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For your point 1 above, the issue seems to be whether pi is truly random or not, so let's swap out pi for any infinitely long string of digits which we know for certain is random. Let's pretend it is being generated by a monkey typing randomly on a typewriter that only has keys for digits 0 to 9, with any key press equally likely at any given point.
Given that, can't we conclude (according to Infinite Monkey Theory) that somewhere in that string of infinite digits there MUST be the string of 777,777,777,777 consecutive 7's? It literally MUST be there, and it MUST be there infinite times, according to that Wikipedia article.
I did say at the beginning that I was just choosing pi as an example of a random string of digits, but if there is still argument whether pi is truly random or not, then we don't have to use that. So the point that Aris' mathematician friend was making, that a specific FINITE string of digits doesn't have to appear within a random and infinite string of digits, seems to be in error?
Then the interesting thing (if I have this right and if we assume that pi is truly random, infinite and non-repeating) is that sure, we can say that an infinite string of 7's cannot appear in pi, but as long as we keep the string of 7's finite in size, then it MUST appear within pi ???
This is getting back to the nature of the conundrum I initially was describing. By the way, you mention pi has "so far passed all tests" for randomness... and as you know, proving a negative is often impossible... so what in your opinion would it take for it to be proven that pi is definitively NOT random?
So you seem to be changing the discussion from Pi to "pi as an example of a random string of digits", and then moving on to "random and infinite string of digits", apparently in a general sense.
That's fine in a way, but then the math expertise I secured (regarding Pi) may understandably not be applicable as per Pi. From reading these posts, you seem to be starting with a personal conclusion about series of consecutive digits, and then looking for ways to support that. When I read the expert math reply above, it resonates more with me than the Infinite Monkey Theory.
So, I don't think I can help this discussion any more. All the best with this quest, love passionate curiousity!
So you seem to be changing the discussion from Pi to "pi as an example of a random string of digits", and then moving on to "random and infinite string of digits", apparently in a general sense.
That's fine in a way, but then the math expertise I secured (regarding Pi) may understandably not be applicable as per Pi. From reading these posts, you seem to be starting with a personal conclusion about series of consecutive digits, and then looking for ways to support that. When I read the expert math reply above, it resonates more with me than the Infinite Monkey Theory.
So, I don't think I can help this discussion any more. All the best with this quest, love passionate curiousity!
Sorry, Aris, I always meant pi to be an example of an infinite, random, non-repeating string of digits. I think in my original post I left out the word "random" but I meant it to be there.
The point wasn't to prove anybody right or wrong or support anything at all, but to get myself and others thinking, something to do in these lockdown times. And yes, your math friend may have been thinking specifically of pi as maybe being non-random and so might not have been "wrong" at all, so let's forget about all of that.
Where we seem to be right now is that in a truly random, infinite, non-repeating sequence of digits (and for now, let's just assume pi is one of these), a specific finite string of digits (of ANY specific finite length) MUST appear infinite times. But as soon as you make the search string of digits infinite in length, it cannot appear at all. Although Garland Best seems to be most knowledgeable on this type of thing and may reply that I still haven't got it right ?
Then the interesting thing (if I have this right and if we assume that pi is truly random, infinite and non-repeating) is that sure, we can say that an infinite string of 7's cannot appear in pi, but as long as we keep the string of 7's finite in size, then it MUST appear within pi ???
This is getting back to the nature of the conundrum I initially was describing. By the way, you mention pi has "so far passed all tests" for randomness... and as you know, proving a negative is often impossible... so what in your opinion would it take for it to be proven that pi is definitively NOT random?
As long as you are saying FINITE, and assuming an "ideal" random number generator then I would say yes. But DON'T take that and use it as proof that INFINITE strings can also exist. Any finite string relative to an infinitely long string may as well be zero in length. I hope my other posts made this clear.
As long as you are saying FINITE, and assuming an "ideal" random number generator then I would say yes. But DON'T take that and use it as proof that INFINITE strings can also exist. Any finite string relative to an infinitely long string may as well be zero in length. I hope my other posts made this clear.
Yes, Garland, and thank you for your input. Your demonstration of Cantor's proof was very good, I'm glad you entered it here for everyone to see without having to search it out.
Here is your issue. This statement cannot be true. Only finite length sequences can occur. The sequence of digits in pi constitute a "countable infinity" and cannot contain an infinite number of infinities inside it.
You make a trivial logical error. Any infinite set contains an infinity of equally infinite sets.
The set of all integers is a countable infinity yet it contains an infinite number of other infinities within it. For instance the set of all even numbers is infinite, as is the set of all odd numbers, the set of all numbers divisible by three, by four and so on to infinity. An infinity of infinities lies within the countable set of the integers.
Last edited by Ed Seedhouse; Thursday, 11th June, 2020, 09:26 PM.
You make a trivial logical error. Any infinite set contains an infinity of equally infinite sets.
The set of all integers is a countable infinity yet it contains an infinite number of other infinities within it. For instance the set of all even numbers is infinite, as is the set of all odd numbers, the set of all numbers divisible by three, by four and so on to infinity. An infinity of infinities lies within the countable set of the integers.
After I read last night about Infinite Monkey theory, Aris, I believe your math nerd friend is in error. The Infinite Monkey Theory is summed up as follows:
"A monkey hitting typewriter keys at random for an infinite time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey will almost surely type every possible finite text an infinite number of times."...
Using Cantor's method, I will generate an infinite string of digits that cannot exist in Pi.
