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Are All Infinities The Same Size?
23 Answers
There are an infinite number of numbers between 1 and 2. There are an infinite number of numbers, thus the latter is bigger than the former. This is just my example but don'y take my word for it...........
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Answers
To show that there are more real numbers between, say, 1 and 2 than there are whole numbers between 1 and infinity, you can try the following exercise: 1. List all the whole numbers (obviously this is impossible, but the point is that you could, in principle, list them: 1, 2, 3... and so on). 2. For each whole number you've listed, pick a real number between 1 and 2...
19:20 Sun 09th Jul 2023
To show that there are more real numbers between, say, 1 and 2 than there are whole numbers between 1 and infinity, you can try the following exercise:
1. List all the whole numbers (obviously this is impossible, but the point is that you could, in principle, list them: 1, 2, 3... and so on).
2. For each whole number you've listed, pick a real number between 1 and 2 at random, until every whole number has a given assigned real number.
3. Now, create a new real number not on the list in the following way: for your first listed number, add one to its first decimal digit; for your second listed number, add one to its second digit; and so on.
For example, suppose you had the following list:
1 1.0 0320667590275902984897...
2 1.6 8 492037575839828927098...
3 1.20 3 85789860827364987647...
4 1.676 7 38381974673499494499...
5 1.9996 8 6886377477110000003...
...
etc. The new number you generate by the rule I listed above would become 1.19489... and is different from every other umber on the list in at least one place, by construction. Therefore, it doesn't fit on the list, and since you've literally exhausted all countable numbers between 1 and infinity and still can find "more" real numbers between 1 and 2.
For more information, look up Cantor's diagonal argument -- the version above is a sketch, and is probably not fully convincing, but the argument is sound and proves that the real numbers are uncountable.
On the other hand, to prove that there are as many real numbers between 1 and 2 as there are between, say, 1 and infinity, then we need to find a way of taking all numbers in one set and mapping them to all numbers in the other set.
The following will do:
1. Let y = tan x, for x from -pi/2 to pi/2. This function is continuous, and it's not difficult (by using normal trigonometry) to show that tan 0 = 0, tan pi/2 "=" infinity, and tan (-pi/2) "=" -infinity (where "=" really means "tends to", rather than literally equalling).
2. Because the function is continuous, and always increasing, then for each number between -pi/2 and pi/2 there's a unique real number output between -infinity and infinity.
3. All we need to do is shift the input from 1 to 2 and the output from 1 to infinity. We can sort that by rewriting:
y = 1 + tan ((x-1)*(pi/2))
and restricting x to between (and including) 1 and 2.
I realise this looks like magic (if it looks like anything at all beyond incomprehensible), and to an extent it is, but the point is that we've got a relation between every real number between 1 and 2, and every real number between 1 and infinity, that has a unique output for each input. So you can "pair off" all the numbers in each range, and so each range is the same size.
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1. List all the whole numbers (obviously this is impossible, but the point is that you could, in principle, list them: 1, 2, 3... and so on).
2. For each whole number you've listed, pick a real number between 1 and 2 at random, until every whole number has a given assigned real number.
3. Now, create a new real number not on the list in the following way: for your first listed number, add one to its first decimal digit; for your second listed number, add one to its second digit; and so on.
For example, suppose you had the following list:
1 1.0 0320667590275902984897...
2 1.6 8 492037575839828927098...
3 1.20 3 85789860827364987647...
4 1.676 7 38381974673499494499...
5 1.9996 8 6886377477110000003...
...
etc. The new number you generate by the rule I listed above would become 1.19489... and is different from every other umber on the list in at least one place, by construction. Therefore, it doesn't fit on the list, and since you've literally exhausted all countable numbers between 1 and infinity and still can find "more" real numbers between 1 and 2.
For more information, look up Cantor's diagonal argument -- the version above is a sketch, and is probably not fully convincing, but the argument is sound and proves that the real numbers are uncountable.
On the other hand, to prove that there are as many real numbers between 1 and 2 as there are between, say, 1 and infinity, then we need to find a way of taking all numbers in one set and mapping them to all numbers in the other set.
The following will do:
1. Let y = tan x, for x from -pi/2 to pi/2. This function is continuous, and it's not difficult (by using normal trigonometry) to show that tan 0 = 0, tan pi/2 "=" infinity, and tan (-pi/2) "=" -infinity (where "=" really means "tends to", rather than literally equalling).
2. Because the function is continuous, and always increasing, then for each number between -pi/2 and pi/2 there's a unique real number output between -infinity and infinity.
3. All we need to do is shift the input from 1 to 2 and the output from 1 to infinity. We can sort that by rewriting:
y = 1 + tan ((x-1)*(pi/2))
and restricting x to between (and including) 1 and 2.
I realise this looks like magic (if it looks like anything at all beyond incomprehensible), and to an extent it is, but the point is that we've got a relation between every real number between 1 and 2, and every real number between 1 and infinity, that has a unique output for each input. So you can "pair off" all the numbers in each range, and so each range is the same size.
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chrissakes only Clarabell has heard of Cantor's diagonal argument - there is a lot on You tube
You make a list of ( the unlistable) and then change the value on the diagonal ( 2nd place in the second on the list, 3rg place in the third) and generate a new member of the list !
various incredz results - an infinite stream of 1,0 s is unlistable. Hang on, you just list them as they start.....
but Cantors argument generates a new one - hence unlistable
Now IF ( big if) all proofs are listable ( clearly they are) and all the things to be proved ( theorems) are unlistable
then within 3 1/2 seconds, you have arrived at Godel's theorem
Namely - there are some true theorems that cannot be proven
well that is OK for an evenings work, innit ?
You make a list of ( the unlistable) and then change the value on the diagonal ( 2nd place in the second on the list, 3rg place in the third) and generate a new member of the list !
various incredz results - an infinite stream of 1,0 s is unlistable. Hang on, you just list them as they start.....
but Cantors argument generates a new one - hence unlistable
Now IF ( big if) all proofs are listable ( clearly they are) and all the things to be proved ( theorems) are unlistable
then within 3 1/2 seconds, you have arrived at Godel's theorem
Namely - there are some true theorems that cannot be proven
well that is OK for an evenings work, innit ?
You could perhaps still define a different way to measure set sizes, which is in a way more intuitive, by taking the difference between the two "ends" of a set. So in that sense [0,2] *would* be twice as large as [0,1]. But if by size of sets you mean how many things are inside each set, then they have the same size by using the arguments above.
