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Chris Maunder wrote: There are actually many different "infinities". The numbers 1,2,3...is an infinite set. The set of real numbers between 1 and 2 (eg 1.1, 1.01, 1.001 and on and on) is also infinite, and large than the set of integers. One infinity can be bigger than another infinity. Even though they are both infinite.
Actually, I've never been totally convinced of this, although I'm open to it being proved in some way.
The only way I've ever seen is Cantor's Diagonalization, which says to take a list of all the Real (Rational plus Transcendental, etc) numbers between zero and one then to create a new number by taking the first decimal digit of the first number, second decimal digit of the second number, third of the third, etc. The argument is that this number cannot be on the list, so therefore you can put the infinite number of Reals into correspondence with the Integers so there there is at least a Countable infinity (number of integers) and an Uncountable infinity (number of Reals).
My scepticism comes from the statement about creating the list of Reals. I'd like to use the following pseudo code:
reals = New List<Real>
reals.Add(0.1)
reals.Add(0.5)
reals.Add(pi)
// as many as you want
repeat forever // until list is complete
for each number r in reals
x = new real
for each decimal digit p of r
digit p of x = not_the_same_as(digit q of r) // function elsewhere
if x not in reals
reals.Add(x)
else
terminate // countable list of reals is completeMy point is that this procedure *is* Cantor's Diagonalization so if that can find another Real to add, then the list building shouldn't have terminated and, if it can't, then it's not been proved that there are more Reals than Integers. It may be true, but this doesn't prove it.
I feel sure there must be an alternative proof to Cantor's, but I've never found it. Perhaps it relies on maths I've never encountered and would need a degree in Mathematics to understand! (Mine was Physics!  )
FYI, I recall I once saw a proof that there are more Transcendental numbers (like pi or e) than Rational numbers (like 1/5, 3/7, etc) but can't recall if it was also based on Cantor's method.
modified 5-Nov-23 13:24pm.
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StarNamer@work wrote: It may be true, but this doesn't prove it.
Interesting argument.
However it fails from the concept of terms versus proof.
An infinite series is just a concept. Since of course you can't reach the end. As you are suggesting. So one must accept the concept without enumerating the set.
But once one accepts the concept then one can discuss it. Thus there is a set that does have all of the numbers (but still conceptually infinite) and thus there can be a comparative enumeration using the other set.
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Infinity is a very tricky concept, and you have misunderstood it.
You cannot add anything to the "end" of an infinite sequence - it has no "end". It is as ridiculous as claiming that your password is the last eight digits of Pi.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
-- 6079 Smith W.
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Daniel Pfeffer wrote: You cannot add anything to the "end" of an infinite sequence - it has no "end".
Start with the 3 and just prefix it repeatedly!
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Daniel Pfeffer wrote: your password is the last eight digits of Pi.
Time to change my password again. 
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You may just got confused with infinity (which is an idea and can not be use as a number) and a fraction that has infinite decimal digits...
Obviously you cannot compute anything with the decimal representation of that fraction as by its nature it will take up infinite time to do so...
The problem is that you decided to cut the flow of infinite decimal digits and make a computation based on that... Depending on where you stop counting the digits you will have different results...
0.1428573 * 7 = 1.0000011
0.14285731428573 * 7 = 1.00000120000011
---
0.142857 * 7 = 0.999999
0.142857142857 * 7 = 0.999999999999
And so on... There is no justification at any point to say those two numbers are the same...
(Think about the division by zero)
"If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization." ― Gerald Weinberg
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I missed out all the boilerplate phrases along the lines of "As you increase the number of digits 7*0.14285714... tends towards 1 so, in the limit, is assumed to be 1...", etc.
Kornfeld Eliyahu Peter wrote: 0.1428573 * 7 = 1.0000011
0.14285731428573 * 7 = 1.00000120000011
---
0.142857 * 7 = 0.999999
0.142857142857 * 7 = 0.999999999999
You've misread the first number (second in my message). It's...
........2857142857142857142857142857143
or
........2857142857142857142857142857142857142857142857142857142857143
That is, an integer with an infinite number of repetitions of ...285714... followed by 3. Although there's a notation for recurring decimals, I don't know of a shorthand for a p-adic number (which is what this is).
The point is that I (and the video) didn't suggest stopping the calculation at any point. Obviously, if you do they aren't the same and, in the case of the 'infinite' integer, you don't actually have a result! It's only if you project to the theoretical limit that the results are equivalent.
Many years ago, I researched for a PhD in Nuclear Structure Physics and studied some High-Energy (particle) Physics so am aware of renormalization to get rid of infinites in theories, the meaning (or lack of it) of anything divided by zero, etc.
I'd just never encountered p-adic numbers[^] before I watched that video[^].
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StarNamer@work wrote: Many years ago, I researched for a PhD in Nuclear Structure Physics and studied some High-Energy (particle) Physics so am aware of renormalization to get rid of infinites in theories, the meaning (or lack of it) of anything divided by zero, etc.
