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The Mathematics of Infinity. Georg Cantor’s Theory of Sets. To see the world in a grain of sand. And heaven in a wildflower: Hold infinity in the palm of your hand, And eternity in an hour . --William Blake.
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The Mathematics of Infinity Georg Cantor’s Theory of Sets To see the world in a grain of sand. And heaven in a wildflower: Hold infinity in the palm of your hand, And eternity in an hour. --William Blake
“From the paradise created for us by Cantor; no one will drive us out.”David Hilbert Georg Cantor
An Introduction to Sets • A = {2, 3, 5, 7, 11, 13, 17, 19} • N = {1, 2, 3, 4, … }N is the set of Natural numbers or the set of counting numbers • Z = { … -3, -2, -1, 0, 1, 2, 3, … }Z is the set of Integers • Q = { a/b | a, b Z, b 0 }Q is the set of Rational Numbers • A N, N Z, and Z Q
Equal and Equivalent Sets • Two sets are Equal if they contain exactly the same elements.A = {2,4,6}, B = {4,2,6}; so A = B • Two sets are Equivalent if there exists a 1-1 correspondence between their elements • A and B above are certainly equivalent, but C = {1,2,3} is also equivalent to each of them.
What does Infinity mean? The known is finite, the unknown infinite; intellectually we stand on an island in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land. --Thomas H Huxley
Infinite Sets • A set A is finite if it is empty or if there is a natural number n such that A is equivalent to {1,2,3, … ,n} • Thus a set is infinite if its elements cannot be counted completely. • An alternate but equivalent definition:A set is infinite if it is equivalent to a proper subset of itself.
Question Are all infinite sets equivalent? Or, in other words, can two infinite sets always be put in 1-1 correspondence? Consider the Natural numbers and the even positive integers: A 1-1 correspondence is shown so these are equivalent sets. 1, 2, 3, 4, … , n, ... 2, 4, 6, 8, … , 2n, ...
N is equivalent to Z 1, 2, 3, 4, 5, 6, 7, 8, 9, … Here is a 1-1 correspondence between the Natural numbers and the Integers. 0, 1, -1, 2, -2, 3, -3, 4, -4, … Every even Natural number, n, is paired with n/2 Every odd Natural number, n, is paired with (1 – n)/2. It is even possible to show a 1-1 correspondence between the Natural numbers and Q, the rational numbers. So N is equivalent to Q as well.
More about Rational Numbers • By definition, any number which may be expressed as a fraction is a Rational number. • It may also be shown that any decimal which terminates or repeats may be made into a fraction and is thus a Rational number.
The Irrationals and the Reals • The Irrational numbers comprise all decimal numbers which are not rational, i.e., cannot be made into fractions. • The Irrational numbers together with the Rational numbers include all possible decimals and form the Real numbers R. • The Real numbers may be shown to be in 1-1 correspondence with the points on a number line. Because of this they are sometimes called the Continuum.
An Answer and More Questions: • N is not equivalent to R. • Proof by contradiction:Assume N is equivalent to R. Then there must be a 1-1 correspondence between them. • This correspondence will pair each Natural number with a Real number. • We will show that this assumption leads to a contradiction by constructing a Real number which has not been included in the correspondence.
A possible correspondence: N [0,1] 1 0 . 3 0 1 2 5 9 4 … 2 0 . 1 6 6 5 2 1 8 … 3 0 . 4 1 1 2 1 0 7 … 4 0 . 2 0 5 0 9 6 3 … . . . . . .
Constructing a new number(not in the correspondence) N [0,1] 1 0 .(3)0 1 2 5 9 4 … 2 0 . 1(6)6 5 2 1 8 … 3 0 . 4 1(1)2 1 0 7 … 4 0 . 2 0 5(0)9 6 3 … . . . . . . Construct a number w which will consist of digits di where the ith digit will be either a 1 (if the ith digit in row i is not a 1) or a 2 (if the ith digit in row i is a 1)
Constructing a new number(not in the correspondence) N [0,1] 1 0 .(3)0 1 2 5 9 4 … 2 0 . 1(6)6 5 2 1 8 … 3 0 . 4 1(1)2 1 0 7 … 4 0 . 2 0 5(0)9 6 3 … . . . . . . So, for this correspondence, the number w, would start: w = 0.1121......
A Contradiction!! So w is different from every Real number in the original correspondence. Since w is not part of the proposed correspondence or any other possible correspondence, no correspondence is possible. Thus, N is not equivalent to R. This means that there are at least two sizes or classes of infinite sets.
