A Piece of Pi For Dessert During Covid Shutdown

[Fred Harvey]

Pargat....I wasn't criticizing anyone, because as I said I can no longer pick out the contributors from the idiots ! I certainly give you the prize for starting one of the best threads on here for a long time.

[Aris Marghetis]

You really don't like people with certain types of names, do you? (you were responding to Dilip)

I genuinely did not mean to insult you (or Dilip). I guess I got fatigued of you saying something was wrong just because you thought it was, rather than proving it. If I went overboard in my responses, that's my bad, my regrets. But then, your last quote here seems to be playing the race card. I cannot tolerate that, no one should. That's my values. I wish you all good health.

[Garland Best]

[QUOTE=Dilip Panjwani;n206675]The 'infinity' in the above example is another way of saying that the decimal numerical system cannot accurately adapt to the fraction under consideration ./QUOTE]

Well, so much for trying to use simple a example to demonstrate! Yes, a fraction is a more convenient way to express my example, but it does not change that fact that to express the fraction as a decimal requires an infinitely long string of digits.

And before you try to argue that then "just use fractions for everything", there are plenty of numbers, like the square root of 2, e and pi, that cannot be expressed as a fraction, only approximately as one.

Ok, try this one. I'm trying to say that infinity as an abstract concept exists. You are trying to say it does not. But by simply trying to argue it does not exist implies that you can CONCEIVE what infinity is to argue against it. Therefore infinity exists as a concept. QED.

[Pargat Perrer]

I genuinely did not mean to insult you (or Dilip). I guess I got fatigued of you saying something was wrong just because you thought it was, rather than proving it. If I went overboard in my responses, that's my bad, my regrets. But then, your last quote here seems to be playing the race card. I cannot tolerate that, no one should. That's my values. I wish you all good health.

Ok, I apologize for what I wrote about names and I take it back, maybe I got too sensitive for a moment there.

I think I DID prove Cantor's proof doesn't work... at least going by Garland's explanation of it. But maybe I need to get a little more rigorous about it.

[Pargat Perrer]

Sigh.

No, Cantor did NOT say "change the digit at the end of it". His proof is based his constructed number having at least one digit different from every number listed in the original infinite list. Proof by contradiction.

1/9 can be written as 0.1111111... to infinity
2/9 can be written as 0.222222... to infinity.

I know that every digit after the decimal place is different in these two numbers. I don't have to go to the "end" of either number to know this.

Now wait, Garland, you changed something there. Those two fractions turn into infinite REPEATING strings of digits. Pi is not one of those, and I wasn't referring to such repeating strings. Obviously I'm NOT going to theorize that somewhere within (0.111111....to infinity) there must be a string of infinite 7's, right?

Now, I'm still in agreement that the infinite string of 7's cannot appear within pi after all, but only because I realize the infinite string of 7's can never be created, as it never ends. Not because of Cantor's proof.

So now about Cantor's proof as you explained it and referring to infinite NON-REPEATING strings of digits such as pi (just one example):
Here's what you wrote in your explanation of Cantor's proof, the very first line you wrote:

"Using Cantor's method, I will generate an infinite string of digits that cannot exist in Pi."

Do you understand that you simply CANNOT create this infinite string of digits? Even ignoring time.

I did make a mistake in the post of mine that you quoted, where I said "then change the digit at the end of it", what I meant to say was "then change the nth digit" where n is as follows (again quoting your explanation of Cantor's method):
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...

So item n would start from the nth spot after the decimal point in pi. As such n is a finite number. Doesn't matter how big it is, still finite. You can never "arrive" at the infinite'th n. I hope that makes sense. Therefore you can never construct this new number that cannot appear in pi. You have to get to the infinite'th n so that you can change the nth digit (by adding 1 to it), but the infinite'th n can never be reached.

