Salem State College MAT 420 Set Theory L. Pedro Poitevin October 25, 2007 1 Foundations De¯nition (Informal). We de¯ne a set A to be any unordered collection of objects, e.g., f3; 8; 5; 1g is a set. If x is an object, we say that x is an element of A or x 2 A if x lies in the collection; otherwise we say that x is not an element of A and write x2 = A. For instance, 3 2 f1; 2; 3; 4; 5g but 7 2= f1; 2; 3; 4; 5g. Axiom 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is an element of B. De¯nition (Equality of sets). Two sets A and B are equal|in notation A = B|if and only if every element of A is an element of B and vice versa. Axiom 2 (Empty set). There exists a set ;, known as the empty set, which contains no elements, i.e. for every object x we have x2 = ;. Exercise. Prove that there is only one empty set. Lemma 1 (Single choice). Let A be a nonempty set. Then there exists an object x such that x 2 A. Axiom 3 (Singletons). If a is an object, then there exists a set fag whose only element is a. Axiom 4 (Pairwise union). Given any two sets A and B, there exists a set A [ B called the union of A and B, whose elements are all objects which are elements of A or B. In symbols, x 2 A [ B () x 2 A _ x 2 B. Remark. This axiom allows us to de¯ne pair sets, triplet sets, quadruplet sets, and so forth. If a; b are two objects, we de¯ne fa; bg := fag [ fbg; if a; b; c are three objects, then fa; b; cg = fag [ fbg [ fcg, and so on. Lemma 2. The union operation above described is commutative and associative. Remark. Because of the associativity of the union operation, we do not need to use parentheses to denote multiple unions; thus, for instance, we can write A[B [C instead of (A[B)[C or A[(B [C). Similarly for unions of four sets A [ B [ C [ D, etc. De¯nition (Subsets). Let A and B be sets. We say that A is a subset of B and write A ⊆ B if every element of A is an element of B. We say that A is a proper subset of B and write A ½ B if A ⊆ B and A 6= B. Proposition 1 (Sets are partially ordered by set inclusion). Let A; B; C be sets. The following three conditions are satis¯ed: 1. If A ⊆ B and B ⊆ A, then A = B. 2. If A ⊆ B and B ⊆ C, then A ⊆ C. 3. If A ½ B and B ½ C, then A ½ C. Axiom 5 (Speci¯cation). Let A be a set, and for each x 2 A, let P (x) be a property pertaining to x (i.e., P (x) is either a true statement or a false statement). Then there exists a set fx 2 A : P (x) is trueg (or simply fx 2 A : P (x)g for short), whose elements are precisely the elements x in A for which P (x) is true. In symbols, for any object y, y 2 fx 2 A : P (x)g () y 2 A ^ P (y) is true. De¯nition (Pairwise intersection). The intersection S1 \ S2 of two sets S1;S2 is de¯ned to be the set S1 \ S2 := fx 2 S1 : x 2 S2g. In other words, x 2 S1 \ S2 () x 2 S1 ^ x 2 S2. Remark. One should be careful with the English word \and": rather confusingly, it can mean either union or intersection, depending on the context. For instance, if one talks about a set of \boys and girls," one means the union of a set of boys and a set of girls, but if one talks about the set of people who are single and male, then one means the intersection of the set of single people with the set of male people. (Can you work out the rule of grammar that determines when \and" means union and when \and" means intersection?) Another problem is that \and" is used in English to denote addition. So, for instance, one could say that \2 and 3 is 5," while also saying that the elements of f2g and the elements of f3g form the set f2; 3g" and \the elements of f2g and f3g form the set ;." This can certainly get confusing! One reason to use mathematical symbols instead of English words such as \and" is that mathematical symbols almost always have a precise and unambiguous meaning, whereas one must often look very carefully at the context in order to work out what an English word means. De¯nition. Two sets A and B are said to be disjoint if A \ B = ;. Remark. Note that being disjoint is not the same as being distinct. If A and B are disjoint, then they are possibly distinct (¯nd an exception!) but if they are distinct, they may not be disjoint. De¯nition (Set di®erence). Given two sets A and B, we de¯ne the set A n B to be the set fx 2 A : x2 = Bg. Proposition 2 (Sets form a Boolean algebra). Let A; B; C be sets, and let X be a set containing A; B; C as subsets. (a) A [; = A and A \; = ;. (b) A [ X = X and A \ X = A. (c) A \ A = A and A [ A = A. (d) A [ B = B [ A and A \ B = B \ A. (e) (A [ B) [ C = A [ (B [ C) and (A \ B) \ C = A \ (B \ C). (f) A \ (B [ C) = (A \ B) [ (A \ C) and A [ (B \ C) = (A [ B) \ (A [ C) (g) A [ (X n A) = X and A \ (X n A) = ;. (h) X n (A [ B) = (X n A) \ (X n B) and X n (A \ B) = (X n A) [ (X n B). Axiom 6 (Replacement). Let A be a set. For any object x 2 A and any object y, suppose we have a statement P (x; y) pertaining to x and y, such that for each x 2 A there is at most one y for which P (x; y) is true. Then there exists a set fy : P (x; y) is true for some x 2 Ag, such that for any object z, z 2 fy :P (x; y) is true for some x 2 Ag () P (x; z) is true for some x 2 A. Axiom 7 (In¯nity). There exists a set N whose elements are called natural numbers, as well as an object 0 2 N and an object n ++ assigned to every natural number n 2 N, such that the Peano axioms hold. Before the development of modern axiomatic set theory, logicians and mathematicians by and large accepted a single, unifying principle, called universal speci¯cation, which implies most of the axioms above described. This principle is stated as follows: \Suppose for every object x we have a property P (x) pertaining to x. Then there exists a set fx : P (x) is trueg such that for every object y, y 2 fx : P (x) is trueg () P (y) is true." (1) The principle of universal speci¯cation is also known as the axiom of comprehension, and it was at the heart of the foundation of mathematics, as understood by brilliant philosophers and mathematicians at the onset of the XX century, particularly Gottlieb Frege. Unfortunately for Frege, and for many mathematicians of his time, this so-called axiom was erroneous, as the venerable philosopher, mathematician, peace-activist, Nobel-prize winning writer, and fantastic educator Lord Bertrand Russell found out. Bertie Russell, whose exploits make him the grandfather of modern mathematical logic, considered the statement P (x) that says \x is a set, and x2 = x." Such a statement looks innocent enough, and it is easy to verify that most sets satisfy that statement. The exceptions seem problematic, true enough, but he concentrated his attention not on the exceptions but on the innocent looking sets x satisfying P (x). (Example: P (f0; 1g) is true, because f0; 1g is not an element of f0; 1g. On the other hand, P (fx : x is a setg) is presumably not true, because a priori fx : x is a setg is itself a set, and so it is an element of itself.) Bertie used the principle of universal speci¯cation to produce the very large set ­ de¯ned below: ­ := fx : P (x) is trueg = fx : x is a set and x2 = xg, i.e., the set of all sets which do not contain themselves (as elements). Bertie ¯nally asked himself (and all of you) the following mind-boggling question, which e®ectively destroyed Frege's axiomatic edi¯ce for all of mathematics: Question (Russell's paradox). Is ­ 2 ­? The problem with the principle of universal speci¯cation is, in some sense, that it creates sets that are \too large." For example, one can talk about the set of all sets, and so one has some sets that contain themselves as elements, which is not an ideal thing. Now that we have seen just how awful the situation can get pretty easily, we add one axiom below to ensure that we do not get any ugly paradoxes like Russell's paradox above.