Infinite numbers are strange
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I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
So how do you put a 3 at the non-existent 'end' of an infinite sequence?
There are no solutions, only trade-offs.
- Thomas SowellA day can really slip by when you're deliberately avoiding what you're supposed to do.
- Calvin (Bill Watterson, Calvin & Hobbes) -
I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
StarNamer@work wrote:
But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3.
But you can't. If you repeat it infinitely that means there's always another digit. When you try and add it to the "end", there's always another digit after that spot, so you're not at the end.
StarNamer@work wrote:
That is clearly an infinity (it has infinitely many digits!),
Not quite. It doesn't mean the result is infinite, just that there's no finite representation in base 10. 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. This is why mathematicians never need to do drugs.
cheers Chris Maunder
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I would not consider 1/7 or its decimal equivalent an infinite number. I think technically it is a rational number. Unless you are using a special library, then computers (and especially databases) don’t deal that well with these types of numbers. This is why there are fixed decimals that always round in favor of the bank.
englebart wrote:
This is why there are fixed decimals that always round in favor of the bank.
This is incorrect. Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down). If the residue is exactly 0.5 cents, the number is rounded "away" - up for positive, down for negative. If you are running a credit, this gives you a tiny statistical advantage. If you are running a debit, this gives the bank a tiny statistical advantage. In neither case is this likely to have a measurable effect, unless you aggregate over billions of operations a day.
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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I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
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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So how do you put a 3 at the non-existent 'end' of an infinite sequence?
There are no solutions, only trade-offs.
- Thomas SowellA day can really slip by when you're deliberately avoiding what you're supposed to do.
- Calvin (Bill Watterson, Calvin & Hobbes)Start with the 3 and just prefix it repeatedly!
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StarNamer@work wrote:
But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3.
But you can't. If you repeat it infinitely that means there's always another digit. When you try and add it to the "end", there's always another digit after that spot, so you're not at the end.
StarNamer@work wrote:
That is clearly an infinity (it has infinitely many digits!),
Not quite. It doesn't mean the result is infinite, just that there's no finite representation in base 10. 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. This is why mathematicians never need to do drugs.
cheers Chris Maunder
Chris Maunder wrote:
But you can't. If you repeat it infinitely that means there's always another digit. When you try and add it to the "end", there's always another digit after that spot, so you're not at the end.
Start with the 3 and just prefix it repeatedly!
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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.
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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I was told there would be no math.
Check out my IoT graphics library here: https://honeythecodewitch.com/gfx And my IoT UI/User Experience library here: https://honeythecodewitch.com/uix
I think this is closer to philosophy than maths. Certainly a long was from the arithmetic I learned in school!
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I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
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 was told there would be no math.
Check out my IoT graphics library here: https://honeythecodewitch.com/gfx And my IoT UI/User Experience library here: https://honeythecodewitch.com/uix
Instead, realize that there is no math spoon
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
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StarNamer@work wrote:
But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3.
But you can't. If you repeat it infinitely that means there's always another digit. When you try and add it to the "end", there's always another digit after that spot, so you're not at the end.
StarNamer@work wrote:
That is clearly an infinity (it has infinitely many digits!),
Not quite. It doesn't mean the result is infinite, just that there's no finite representation in base 10. 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. This is why mathematicians never need to do drugs.
cheers Chris Maunder
Chris Maunder wrote:
This is why mathematicians never need to do drugs.
Are you sure? It would be possible that they are always under the effect of drugs... like Obelix, but instead of falling in the cauldron of magic potion, they fell into the cauldron of LSD :rolleyes: :laugh:
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
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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.
Daniel Pfeffer wrote:
your password is the last eight digits of Pi.
Time to change my password again. :D
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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
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:
But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3.
But you can't. If you repeat it infinitely that means there's always another digit. When you try and add it to the "end", there's always another digit after that spot, so you're not at the end.
StarNamer@work wrote:
That is clearly an infinity (it has infinitely many digits!),
Not quite. It doesn't mean the result is infinite, just that there's no finite representation in base 10. 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. This is why mathematicians never need to do drugs.
cheers Chris Maunder
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
reals.Add(0.1)
reals.Add(0.5)
reals.Add(pi)
// as many as you wantrepeat 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.
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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[^].
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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I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
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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You cannot add a 3 to the end of an infinite number, as that converts it to a finite one.
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englebart wrote:
This is why there are fixed decimals that always round in favor of the bank.
This is incorrect. Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down). If the residue is exactly 0.5 cents, the number is rounded "away" - up for positive, down for negative. If you are running a credit, this gives you a tiny statistical advantage. If you are running a debit, this gives the bank a tiny statistical advantage. In neither case is this likely to have a measurable effect, unless you aggregate over billions of operations a day.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Daniel Pfeffer wrote:
Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down)
I couldn't find any regulation that specifies how banks round numbers. And given 'banks' exist throughout the world I suspect certainly in some places rounding might use different rules. I worked for a financial company (not a bank) and the rounding was decided by me. There are multiple rules. Last time I looked (and can recall) there are three different types of rounding suited to financial transactions. Rather than scientific. Although one of those might also be scientific. Two of them provide 'better' results than just rounding up on '0.5' otherwise down. Which I suspect you are referring to. That specific method tends to favor a specific result. Because there are 6 digits from 5-9 but only 4 from 0-4.
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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
reals.Add(0.1)
reals.Add(0.5)
reals.Add(pi)
// as many as you wantrepeat 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.
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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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.
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.
