Naive Set Theory and Cantor’s Transfinite Realm

Naive Set Theory and Cantor’s Transfinite Realm

Waseda University, SILS, History of Mathematics Naive Set Theory and Cantor’s Transfinite Realm Outline

Introduction Set Theory Cardinality

The Cardinality of Transfinite Sets Some Transfinite Sets The Cardinality of Power Sets Naive Set Theory and Cantor’s Transfinite Realm Introduction Set Theory Set Theory

Set theory is now the theoretical foundation for many branches of mathematics. Although set theory was first developed by Cantor and his contemporaries, they developed what is now called “naive" set theory, relying only on our intuitive concepts of sets and their relations.

In the early 20th century, among growing concerns about various foundational issues, it was noticed that there are a number of paradoxes that can be produced by infinite sets.

Throughout the 20th century, mathematicians developed various axiomatic systems to make set theory more rigorous and to free it from these paradoxes. These are the non-naive theories that are used now. Naive Set Theory and Cantor’s Transfinite Realm Introduction Set Theory Basic Ideas of Set Theory, 1

§ A set is an undefined, primitive concept, but we may understand it as some collection of things. § If the thing x belongs to the set A, we write x P A and say x is a member of A, or x an element of A. We write the members of a set in curly braces, as N “ t1, 2, 3, ...u. § We can define sets with rules. For example, the prime numbers are

tx P N|x ą 1, and y P N, y ą 1, y ‰ x, y ffl xu.

§ A “ B if and only if every element of A belongs to B and every element of B belongs to A – the members are the same. § A is a proper subset of B, written A Ă B, if and only if x P A ùñ x P B but A ‰ B. Naive Set Theory and Cantor’s Transfinite Realm Introduction Set Theory Basic Ideas of Set Theory, 2

§ The set that has no members is called the empty set and is written { }, or H. § The power set of a given set A, written PpAq, is the set formed from all the subsets of A, including A itself and the empty set. For example, if A “ ta, b, cu, then

PpAq “ tt u, tau, tbu, tcu, ta, bu, ta, cu, tb, cu, ta, b, cuu. Naive Set Theory and Cantor’s Transfinite Realm Introduction Set Theory Basic Operations of Set Theory

§ The union of set A and set B, A Y B, is the set of all the members of set A, all the members of B, and any members that are in both sets. (The logical “or.”) § The intersection of set A and set B, A X B, is the set of all elements that are members of both A and B. (The logical “and.") Naive Set Theory and Cantor’s Transfinite Realm Introduction Cardinality Comparing Cardinality

If A and B are sets, the cardinality of A is less than or equal to s s the cardinality of B, A ő B, when we can make a one-to-one correspondence between all of the elements of A and some subset of the elements of B. (For example, we make such a correspondence between x P N and 1{x P Q.)

If A and B are sets, the cardinality of A is strictly less than the s s s s cardinality of B, A ă B, when A ő B, but there is no one-to-one mapping between all the members of B and a subset of the s s members of A, namely B ř A. (For example, we can map all x P N to, say, 1{xπ P R, but we cannot map all y P R Ñ N, because of the nondenumerability of R.) Naive Set Theory and Cantor’s Transfinite Realm Introduction Cardinality The Equality of Cardinality

Shröder-Bernstein theorem: If A and B are sets, the cardinality s s s s of A is equal to the cardinality of B, A “ B, when A ő B and s s B ő A. (The proof is fairly difficult.)

This gives a manageable way to check whether or not two sets have equal cardinality that relies on the basic concept of one-to-one correspondence. Naive Set Theory and Cantor’s Transfinite Realm The Cardinality of Transfinite Sets Some Transfinite Sets Irrationals and Reals, Planes and Lines

Irrationals and Reals: We can show that the irrational numbers, I, have the same cardinality as the real numbers, R. s s Since I Ă R, I ő R. Then we produce a mapping of all the real numbers onto specially constructed irrational numbers Rs sI (a0.a10a211a3000a41111a500000...), so that ő .

Plane and line: We can show that a plane has the same cardinality as a line. We consider a square of ordered points px, yq, where 0 ă x ă 1 and 0 ă y ă 1, and a line of points z in an open interval p0, 1q. We map the line to, say, pz, 1{2q. Then we map each point of the square, px, yq, where x “ 0.a1a2a3... and y “ 0.b1b2b3... to some point z “ 0.a1b1a2b2a3b3... of the line. Naive Set Theory and Cantor’s Transfinite Realm The Cardinality of Transfinite Sets The Cardinality of Power Sets The Cardinality of Power Sets, 1

s Ę Theorem: If A is any set, then A ă PpAq. s Ę Part 1: We first prove that A ő PpAq. For example, let A “ ta, b, c, d, ...u, then we could map

A = { a, b, c, d, e, f, g, . . . } ÙÙÙÙÙÙÙ B = { tau, tbu, tcu, tdu, teu, tf u, tgu,...} s Ę Since B is clearly a subset of PpAq, then A ő PpAq. Naive Set Theory and Cantor’s Transfinite Realm The Cardinality of Transfinite Sets The Cardinality of Power Sets The Cardinality of Power Sets, 2

Part 2: We prove that there is no one-to-one mapping between Ę all of A and all of PpAq. Proof by contradiction. We assume there is such a mapping, say

A PpAq a Ø tb, cu b Ø tdu c Ø ta, b, c, du d Ø t u ......

such that all of the elements of A are mapped to all the elements of PpAq. Then we consider the following element of PpAq: B is the set of every element in A that is not a member of the subset of PpAq to which it is mapped. Naive Set Theory and Cantor’s Transfinite Realm The Cardinality of Transfinite Sets The Cardinality of Power Sets The Cardinality of Power Sets, 3

Therefore, B Ă A and it must be that B P PpAq. Therefore, B is in the list, hence there is some y P A that is matched to B. Is y P B or y R B?

Case 1: Suppose y R B. Therefore by the definition of B, y P B, which is a contradiction.

Case 2: Suppose y P B. But B contains only elements which are not matched to it, therefore it must be that y R B, which is a contradiction. s Ę Therefore, A ő PpAq but there is no one-to-one mapping s Ę between A and PpAq, so A ă PpAq. Naive Set Theory and Cantor’s Transfinite Realm The Cardinality of Transfinite Sets The Cardinality of Power Sets The Transfinite Realm

In this way, Cantor claimed that given any set, another set of greater cardinality could always be created by taking the power set.

So that we can create a series of infinite sets with increasingly greater cardinality, s s Ę Ğ N ă R ă PpRq ă PpPpRqq...

or, ℵ ℵ ℵ 0 ă 1 ă 2 ă ...