As you are no doubt aware, there are no infinities in nature. Any infinity that does turn up in our theories is a sign that the theory is incomplete. Renormalization is a clever mathematical trick that allows us to make predictions without having a complete theory, but one must remember its limitations - it is a mathematical trick, not part of a correct description of nature.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
-- 6079 Smith W.
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Daniel Pfeffer wrote: As you are no doubt aware, there are no infinities in nature
Not sure I agree with that.
Time exists because the universe exists.
If the universe ends then time ends.
If the universe never ends then time never ends. It becomes infinite.
Your statement above is an affirmative belief (not a proof) that the universe must end.
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There are only two possibilities: (a) the Universe has always existed, or (b) the Universe has a finite age.
If the Universe has always existed, its age is infinite. Adding any finite amount to that value will still give infinity.
If the Universe began at a finite point in the past, its age is finite. Adding any finite amount, however large, to that finite number will still yield a finite number.
If our theories are correct, the Universe is approximately 14 billion years old. Adding a googolplex (1010100 years) to that will yield a very large, but still finite, number. Note that even the most far-out theories cannot say anything about how the Universe will look at that advanced age; the extrapolation from our current observations is simply too extreme.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
-- 6079 Smith W.
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Daniel Pfeffer wrote: There are only two possibilities: (a) the Universe has always existed, or (b) the Universe has a finite age.
I do not agree. That is a binary statement that is not necessarily true.
The universe could have 'started' in the sense of the word meaning that before that point time did not exist and after that point time did.
Daniel Pfeffer wrote: If the Universe has always existed, its age is infinite.
Yet you said "there are no infinities in nature."
Are you claiming that time is not part of nature?
Daniel Pfeffer wrote: If the Universe began at a finite point in the past, its age is finite
I don't agree with that at all.
There are an infinite number of positive integers. Positive integers, specifically and by definition, start at zero. But they never end.
The universe itself can only 'begin' when time starts. That is the nature of the word 'begin'. But the fact that it started does not mean that it must end.
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Finally I had time to watch the video... Now I understand you better...
However I have to say that, that video cuts corners in a very horrible way IMHO...
It seems that its simplifications and inclusions are chosen to server a specific end result, but not clear and whole or precise...
"If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization." ― Gerald Weinberg
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Obligatory SMBC[^]. 
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
- Homer
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Wordle 868 3/6
🟨⬛⬛🟨⬛
⬛⬛⬛🟨🟨
🟩🟩🟩🟩🟩
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Wordle 868 5/6
🟨⬜🟨⬜⬜
⬜⬜🟨🟩⬜
⬜🟩⬜🟩⬜
⬜🟩⬜🟩⬜
🟩🟩🟩🟩🟩
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Wordle 868 3/6
⬛⬛🟨🟨⬛
⬛⬛⬛🟨🟨
🟩🟩🟩🟩🟩
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⬜⬜🟨🟨⬜
🟩🟩🟩🟩🟩
In a closed society where everybody's guilty, the only crime is getting caught. In a world of thieves, the only final sin is stupidity. - Hunter S Thompson - RIP
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Wordle 868 5/6
🟨⬛🟨⬛⬛
🟨⬛🟨⬛⬛
⬛🟨🟩⬛🟩
⬛⬛🟩🟩🟩
🟩🟩🟩🟩🟩
Ok, I have had my coffee, so you can all come out now!
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Can't work out what you used for your fourth go !
In a closed society where everybody's guilty, the only crime is getting caught. In a world of thieves, the only final sin is stupidity. - Hunter S Thompson - RIP
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You know what? I cannot remember myself! 
Ok, I have had my coffee, so you can all come out now!
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Wordle 868 5/6
⬜⬜🟨🟨⬜
⬜🟨🟩⬜⬜
⬜⬜🟩🟩🟨
🟩🟩🟩🟩⬜
🟩🟩🟩🟩🟩
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Wordle 868 5/6
⬜⬜🟨⬜⬜
⬜🟨⬜⬜⬜
⬜🟩⬜🟩🟨
⬜🟩⬜🟩⬜
🟩🟩🟩🟩🟩
"A little time, a little trouble, your better day"
Badfinger
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Really interesting interview, very long but fascinating.
Quote: The Adobe Corporation has announced that cofounder John Warnock died at the age of 82 on August 19, 2023. His death will be a loss to many, most of all his closely knit family, but extending as well to many professional colleagues and, indeed, to us at the Computer History Museum.
In Memoriam: John Warnock (1940–2023) - CHM[^]
I don't think before I open my mouth, I like to be as surprised a everyone else.
PartsBin an Electronics Part Organizer - Release Version 1.3.0 JaxCoder.com
Latest Article: SimpleWizardUpdate
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Mike Hankey wrote: Adobe Corporation
I'm just hoping for his sake he won't primarily be remembered for starting a company that devolved into one that produced unwanted nagware/bloatware.
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Hopefully not, and not responsible for the subscription model.
Whoever came up with that out to publicly flogged.
I don't think before I open my mouth, I like to be as surprised a everyone else.
PartsBin an Electronics Part Organizer - Release Version 1.3.0 JaxCoder.com
Latest Article: SimpleWizardUpdate
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