I always wanted to know if I could compare infinity and in the first year Advanced Mathematics, I learnt that since infinity is a not a number therefore it does not follow any axioms and so is not ordered e.g. you can not say $\infty^{2}>\infty$ but I just came across This which concluded you can compare infinity and so I researched about it and here there are the two answers that I have so far:
------------------------------------------------------------------------------------
By Dr. Rod Yager:
Basically, there is more than one infinity - and some are bigger than
others.
To put it simply, two sets have the same size if you can find a one-to-
one and onto function between them. [That is, find a way of pairing
off the elements, so each of the elements in one set is matched to
precisely one element in the other set].
This works just as well with infinite sets as with finite sets. But it
does give surprising results. For instance, the number of fractions is
exactly the same as the number of natural numbers.
You can match them up as follows:
1 <---> -1
2 <---> 0
3 <---> 1
4 <--> -2
5 <--> -3/2
6 <--> -1/2
7 <--> 1/2
8 <--> 3/2
9 <--> 2
10 <--> -3
11 <--> -8/3
12 <--> -7/3
13 <--> -5/3
14 <--> -4/3
15 <--> -2/3
16 <--> -1/3
17 <--> 1/3
etc.
[Here, the fractions are listed by first listing those between -1 and
1 with denominator <=1, then those between -2 and 2 with denominator
<=2, then between -3 and 3 with denominator <=3, leaving out anything
which has occurred earlier. Clearly, each fraction will occur exactly
once in this list.]
Notice this is surprising. There are an infinite number of fractions
between 0 and 1, and infinite number between 1 and 2 etc, but here
\inf * \inf is just the same as the number of natural numbers (which
is the smallest possible infinity).
However, there are bigger infinities. For instance, the number of real
numbers is greater than the number of natural numbers.
And for any set S, the number of subsets of S is always greater than
the number of elements in S. [So if you start with an infinite set,
repeating this over and over lets you build bigger and bigger
inifinities].
One of the unanswered questions in mathematics is whether there is an
infinity BETWEEN the number of natural numbers and the number of real
numbers. No-one knows yet - and some suspect that this is a question
that actually can't be answered.
---------------------------------------------------------------------------------------------------
By Dr. Gerry Myerson:
If you have two finite sets and want to know which one has more elements
then you can pair off elements in one with elements in the other
and if there are elements left over in one set when you've exhausted
the other
then the set with elements left over has more elements.
For example, if A = {North, East, South, West}
and B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
and you pair off
North <-> Sunday, East <-> Monday, South <-> Tuesday, West <->
Wednesday,
then you've exhausted A and still have things left over in B,
so B has more elements.
In the late 1800s a mathematician named Cantor showed that we can do
pretty much the same thing with infinite sets. If there is a way
to pair off two sets so that nothing is left over in either set,
then we say the two sets have the same cardinality
(a fancy word for "number of elements").
If no matter how we try to pair up the two sets
the first one always has elements left over,
we say the first set has greater cardinality than the second.
So, for example, the odd numbers {1, 3, 5, 7, ...}
and the even numbers {2, 4, 6, 8, ...} have the same cardinality
because we can pair each odd number n with the even number n + 1
(1 <-> 2, 3 <-> 4, 5 <-> 6, etc.)
and nothing is left over in either set.
But the even numbers also have the same cardinality
as the set of all positive integers {1, 2, 3, 4, ...}
because we can pair each positive integer n with the even number 2n
(1 <-> 2, 2 <-> 4, 3 <-> 6, etc)
and nothing is left over in either set.
If we write N for {1, 2, 3, 4, ...}
and N^2 for the set of all pairs of positive integers
(that is, N^2 is the set whose elements are all the ordered pairs (a, b)
where a and b are positive integers)
then we can list the elements of N^2 as follows:
(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), (1, 4), ....
that is, in increasing order of a + b, and, among pairs that have
the same value of a + b, list them in increasing order of a
(so (3, 13) will come after (5, 10) because 3 + 13 > 5 + 10,
but (3, 13) will come before (5, 11) because 3 + 13 = 5 + 11 and 3 < 5).
Now we can pair each element in our list of N^2
with the corresponding element in the list of N:
(1, 1) <-> 1, (1, 2) <-> 2, (2, 1) <-> 3, (1, 3) <-> 4, (2, 2) <-> 5,
etc.,
and nothing will be left out in either set,
so N has the same cardinality as N^2.
A similar argument can be used to show that if Q is the set of
fractions,
that is, Q is everything of the form a / b where a and b are positive
integers
and a / b is in lowest terms,
then the cardinality of Q is the same as that of N.
Now let I be the set of decimal numbers between 0 and 1.
The argument on the website shows that no matter how you try to pair off
the elements of I with the elements of N = {1, 2, 3, 4, ...}
there will always be elements of I left over.
That proves that the cardinality of I is greater than the cardinality
of N.
Summary:
When we say "infinity is not a number," we mean it is not a decimal
number
like 3 or 2/7 or square root of 2 or pi, and it doesn't satisfy the
axioms
the decimal numbers satisfy.