Each finite n that you do reach (and change by adding 1) adds 1 to the length of this number you are creating, but each time you do that, your new number is FINITE.
141592653589793238462643383...
415926535897932384626433832...
159265358979323846264338327...
592653589793238462643383279...
926535897932384626433832795...

now stringing the highlighted digits together and adding 1 to each of them (adding 1 to 9 gives 0):
22074

For example, above you have just changed the 5th digit and created the number 22074. A finite string of digits, guaranteed to appear infinite times within pi.
You can NEVER arrive at the infinite'th n to create your infinitely-long new number that you propose cannot appear in pi. Therefore it is impossible to create a string of digits that cannot appear within pi.

That is a clear-cut refutation of Cantor's proof, unless something has been left out of his method that I don't know about. The key to this is that every FINITE string of digits you create is guaranteed to exist within Pi, not just once, but infinite times.

[Garland Best]

You can NEVER arrive at the infinite'th n to create your infinitely-long new number that you propose cannot appear in pi. Therefore it is impossible to create a string of digits that cannot appear within pi.

Why are you allowed to generate an infinitely long sequence of digits for Pi, while I am not allowed to form an infinitely long sequence of digits for my new number?

[Pargat Perrer]

Why are you allowed to generate an infinitely long sequence of digits for Pi, while I am not allowed to form an infinitely long sequence of digits for my new number?

Because Pi is pre-existing. It already exists, we didn't generate it.

The number you are creating does not pre-exist, you are creating it by going to the nth digit of pi and changing that digit. That is the method of Cantor's proof, and that is why I say Cantor in his proof disrespected the nature of infinity.

As you go on into infinity creating longer and longer number trying to reach the "magic number" that cannot appear within pi, each time you change the nth digit, you create a FINITE number. And as was shown earlier in this thread, any finite string of random non-repeating digits MUST appear infinite times within pi. Maybe Cantor didn't realize that in his time.

[Aris Marghetis]

Ok, I apologize for what I wrote about names and I take it back, maybe I got too sensitive for a moment there.

I think I DID prove Cantor's proof doesn't work... at least going by Garland's explanation of it. But maybe I need to get a little more rigorous about it.

I apologize as well. When I look back at my language, I can see how it could be inflammatory. My regrets.

Ironically, overnight, an old friend of mine sent me some material that counter-argues against Cantor's work, and after some time with it, I see that point. I don't know if I'm ready to accept it, but it did get me thinking, which was probaby the initial intent of your thread.

Are we good to continue from here?

[Garland Best]

Because Pi is pre-existing. It already exists, we didn't generate it. The number you are creating does not pre-exist, you are creating it by going to the nth digit of pi and changing that digit.

Then by that argument, I can't generate the list of all possible number strings in Pi, because I generated it by lopping off successive digits in the number Pi and listing them. I will never get to my end of my list.

Also by that argument, I can't generate the infinite string of digits in Pi because it is impossible for me to write it down completely. There will always be another digit to write down.

If you are going to allow me to generate an infinite sequence in even one of these processes, then you need to allow me to do it everywhere. You cannot pick and choose.

[Pargat Perrer]

Then by that argument, I can't generate the list of all possible number strings in Pi, because I generated it by lopping off successive digits in the number Pi and listing them. I will never get to my end of my list.

Also by that argument, I can't generate the infinite string of digits in Pi because it is impossible for me to write it down completely. There will always be another digit to write down.

If you are going to allow me to generate an infinite sequence in even one of these processes, then you need to allow me to do it everywhere. You cannot pick and choose.

I did some further reading last night and one thing I found is that what we are really talking about is the decimal representation of pi. This is where the infinite string of non-repeating digits appears. If we chose pi itself as the base of our number system, then pi = 10.

So pi exists and it is an irrational and finite number and its decimal representation is a string of infinite non-repeating digits, that much we know for certain.

Ok, so the first 2 paragraphs of your post: I agree completely, 100%. You cannot generate that list. Yet that is part of Cantor's method. Cantor didn't even realize that the list can never be generated? To me, his method ends right there. It's like you tell someone you know how to fly up from the Earth using anti-gravity, and your first instruction is: while standing on the surface of the Earth, turn off gravity.

So your first 2 paragraphs are the very reasons why Cantor's proof doesn't work. He ignored the real world.