1) Generate a list of all possible sequences of digits in Pi as follows.
a) item 1 in our list starts from the first spot after the decimal place: 141592653589793238462643383...
b) item 2 in our list starts from the second spot after the decimal place: 415926535897932384626433832...
c) item 3 in our list starts from the third spot after the decimal place: 159265358979323846264338327...
d) item 4 in our list starts from the fourth spot after the decimal place: 592653589793238462643383279...
e) item 5 in our list starts from the fifth spot after the decimal place: 926535897932384626433832795...
and so on.
2) Generate our new number by taking the nth digit in each number in our list (so 1 from item 1, 1 from item 2, 9 from item 3, 6 from item 4, 3 from item 1), and increasing the value by one. The pattern below will make this clear. 141592653589793238462643383...
415926535897932384626433832...
159265358979323846264338327...
592653589793238462643383279...
926535897932384626433832795...
..... 22074....
This new number:
a) Does not match the first number, because the first digit is different
b) Does not match the second number, because the second digit is different
c) Does not match the third number, because the third digit is different
.....
n) Does not match the nth number because the nth digit is different.
And onwards to infinity. Therefore this infinite string of digits that I created CANNOT exist in Pi. And I can create similar numbers using any number of different ways, as long as I ensure that given that the nth digit is different than the matching digit in the list.
Given that I can create an infinite number of infinite strings of digits that I can prove do not exist in Pi, it stands to reason that your pattern of infinite 7's does not have to exist in Pi.
Thanks again for explaining all of that in great detail, Garland. Now.... I think I see a fallacy in this proof. The key statement is "Therefore this infinite string of digits that I created...". The problem is, you can't ever actually create it. In order to create it, you need to get to the Nth infinite digit and add 1 to it. You would need infinite time to get there, and you would never arrive. So you can't add the 1.
This means a string that cannot exist in pi or any other infinite random not-repeating string of digits can never be created. If you reach some finite Nth digit and add 1, that is a finite string and that COULD exist elsewhere in pi because pi goes on for infinity. And in fact, as our other discussions have concluded, any finite string, no matter how long, MUST exist infinite times within pi (I am assuming pi IS random and is non-repeating for infinity).
This doesn't change anything with respect to my infamous infinite string of 7's, that string can also never be created and so we can't really say anything about it other than it never terminates so it cannot appear somewhere within pi. I'm ok with that, I wasn't trying to "prove" that it must be in pi, and I erroneously assumed at the beginning of this thread that it must be somewhere in pi. So you have dissuaded me of that, Garland :)
With respect to what Ed Seedhouse wrote about the set of infinite even numbers existing within the set of integers, he's correct that the SET of those numbers must exist within, but the actual ARRANGEMENT of those numbers of course does not exist within the set of integers. So I think that was comparing apples to oranges, because I was saying (again, in error) that the arrangement of infinite 7's must appear within pi. But if we terminate the string of 7's, even if it's a gazillion digits long, then that arrangement of 7's must appear somewhere within pi and infinite times within pi.
Perhaps this is all helping us to realize this covid stuff has only been happening for a very short time... no matter how long it may feel.
Thanks again for explaining all of that in great detail, Garland. Now.... I think I see a fallacy in this proof. The key statement is "Therefore this infinite string of digits that I created...". The problem is, you can't ever actually create it. In order to create it, you need to get to the Nth infinite digit and add 1 to it. You would need infinite time to get there, and you would never arrive. So you can't add the 1.
Mathematics assumes infinite procedures happen instantaneously and not over time. If you bring in time you are doing physics, not math. It is unfair and illogical to insist that mathematics should apply to the "real" world.
Mathematics is it's own world having nothing to do with what we call the "real" world. It is all the more surprising then, that math has such useful applications to reality.
Last edited by Ed Seedhouse; Friday, 12th June, 2020, 08:39 PM.
Mathematics assumes infinite procedures happen instantaneously and not over time. If you bring in time you are doing physics, not math. It is unfair and illogical to insist that mathematics should apply to the "real" world.
Mathematics is it's own world having nothing to do with what we call the "real" world. It is all the more surprising then, that math has such useful applications to reality.
I thought calculus was a part of mathematics that deals with time, but anyway... I don't know whether math can always apply to the real world or not, that's for a mathematician to say (and you may be one, Ed, I have no idea).
But I would move past the time issue to say that even if you could get to the Nth element of a string of digits in instantaneous time, you still can't get to the INFINITE Nth element. As soon as you claim to have "arrived" at that element, the infinite string has instantaneously filled the entire universe with new digits, and maybe filled infinite other universes as well.
So there is no getting to this mythical Nth element, and you have to get to it because you don't know what the digit is going to be before you add 1 to it. You have to discover what that digit is, then add the 1 to it. It can't be done. This string of digits that can't appear in pi also can't be constructed, just like my infinite string of 7's.
I don't know, but I'd say offhand that this blow's Cantor's theory out of the water.
I thought calculus was a part of mathematics that deals with time, but anyway... I don't know whether math can always apply to the real world or not, that's for a mathematician to say (and you may be one, Ed, I have no idea).
But I would move past the time issue to say that even if you could get to the Nth element of a string of digits in instantaneous time, you still can't get to the INFINITE Nth element. As soon as you claim to have "arrived" at that element, the infinite string has instantaneously filled the entire universe with new digits, and maybe filled infinite other universes as well.
So there is no getting to this mythical Nth element, and you have to get to it because you don't know what the digit is going to be before you add 1 to it. You have to discover what that digit is, then add the 1 to it. It can't be done. This string of digits that can't appear in pi also can't be constructed, just like my infinite string of 7's.
I don't know, but I'd say offhand that this blow's Cantor's theory out of the water.
All utterly beside the point. And beating a dead horse to boot.
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