Cardinality of Sets. P. Danziger

Size: px

Start display at page:

Download "Cardinality of Sets. P. Danziger"

Baldric Austin
6 years ago
Views:

1 MTH Cardinality of Sets P Danziger Cardinal vs Ordinal Numbers If we look closely at our notions of number we will see that in fact we have two different ways of conceiving of numbers The first is as an order relation (ordinality), the second is as method of counting (cardinality) When we count a set of n objects we assign to each object an integer in sequence, starting from one So each object is assigned a unique number from to n Such an assignment is called a one to one correspondence (or in mathematical parlance a bijection) Definition A one to one correspondence, or bijection is a function, f, which maps a set S to a set T (we write f : S T ) in such a way that Each element of S is mapped to a unique element of T Every element of T is the image of some element of S In a similar way we use such bijections to measure distance, by putting space in one to one correspondence with the real numbers Each distance corresponds to exactly one unique real number Of course we are ignoring any imprecisions of the physical act of measurement, in our ideal mathematical world this is the way things would be This kind of view of number is known as cardinality The other notion of number is ordinality, in this notion we are only concerned with statements like x is bigger than y This is actually the notion of number seen in the discussion of limits Nowhere do we consider actual magnitudes, only relations of magnitudes While this distinction may seem pedantic it has two important ramifications The first is in the field of measurement, If we are not able to give quantitative measure to an object of study we can still get some information from an ordinal measure, this is often the case in the social sciences Thus we may say John is more emotional than Mary, though we may not quantify by how much, since we have no scale of measurement The second ramification is in the notion of infinity With an ordinal representation of infinity we may only talk about infinity representing an unbounded quantity, as in the discussion of limits The ordinal interpretation of infinity has a safety valve for dealing with the infinite The infinities encountered when considering limits are potential infinities, meaning that they represent potentially unbounded quantities A function never actually equals infinity, it merely gets larger and larger At the point where it would become infinite it actually becomes undefined Thus calculus avoids dealing directly with infinity On the other hand with a cardinal viewpoint we may consider a bijection between two infinite sets, the existence of such a bijection would imply that they have the same size Thus we may

2 MTH Cardinality of Sets P Danziger consider sets with infinite cardinality, for example the set of natural numbers, or the set of real numbers between 0 and This gives a much finer representation of the notion of infinity than an ordinal representation The notion of an actual infinity has always been contentious Definition We say that two sets X and Y have the same cardinality if there exists a bijection f : X Y If Y = {,,, n}, for some fixed n Z +, and there exists a bijection f : X Y, then we say that X is of size n, and write X = n Note that there exists a bijection f : X Y if and only if there exists a bijection g : Y X (g = f ) What happens if we extend the first part of this definition to infinite (unbounded) sets? Cantor and Countable Infinity The first person to really consider the notion of infinity in this way was Georg Cantor (845-98) Cantor was born in St Petersburg, Russia, but moved to Berlin when he was Cantor studied mathematics at Zurich, and ended up teaching at Halle university in Germany Cantor never gained the recognition he felt he deserved, and during his life his ideas were ridiculed by the mathematical community This effected him deeply and he spent much of his life in and out of mental institutions, suffering repeated breakdowns However by the time of his death in 98 his ideas were becoming widely accepted and his genius started to be recognised Cantor s basic idea was to use cardinality as a definition of infinity Thus we may talk about the number of natural numbers, or the number of rational numbers and so on If two infinite sets can be put into one to one correspondence with each other, then they represent the same infinity, just as any two sets which can be put into one to one correspondence with each other represent the same number Definition A set X is said to be countably infinite, or countable, if it has the same cardinality as Z + Thus a set is countable if and only if there is a bijection from it to Z + Note that compositions of bijections are bijections So once we know a set, X, is countable finding a bijection from another set, Y to X shows that Y is countable as well