And again, I have to repeat this because it's the most important thing: you can't go to the infinite'th n and change it, BUT you can go to the gazillionth n and change that, and what you then have is a finite number, and it is guaranteed to appear somewhere in the decimal representation of pi, infinite times. No matter how big this finite number is, it will appear in pi infinite times because it is a finite number.

This number you are chasing after that cannot appear in pi, you might as well be a puppy chasing its tail. So as far as cherry-picking goes, I'll say this: since we cannot ever generate ANY infinite string of digits, there are NO such infinite strings that can PROVABLY not appear in pi's decimal representation. So even my infinite string of repeating 7's, again you cannot actually get to the infinite'th digit to represent it, so again you can't prove that it doesn't exist in pi. I think we both UNDERSTAND that such infinite strings are not going to appear in the decimal representation of pi, but there is no PROVING it. That is what I am saying, in essence. Cantor's proof is invalid.

[Garland Best]

since we cannot ever generate ANY infinite string of digits, there are NO such infinite strings that can PROVABLY not appear in pi's decimal representation. So even my infinite string of repeating 7's, again you cannot actually get to the infinite'th digit to represent it, so again you can't prove that it doesn't exist in pi. I think we both UNDERSTAND that such infinite strings are not going to appear in the decimal representation of pi, but there is no PROVING it. That is what I am saying, in essence. Cantor's proof is invalid.

And here is where we are at an impasse. You claim that it is impossible to generate an infinite string, while I say that yes, I can conceptualize it's existence so it exists. Until one of us change our minds we are not going to resolve this. That's ok. Extrapolating from the finite to the infinite is still argued among mathematicans.

For now I will leave you this quote from Lewis Carroll:

“I can’t believe THAT!” said Alice.

“Can’t you?” said the Queen in a pitying tone. “Try again: draw a long breath, and shut your eyes.”

Alice laughed. “There’s no use trying,” she said, “one can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why sometimes I believed as many as six impossible things before breakfast!”

[Ken MacDonald]

And here is where we are at an impasse. You claim that it is impossible to generate an infinite string, while I say that yes, I can conceptualize it's existence so it exists. Until one of us change our minds we are not going to resolve this. That's ok. Extrapolating from the finite to the infinite is still argued among mathematicans.

For now I will leave you this quote from Lewis Carroll:

“I can’t believe THAT!” said Alice.

“Can’t you?” said the Queen in a pitying tone. “Try again: draw a long breath, and shut your eyes.”

Alice laughed. “There’s no use trying,” she said, “one can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why sometimes I believed as many as six impossible things before breakfast!”

Yaaaaawwwwwwwnnnnnnnnnnn!!

Look at all that chess content.

[Pargat Perrer]

And here is where we are at an impasse. You claim that it is impossible to generate an infinite string, while I say that yes, I can conceptualize it's existence so it exists. Until one of us change our minds we are not going to resolve this. That's ok. Extrapolating from the finite to the infinite is still argued among mathematicans.

For now I will leave you this quote from Lewis Carroll:

“I can’t believe THAT!” said Alice.

“Can’t you?” said the Queen in a pitying tone. “Try again: draw a long breath, and shut your eyes.”

Alice laughed. “There’s no use trying,” she said, “one can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why sometimes I believed as many as six impossible things before breakfast!”

Nice quote!

Hey Garland, I'm all for imagining impossible things. In fact that's what got me into trouble with my string of infinite 7's, I imagined it as something real.

I agree that your infinite number can be imagined just as my string of 7's. But the problem lies in Cantor's method where it must not just be imagined, but REALIZED, because his method requires that we go to the intinite'th digit and add 1 to it. We have to know what that digit is, and we can never know.

So even if we're at an impasse, the onus is on any supporter of Cantor's method to show how it can be done. You are a scientist, correct? You would know that scientific proof requires some repeatable method that others can perform to validate results. Cantor's method cannot be done even once, ever.

Change the infinite'th digit in an infinite series... turn off gravity at the surface of the Earth... equally impossible things that we can imagine, but never perform.

Thank you for a great exchange of ideas.

[Fred Harvey]

....... while I say that yes, I can conceptualize it's existence so it exists.

This must be the "game, set and match" moment! Who won?