So now, we have at least two sizes of infinity: • Sets equivalent to N, and • Sets equivalent to R. • We say that sets equivalent to N have cardinality 0 (read aleph null). • The collection of all sets equivalent to R are said to have cardinality c where c stands for the cardinality of the continuum. • Are there others?
Sizes of Infinity Just as the cardinal number 3 is used to describe the size of any set equivalent to {a, b, c}, 0 is used to describe the size of any infinite set equivalent to N. And c is used to describe the size of any infinite set equivalent to R. If there are other sizes of infinite sets, then we need other numbers to describe the cardinality of these sets.
I could be bounded in a nutshell, and count myself a king of infinite space. --Shakespeare It is also reasonable to assume that if these sizes are truly different, there must be some sort of ordering of them. e.g. 0 < c. If we use the symbols 1, 2, … to represent the cardinality of these sets we might assume that 0 < 1 < 2 < …
When is a set ‘bigger’ than another? With finite (countable) sets, it is easy to determine if the cardinality of one set is larger than the cardinality of another. For infinite sets we will use the definition:If A and B are sets and |A| and |B| represent the cardinality of sets A and B respectively, then we say |A| > |B| if:1. there exists a 1-1 correspondence between all of set B and a proper subset of A. and2. there does not exist a 1-1 correspondence between B and all of A.
Two Questions • Are there a finite number(perhaps only the two we have seen) or an infinite number of these sizes of infinity? • Where does c, the cardinality of R fit into this scheme? We know it is larger than 0, but is it larger than 1, etc.?
Power Sets The Power Set of set A, denoted by P(A) is the set of all the subsets of A. i.e. P(A) = { X | X A } It is fairly obvious that for all finite sets A, | P(A) | > | A |. For example: If A = {p,q,r} then the Power Set of A, P(A) = { {p},{q},{r},{p,q},{p,r},{q,r},{p,q,r}, } and | P(A) | = 8 > | A | = 3.
Cantor’s Theorem Cantor proved that for infinite sets, also, the set of all subsets of a set is ‘larger’ than the original set. Let A be an infinite set with cardinality | A | Let S be the set of all the subsets of set A. Cantor showed that | A | < | S |. “I can see it, but I don’t believe it.” Georg Cantor
Let A be any set with cardinality |A|. Let S be the set of all subsets of A and have cardinality |S|. • To prove that |A| < |S| we must show that: • A can be put in 1-1 correspondence with a proper subset of S. • A cannot be put in 1-1 correspondence with all of S. • The first step is easy since A can be put in a 1-1 correspondence with all the single element subsets of A. i.e. if x A then x {x} S.
Now assume that there is a 1-1 correspondence between A and S. i.e. every element of A is paired with a subset of A found in S. Now with this matching, it makes sense to ask if, for each pairing, the element from A is contained in the subset from S with which it is matched. If this is not the case then put all such elements of A in a set and call it W. Clearly W is a subset of A. So W must be paired with some element, say z of A. Now either z is in W or it is not. Let’s look at both possibilities.
In an Indirect Proof, we assume the negation of what we wish to prove and look for a contradiction. If such is found, it means the assumption was wrong. If z is in W, then z is contained within the set with which it is matched and by the definition of W cannot be in W, which is a contradiction. If z is not in W, then it is not contained within the set with which it is matched and again by the definition of W, z must be contained in W. This, also gives a contradiction. Thus we are forced to conclude that no 1-1 correspondence was possible between A and S.
There is no smallest among the small and no largest among the large; but always something still smaller and something still larger. --Anaxagoras So we have answered one of the questions. This result means that there are an infinite number of sizes of infinity.Now for the second question. Where does c, the cardinality of the Real numbers fit into this limitless progression of larger and larger infinite sets? 0 , 1 , 2 , ... Cantor showed that c is the size of the set of all the subsets of the Natural numbers and he guessed that c = 1 , i.e. that c was the next smallest size.
The Continuum Hypothesis But Cantor was never able to prove that there was no size of infinity between 0 and c. Thus was born the Continuum Hypothesis:If a set has size 0 ,then the set of all its subsets has size 1 .And more generally, if a set has size n , then the set of all its subsets has size n+1. Many tried to prove this hypothesis, but none succeeded.
Contradictions?? In 1940, Kurt Gödel proved that the Continuum Hypothesis was consistent with the axioms of set theory, i.e. that it will not lead to any contradictions. In 1963, Paul Cohen proved that assuming the Continuum Hypothesis false will not lead to any contradictions within set theory.
A new Axiom This means the Continuum Hypothesis can neither be proven true or false within set theory. So we may treat it as a separate axiom. Thus, much like Euclid’s 5th postulate, we get one kind of set theory if we assume it true and another if we assume it false.
