How does one measure the size of a set, and is there a way of saying one set is bigger than another, even when they are both infinite? What "numbers" do measure size of infinite sets?
Size of A Set
Cardinal Numbers: In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set.
Ordinal Numbers: In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditary transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals.
From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are 0, 1, 2, ..., omega, omega+1, omega+2, ..., omega+omega, omega+omega+1, .... The notation of ordinal numbers can be a bit counterintuitive, e.g., even though 1+omega=omega, omega+1>omega. The cardinal number of the set of countable ordinal numbers is denoted aleph_1 (aleph-1).
This is the standard representation of ordinals. In this representation, 
Every set can be well ordered, so every set has the cardinality of a unique cardinal number. The cardinality of a set S, denoted |S|, is the number of elements in S. If the set has an infinite number of elements, then its cardinality is ∞.
Example: |{1, 7, 3}| = 3
Example: |N| = ∞
Two sets A and B have the same cardinality if there is a bijection f:A->B.
Example: The sets A={red, violet, green, yellow} and B={1, 2, 3, 4} have the same cardinality, since there is a bijection from A to B (find one).
A set A is finite (with cardinality n∈N) if there is a bijection from the set {0, 1, ..., n-1} to A, for some natural number n. A set A is infinite if it is not finite.
A set A is infinite if there exists an injection f:A->A such that f(A) is a proper subset of A. A set is finite if it is not infinite.
One Set is Bigger than One Another Even If They Are Both Infinite!
It is easy to recognize the size of finite set rather than infinite set because we can count the number of elements in finite sizes. THE SET IS COUNTABLE IF IT IS FINITE OR THERE IS A BIJECTION BETWEEN IT AND N(COUNTABLE INFINITE). For instance, if a class holds 40 people and only 35 people attend the class, thus, we know that the number of seats is larger than the number of people since 40 is greater than 35.
However, you will surprise to know that it is no different if we want to compare the size of infinite sets too! Using the same example above, we can still know that the number of seats is larger than the number of people even if we don’t know the number (how many) of them! THE SET IS UNCOUNTABLE IF IT IS NOT COUNTABLE(NEITHER FINITE NOR COUNTABLE INFINITE) So, how to compare their sizes? Here is a technique introduced by Georg Cantor, a great Mathematician, in which we should ask everyone to sit down and if there are number of seats left over, so we can conclude that there are more seats than people.
To explain this mathematically, let assume that X and Y are sets. Each element of X is matched with one and only one element of Y. This process is called as a one-to-one correspondence between sets X and Y. If we can construct the one-to-one correspondence between them, then we can say X and Y have the same size (X and Y have the same cardinality).
However, IF every trial to construct the one-to-one correspondence leaves X with elements that are not matched with elements of Y, THEN we can say that X is larger than Y. (The cardinality of X is larger than the cardinality of Y)
As a conclusion, check out this video.
Example
Take an example with the set of natural numbers N and the set of positive even numbers A, let ‘n’ is an element in N ,then we can find corresponding ‘a’ in A such that a=2n. That is, N and A are one to one and have the same cardinality. Therefore, we can say N and A have the same size.
| N |
1 |
2 |
3 |
... |
.. |
n |
... |
... |
| A |
2 |
4 |
6 |
... |
... |
2n |
... |
... |
But that may be confused, because A is a part of N. Why is the size of N and A the same? Let’s see another example.
Suppose we have a big bag and inside the big bag there are infinite small bags that marks natural number 1, 2, 3…and the first small bag has 1 bean, the second one has 2 beans, and the nth has n beans. Now we take out the first small bag and let you know we did that. Then you may be saying the size of the big bag has changed. However, if we take out the first small bag and 1 bean from each of the remaining small bags, then the second small bag has 1 bean, the third small bag has 2 beans. We remarked the small bags as 1, 2, 3…and do not tell you we did this. You will not find the difference. Thus the number of the small bags is the same.
From the example above, you may have that feeling: For the infinite set, the part can form the whole. Just like the example above, take out the first small bag and 1 bean from each remaining small bags, and remark the remaining small bags, then we get the original big bag. That hints one trait of infinite set: In a sense, infinite set is the same as a part of itself. In fact, one definition of infinite set is “An infinite set as one that can be placed into a one-to-one correspondence with a strict subset (wikipedia).” Therefore, that the size of N and A is the same is not strange; this is the trait of infinite set.
The size of some infinite set (Just my thinking, maybe not accurate):
| N(natural) | Z(integer) | Q(rational) | Q'(irrational) | R(real) | C(complex) |
| 1 lv(lowest) | 1 lv | 1 lv | 2 lv | 2 lv | 2 lv |
References:
Wikipedia. Definition of infinitive set. http://en.wikipedia.org/wiki/Cardinality
How to Measure the Size of Infinite Set?
Georg Cantor coined the infinite cardinal numbers, which help to measure the size of infinite numbers. It might be intuitive to say that the size of infinite number set is " infinite". However, it would be wrong. Lets consider the Set {1,2,3 ...} which one might say "infinite size", but there exist other set of infinite numbers whose sizes are bigger. For instance, the size of all Real numbers are bigger than the size of the Integer numbers. Therefore, there is no single "infinity" concept, but a lot of different ones corresponding to different sizes, and infinite cardinal numbers helps to measure the sizes of the sets .
( No, aleph-null) The size of the set of integers which is the same as the size of the set of odd integers, which is the same as the size of the set even integers, which is the same as the size of the {1,2,3...} natural numbers.
The reason why these sets are the same is proved by the bijection.
Definition: bijection means that any element in set A can be matched with one and only one element in set B, and vise versa. For example, the set A ={red ,green ,yellow} and B = {apple, watermelon, banana }, can be easily matched as red => apple, green => watermelon , and yellow => banana. Therefore, we can see that in previous example of (aleph-null) we can match each pairs of elements in any set with each other.
1 <=> 2
3 <=> 4
5 <=> 6
....
In fact, there are infinitely many different infinite cardinal numbers, and the notation is N0, N1, N2, N3 ....
infinite_cardinal_Toronto_math
Additional References / Sources
Bartle R. G. and Sherbert D. R. (2011). Introduction to Real Analysis (4th ed., ). Illinois: John Wiley & Sons, Inc.
http://people.umass.edu/gmhwww/382/pdf/09-infinite%20sizes.pdf
http://math.arizona.edu/~eacosta/pdfs/talks/OrdinalsPart2.pdf
http://www.cs.utexas.edu/~eberlein/cs336/cardinality.html
