think either of us said our way is the only way to look at it.

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Think about how monumentally unlikely it is that human beings appeared in the universe
The odds are 1. It happened.


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or that Leicester won the English Premier League last season.
I plead ignorance, being a citizen of the colonies. I have my musket ready in case any redcoats show up. But if it happened in the past, the odds are either 1 or 0 depending on whether it happened or not.

You are now discussing the philosophy of probability. Is probability inherent in the event itself? Or only a measurement of our ignorance? If I flip a coin and it lands on the ground, what are the odds it's heads before I look at it? What are the odds for you if you look at it? If the two numbers are different (1/2 for me, certainty one way or the other for you) then odds reflect the state of our knowledge and say nothing about events themselves.

I must say I don't see the relevance of this line of thought to the subject. But I don't know much about Collatz or probabilistic proofs.

Last edited by Maschke; April 5th, 2017 at 03:08 PM.
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April 5th, 2017, 03:45 PM   #42
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Originally Posted by Maschke View Post
Is this for me or the OP? I don't think either of us said our way is the only way to look at it.
You. The other was for the OP.

You explicitly stated that he's making an error. There's nothing in his view of the problem that stops it being Collatz, is there? You both have the same operations in the same order, just grouped differently.

Clearly the odds I was talking about were the prior odds in both cases. Nothing we know about the universe makes higher life forms remotely likely.
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April 5th, 2017, 08:40 PM   #43
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Maschke, I don't think invoking (3n+1)/2 is confusing in any way. It is just a different way of stating the problem other than just phrasing 3n+1 yet we know very well the next number would be even. what we don't know is the outcome after we divide by 2. That's where probability plays its role because the number could either be odd or even. That's what we are sure about. And the formula is what would determine it.
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April 5th, 2017, 09:20 PM   #44
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Originally Posted by v8archie View Post
For any odd number, there is either a 100% chance that the outcome is odd or there is a 100% chance that it is even. It's not a probablistic system, it's deterministic.
That's true, there is 100% chance of being odd and 100% chance of being even. But probability is not what actually is but our perception. It is what is in our minds. The information we have. In the example I used earlier about the cat in the box, anyone that doesn't know which box the cat is in knows that it is 50/50.. anyone who already knows which box contains the cat 100% where the cat is.

On the other point, indeed if we ever flip a coin a million times, indeed there is a chance, however slim, that we would have all heads. But what needs to be proven is that if we flip the coin an infinite number of times, the probability of getting all heads is zero. That can be proven.
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April 5th, 2017, 09:28 PM   #45
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Maschke, I don't think invoking (3n+1)/2 is confusing in any way. It is just a different way of stating the problem other than just phrasing 3n+1 yet we know very well the next number would be even. what we don't know is the outcome after we divide by 2. That's where probability plays its role because the number could either be odd or even. That's what we are sure about. And the formula is what would determine it.
Yes but the thing to be proved is of the form $\forall n P(n)$ where $P(n)$ is the proposition that the sequence collapses to $1$ when started from $n$. So it's not sufficient to prove that the probability of $n$ not going to $1$ is zero; because in an infinite probability space, probability zero events can happen.

If we perform the thought experiment of flipping infinitely many fair coins, one for each natural number, the probability that the first one is heads is $\frac{1}{2}$; the probability the first two are heads is $\frac{1}{4}$; and the probability that the first $n$ are heads is $\frac{1}{2^n}$. As $n$ increases without bound, the probability that the first $n$ are heads goes to zero.

This is true.

It is nevertheless the case that it is conceivable that infinitely many coins may land on heads. Each coin flip is independent of all the others. The coins don't "know" what the other coins are doing. So even though the probability is zero, it could happen. It's just incredibly unlikely.

Another thought experiment is to "throw a dart at the real number line." The probability of hitting a rational is zero; but there are rationals on the line, lots of them. You might get lucky and hit one.

In fact if you flip infinitely many coins, the odds of any particular sequence is zero. Yet some sequence must occur. If you throw a dart at the real line, you must hit SOME real number, but the odds of you hitting that particular number are zero. This is basic infinitary probability theory.

Here is yet another striking example. We can define the asymptotic density of a set of natural numbers as the limit of the percentage of members of that set in the first $n$ numbers, as $n$ goes to infinity.

So for example the asymptotic density of the even numbers is $\frac{1}{2}$ just as we'd expect.

It turns out that the asymptotic density of the primes is zero. In other words if you pick a random natural number, it's not prime. But there are plenty of primes.

[Note -- asymptotic density is not a probability measure for technical reasons, so "picking a natural number at random" must be taken with a grain of salt. There is no uniform probability measure on the natural numbers].

http://math.stackexchange.com/questi...of-primes-zero

tl;dr: Even if you prove that the probability that Collatz fails is zero, it still might fail.
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Last edited by Maschke; April 5th, 2017 at 09:50 PM.
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April 5th, 2017, 09:40 PM   #46
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Originally Posted by v8archie View Post
You. The other was for the OP.

You explicitly stated that he's making an error. There's nothing in his view of the problem that stops it being Collatz, is there? You both have the same operations in the same order, just grouped differently.
IMO the problem is inherently not approachable by probabilistic means, for the reasons outlined in my previous post.

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Clearly the odds I was talking about were the prior odds in both cases. Nothing we know about the universe makes higher life forms remotely likely.
I disagree. Everything I know about the universe makes higher life forms like me inevitable. After all I'm here. You want to talk about "prior probabilities" but that's metaphysics if not outright woo. If I went back in time to the moment of the big bang, knowing what I know, the probability that I'll live is 1. And this is not a scientific question, it's a philosophical one. You can't possibly be using this example to make a point about OP's proof of Collatz.

I disclaim that if I'm wrong I'm wrong, this [the philosophy of probability] is not an area I have expertise in. Someone will need to show me I'm wrong though.

Last edited by Maschke; April 5th, 2017 at 09:48 PM.
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April 5th, 2017, 10:45 PM   #47
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I think we have finally made it. We have the full proof. But still, there is a little more studies to be done on the hailstone sequence, stuff like 2^n numbers. In a different type of proof, one can show that the sequence always has to come to a power of 2.

Also a study of the longest chains of consecutive odds..whether the numbers that trigger them have any special properties or whether they are just random (and a little lucky).
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April 5th, 2017, 11:24 PM   #48
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I think we have finally made it. We have the full proof.
What? Where?
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April 5th, 2017, 11:43 PM   #49
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Joppy, we have shown that...
It is impossible to have any positive integer from which the sequence would infinitely rise either due to generating an infinite consecutive odd numbers or having more odd numbers than even numbers being generated
And also the sequence must always converge.
It holds and the proof is rigorous.
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April 6th, 2017, 07:56 AM   #50
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After all I'm here.
You apparently assume that the universe is deterministic.
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