Despite that, we can compare two infinite sets and say which, if
either, is bigger.
---------------------------------------------------------------------------------------------------
So we actually can not compare infinities, what we compare is actually the cardinality.
------------------------------------------------------------------------------------
By Dr. Rod Yager:
Basically, there is more than one infinity - and some are bigger than
others.
To put it simply, two sets have the same size if you can find a one-to-
one and onto function between them. [That is, find a way of pairing
off the elements, so each of the elements in one set is matched to
precisely one element in the other set].
This works just as well with infinite sets as with finite sets. But it
does give surprising results. For instance, the number of fractions is
exactly the same as the number of natural numbers.
You can match them up as follows:
1 <---> -1
2 <---> 0
3 <---> 1
4 <--> -2
5 <--> -3/2
6 <--> -1/2
7 <--> 1/2
8 <--> 3/2
9 <--> 2
10 <--> -3
11 <--> -8/3
12 <--> -7/3
13 <--> -5/3
14 <--> -4/3
15 <--> -2/3
16 <--> -1/3
17 <--> 1/3
etc.
[Here, the fractions are listed by first listing those between -1 and
1 with denominator <=1, then those between -2 and 2 with denominator
<=2, then between -3 and 3 with denominator <=3, leaving out anything
which has occurred earlier. Clearly, each fraction will occur exactly
once in this list.]
Notice this is surprising. There are an infinite number of fractions
between 0 and 1, and infinite number between 1 and 2 etc, but here
\inf * \inf is just the same as the number of natural numbers (which
is the smallest possible infinity).
However, there are bigger infinities. For instance, the number of real
numbers is greater than the number of natural numbers.
And for any set S, the number of subsets of S is always greater than
the number of elements in S. [So if you start with an infinite set,
repeating this over and over lets you build bigger and bigger
inifinities].
One of the unanswered questions in mathematics is whether there is an
infinity BETWEEN the number of natural numbers and the number of real
numbers. No-one knows yet - and some suspect that this is a question
that actually can't be answered.
---------------------------------------------------------------------------------------------------
By Dr. Gerry Myerson:
If you have two finite sets and want to know which one has more elements
then you can pair off elements in one with elements in the other
and if there are elements left over in one set when you've exhausted
the other
then the set with elements left over has more elements.
For example, if A = {North, East, South, West}
and B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
and you pair off
North <-> Sunday, East <-> Monday, South <-> Tuesday, West <->
Wednesday,
then you've exhausted A and still have things left over in B,
so B has more elements.
In the late 1800s a mathematician named Cantor showed that we can do
pretty much the same thing with infinite sets. If there is a way
to pair off two sets so that nothing is left over in either set,
then we say the two sets have the same cardinality
(a fancy word for "number of elements").
If no matter how we try to pair up the two sets
the first one always has elements left over,
we say the first set has greater cardinality than the second.
So, for example, the odd numbers {1, 3, 5, 7, ...}
and the even numbers {2, 4, 6, 8, ...} have the same cardinality
because we can pair each odd number n with the even number n + 1
(1 <-> 2, 3 <-> 4, 5 <-> 6, etc.)
and nothing is left over in either set.
But the even numbers also have the same cardinality
as the set of all positive integers {1, 2, 3, 4, ...}
because we can pair each positive integer n with the even number 2n
(1 <-> 2, 2 <-> 4, 3 <-> 6, etc)
and nothing is left over in either set.
If we write N for {1, 2, 3, 4, ...}
and N^2 for the set of all pairs of positive integers
(that is, N^2 is the set whose elements are all the ordered pairs (a, b)
where a and b are positive integers)
then we can list the elements of N^2 as follows:
(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), (1, 4), ....
that is, in increasing order of a + b, and, among pairs that have
the same value of a + b, list them in increasing order of a
(so (3, 13) will come after (5, 10) because 3 + 13 > 5 + 10,
but (3, 13) will come before (5, 11) because 3 + 13 = 5 + 11 and 3 < 5).
Now we can pair each element in our list of N^2
with the corresponding element in the list of N:
(1, 1) <-> 1, (1, 2) <-> 2, (2, 1) <-> 3, (1, 3) <-> 4, (2, 2) <-> 5,
etc.,
and nothing will be left out in either set,
so N has the same cardinality as N^2.
A similar argument can be used to show that if Q is the set of
fractions,
that is, Q is everything of the form a / b where a and b are positive
integers
and a / b is in lowest terms,
then the cardinality of Q is the same as that of N.
Now let I be the set of decimal numbers between 0 and 1.
The argument on the website shows that no matter how you try to pair off
the elements of I with the elements of N = {1, 2, 3, 4, ...}
there will always be elements of I left over.
That proves that the cardinality of I is greater than the cardinality
of N.
Summary:
When we say "infinity is not a number," we mean it is not a decimal
number
like 3 or 2/7 or square root of 2 or pi, and it doesn't satisfy the
axioms
the decimal numbers satisfy.
Despite that, we can compare two infinite sets and say which, if
either, is bigger.
---------------------------------------------------------------------------------------------------
So we actually can not compare infinities, what we compare is actually the cardinality.