3 MTH Cardinality of Sets P Danziger Even Numbers As an illustration of Cantorean arguments, consider the set of positive even numbers, E = {, 4, 6, 8, } Consider f : Z E, f(n) = n f is a bijection (Exercise) Natural Numbers Even Numbers Thus the conclusion is that the number of natural numbers has the same cardinality as the positive even numbers, very strange Thus the positive even numbers are countable This leads to strange phenomena, such as Hilbert s hotel: Hilbert s hotel is a hotel with an infinite number of rooms, numbered,, 3, One day the hotel is full, with an infinite number of guests That day an infinite number of buses arrive carrying an infinite number of new guests No problem, says the manager, we can accommodate you all How does the manager accommodate the new guests? For each of the current guests the manager moves the person in room i to room i, thus freeing up an infinite number of rooms for the new guests Z What about the number of integers? { n Consider the function, f : Z + Z, f(n) = n The output of this function goes as follows: if n is odd if n is even n f(n) 0 f is a bijection (Exercise) Thus the integers have the same cardinality as the positive integers, they are countable 3 Q What about the rational numbers, Q? In order to count the rational numbers we create an infinite table on which to count, the denominator increases as we move to the right, and the numerator increases as we move down (see figure ) This enumerates all possible values of numerator and denominator 3

4 MTH Cardinality of Sets P Danziger Figure : Cantor s Counting of the Rationals We now count diagonally starting at the top left, starting at the top left Every time we reach the top we move right, and every time we reach the left hand side we move down This assigns a unique natural number to each rational number n f(n) 3 3 Thus the rational numbers are countable, ie they have the same cardinality as the natural numbers 4 [0, ] We have seen that the natural numbers, the even numbers, the integers and the rational numbers are all countable, ie are the same size of infinity What about the real numbers R? We can simplify things by only considering those real numbers between 0 and, there are certainly an infinite number of them We claim that this infinity is different than the previous ones, that is the set of real numbers between 0 and is not countable This means that there is no one to one correspondence between this set and the set of natural numbers We are thus faced with the problem of showing that something does not exist, always much harder than showing that something does The answer is Cantor s diagonalisation proof This method of proof has been adapted to many circumstances, including the proof that the 4

5 MTH Cardinality of Sets P Danziger problem of deciding whether a machine (computer) will eventually halt on any input is insoluble (the halting problem) Theorem There is no bijection f : Z + [0, ] Proof: (By contradiction) Suppose not, that is suppose that there is a bijection f : Z + [0, ] That is, to each real number between 0 and we can assign a unique positive integer If this were so we could take an infinitely large sheet of paper and write out the positive integers, n corresponding to the real numbers x An example of how part of such a table might look is shown below n x Now we will create a real number between 0 and which cannot be on the table On the ith row we identify the ith digit after the decimal point In the example above this gives the digits, 5, 3, 4, 5, 9, We can consider this as a number between 0 and, namely Now, for each identified digit we change it to something else, for example we might add, changing 9 to 0 (add mod 0) This gives a new number, in this case This number, which lies between 0 and, cannot be on the list The reason is that it differs from the first number on the list in the first position after the decimal point, from the second number on the list in the second position after the decimal point, from the third number on the list in the third position after the decimal point, and so on In general it differs from the ith number on the list in the ith position after the decimal point, since we started with that digit and then changed it Thus we have created a number between 0 and which does not have a natural number assigned to it, if it did it would be on the list 3 The Continuum and Beyond This shows that the number of real numbers between 0 and is not countable, this is a different infinity! This infinity is called the continuum While it was surprising that seemingly different 5

6 MTH Cardinality of Sets P Danziger sized sets have the same infinite number of elements, once we have accepted this fact, it is even more surprising that there should be more than one type of infinity It should be noted that it can be shown that the set of real numbers has the same cardinality as the set of real numbers from 0 to, thus the set of real numbers, R, also has the size of the continuum This naturally raises the question of whether there are more infinities, indeed there are, a countably infinite number of them! For example, consider F = {f f : R R}, the set of functions from R to R It can be shown that F is neither countable, nor the continuum It is evident from the proof above that the size of the continuum is in some sense larger than countable, this raises the question of whether there are any infinities in between these two? Nobody has been able to find one, but nobody has been able to show that one does not exist In fact in 963 Paul Cohen (934 - ) showed that this question is undecidable, and so when dealing with transfinite numbers it is necessary to take as an axiom that there is no infinity between the countable and the continuum, this is known as the continuum hypothesis Of course we can equally well make the opposite assumption that there is such an infinity, each gives rise to a different mathematical system 6

Cantor and Infinite Sets

Cantor and Infinite Sets Galileo and the Infinite There are many whole numbers that are not perfect squares: 2, 3, 5, 6, 7, 8, 10, 11, and so it would seem that all numbers, including both squares and