The concept of infinity (∞) is often misinterpreted as representing a very big number like 10exp10exp52 (10 raised to the power of 10 raised to the power of 52). Infinity isn’t a number at all, but a concept to describe an ‘unbounded’ set of things. Such unbounded sets don’t behave at all like finite sets, regardless of how big.
For example if I take the set of non-negative integers from 0 to 100 {0, 1, 2, 3, ..., 100} and then get rid of all the odd integers {1, 3, 5, 7, ..., 99} it’s pretty clear that the set of even integers I’m left with {0, 2, 4, 6, ..., 100} is about half the size of the set I started with. But what if I start with the infinite set of integers, called the set of natural numbers, which is unbounded, and get rid of all the odd integers in this set? Even though both sets are infinite, is the remaining ‘infinite’ set around half the size of the ‘infinite’ set I started with?
Set of all natural numbers: {0, 1, 2, 3, 4, 5, ...}
Set of all even naturals: {0, 2, 4, 6, 8, 10, ...}
Our common sense seems to suggest that the second set is smaller than the first, because our common sense is trained to deal with finite things, not unbounded sets. The fact is, there is a mathematical way to check the ‘size’ of a set. If I can make a one-to-one mapping from an element of the first set to one and only one element of the second set, then clearly the two sets have the same number of elements. If, on the other hand, I end up with something ‘left over’ in either set, the two sets cannot be the same ‘size’. I can make just such a mapping from the set of all integers to the set of all even integers as follows: take an element of the set of integers, multiply it by 2, to obtain one, and only one, member of the set of even integers:
2 x2 4I can take any arbitrary element of the first set, say 2345, and obtain its corresponding match in the second set, 4690, by multiplying by two. Similarly, I can take any element in the second set, say 12386, and easily obtain its unique match in the first set by dividing by two, 6193. Such a 1 to 1 mapping of elements in the first set to all members in the second leaves nothing ‘left over’ in either set. So for each element of the set “all natural numbers” there is one and only one, matching element of the set “all even integers”. So the sets are the same ‘size’ even though we obtained the second set by getting rid of every other element of the first. Strange yes?
3 x2 6
4 x2 8
5 x2 10
Etc.
It can also be shown that the set of all rational numbers, that is, all numbers that can be expressed as a fraction (a ratio of two integers) can also be mapped one to one to the set of natural numbers. So if we add in all the fractions to the set of integers, the two sets are still of the same ‘size’. Similarly if we add in all the negative rational numbers, the set can be mapped one to one to the set of all positive rationals. The mappings are more complex and I won’t go into them here, but they do exist and can be found in any book of set number theory. In fact, all of these sets can be mapped to the set of natural numbers and are said to have the same cardinality instead of ‘size’. Infinite sets having cardinality (size) the same as the natural numbers are symbolized by Aleph0 (the Hebrew character).
So are all infinite sets the same ‘size’ in this manner? After our experience so far we might think so, but the answer is surprisingly, ‘no’. The sets we have been looking at, although they are infinite are still ‘denumerable’. In other words, one can find rules to list out the elements of these sets in an ordered array without leaving anything out, and then find rules for mapping the elements of the set to the elements of the natural numbers in a one-to-one way. In other words, the sets can be counted. But the set of Real numbers (rational numbers plus non-repeating decimals like √2) is NOT denumerable in this way. To see this, lets assume you make a list of all the Real numbers:
...901234.8767894847...Of course, any element of your set can have any number of digits to the left or right of the decimal, but we can always use leading zeros so the decimal points line up in this way. This is what we did for the third number in the list above. I just limited the number of digits shown before the decimal to seven and after the decimal to ten to make this illustration easier to put on screen (I left my infinite screen at work).
...901234.2937640974...
...901234.5564737222...
...
...000002.8878903489...
Etc.
So let’s assume that you tell me that every single real number is on your list of numbers, not a single one is left out. Assuming you had unlimited room for such a list on your paper and an unlimited amount of time to make it, is your list complete? No. Because I can come up with a rule to make a real number not on your list, no matter how big it is! How? Well I simply choose the nth digit of my number to be different than the nth digit of the nth number on your list. Let’s just look at digits after the decimal. First, I make the first digit different from the ‘8’ in your first number so my number can’t match the first number in your list. I’ll pick ‘2’. Then I make the second digit after the decimal ‘3’ so it doesn’t match the second number in your list which has a ‘9’ in that position. I choose the third digit in my number different from the third digit in your third number, and so on ad infinitum.
Your List Mine ExplanationIn this way I can make a number that doesn’t match any of the numbers on your list, no matter how long your list is. So no ‘list’ of real numbers can contain all the real numbers. Mathematicians say the set of real numbers is not denumerable. It is called the ‘continuum’ set because it represents the set of points on a continuous line. The continuum is an infinite set of higher cardinality than the set of natural numbers. There are many sets that map one-to-one to the continuum set and they are all represented as having cardinality represented by C.
.8767894847... ... .2... It won't match your 1st number
.2937640974... ... .23... It won't match your 2nd number
.5564737222... ... .231... It won't match your 3rd number
Etc.
There is at least one more set of cardinality even higher than the continuum, written 2C, but it involves higher concepts in mathematical theory than I want to cover here. Theoretically there is no limit to the number of different cardinalities and as far as I know and have been able to discover from my research, there is no known ‘maximum’ cardinality. Sets of larger cardinality than 2C seem to exist mostly in symbolic mathematics as “sets of subsets”, without necessarily corresponding to things more “concrete” like the set of natural or real numbers. Whether such infinities really exist as more than concepts, in other words, whether they correspond to real things ‘out there’ in the universe, is an open question.
Infinite numbers are interesting things. I can counter-intuitively take a subset of an infinite set that doesn’t include certain elements of the set, yet still end up with a set the same size I started with. Even so, not all ‘infinities’ are created equal and there are different sizes of ‘infinity’.
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