MAGIC Set theory. lecture 1

Transcription

1 MAGIC Set theory lecture 1 David Asperó University of East Anglia 15 October 2014

2 Welcome Welcome to this set theory course. This will be a 10 hour introduction to set theory. The only prerequisite is some level of mathematical maturity. The course will be mostly self contained but you are and will be invited to look in other sources.

3 Welcome Welcome to this set theory course. This will be a 10 hour introduction to set theory. The only prerequisite is some level of mathematical maturity. The course will be mostly self contained but you are and will be invited to look in other sources.

4 Set theory plays a dual role. It provides a foundation for mathematics and it is itself a branch of mathematics with applications to other areas of mathematics.

5 Reducing everything to sets Set theory was developed / discovered / instigated by Georg Cantor, in the second half of the 19th century, as a result of his investigations of trigonometric series rather than out of foundational considerations. However, set theory would soon become the prevalent foundation. In fact, it was born at a time when mathematicians saw the need to define things carefully (i.e., define the object of their study in a mathematical language referring to reasonably simple and well understood entities) and set theory provided the means to do exactly that. Example: What is a differentiable function? What is a continuous function? What is a function?

6 Reducing everything to sets Set theory was developed / discovered / instigated by Georg Cantor, in the second half of the 19th century, as a result of his investigations of trigonometric series rather than out of foundational considerations. However, set theory would soon become the prevalent foundation. In fact, it was born at a time when mathematicians saw the need to define things carefully (i.e., define the object of their study in a mathematical language referring to reasonably simple and well understood entities) and set theory provided the means to do exactly that. Example: What is a differentiable function? What is a continuous function? What is a function?

7 Reducing everything to sets Set theory was developed / discovered / instigated by Georg Cantor, in the second half of the 19th century, as a result of his investigations of trigonometric series rather than out of foundational considerations. However, set theory would soon become the prevalent foundation. In fact, it was born at a time when mathematicians saw the need to define things carefully (i.e., define the object of their study in a mathematical language referring to reasonably simple and well understood entities) and set theory provided the means to do exactly that. Example: What is a differentiable function? What is a continuous function? What is a function?

8 A case example: A relation is a set of ordered pairs (a, b). And a function f is a functional relation (i.e., (a, b), (a, b 0 ) 2 f implies b = b 0 ). What is an ordered pair (a, b)? Well, given a, b, we can define (a, b) ={{a}, {a, b}} (this definition is due to Kuratowski). Fact Given any ordered pairs (a, b), (a 0, b 0 ), (a, b) =(a 0, b 0 ) if and only if a = a 0 and b = b 0. [Easy exercise: Check] Similarly, for given n, we can define the n tuple (a 0,...,a n, a n+1 )=((a 0,...,a n ), a n+1 ).

9 A case example: A relation is a set of ordered pairs (a, b). And a function f is a functional relation (i.e., (a, b), (a, b 0 ) 2 f implies b = b 0 ). What is an ordered pair (a, b)? Well, given a, b, we can define (a, b) ={{a}, {a, b}} (this definition is due to Kuratowski). Fact Given any ordered pairs (a, b), (a 0, b 0 ), (a, b) =(a 0, b 0 ) if and only if a = a 0 and b = b 0. [Easy exercise: Check] Similarly, for given n, we can define the n tuple (a 0,...,a n, a n+1 )=((a 0,...,a n ), a n+1 ).

10 A case example: A relation is a set of ordered pairs (a, b). And a function f is a functional relation (i.e., (a, b), (a, b 0 ) 2 f implies b = b 0 ). What is an ordered pair (a, b)? Well, given a, b, we can define (a, b) ={{a}, {a, b}} (this definition is due to Kuratowski). Fact Given any ordered pairs (a, b), (a 0, b 0 ), (a, b) =(a 0, b 0 ) if and only if a = a 0 and b = b 0. [Easy exercise: Check] Similarly, for given n, we can define the n tuple (a 0,...,a n, a n+1 )=((a 0,...,a n ), a n+1 ).

11 So we can successfully define the notion of function from the notion of set (and the membership relation 2, of course). And the notion of set is presumably easier to grasp than the notion of function.

12 What about natural numbers, integers, rational, reals and so on? We can define 0 = ; (the empty set, the unique set with no elements). The set ; has 0 members. We can define 1 = {0} = {;}. The set {;} has 1 member. We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2 members.... In general, we can define n + 1 = n [{n}. With this definition n + 1 is a set with exactly n + 1 many members and these members are all natural numbers m such that m apple n.

13 What about natural numbers, integers, rational, reals and so on? We can define 0 = ; (the empty set, the unique set with no elements). The set ; has 0 members. We can define 1 = {0} = {;}. The set {;} has 1 member. We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2 members.... In general, we can define n + 1 = n [{n}. With this definition n + 1 is a set with exactly n + 1 many members and these members are all natural numbers m such that m apple n.

14 What about natural numbers, integers, rational, reals and so on? We can define 0 = ; (the empty set, the unique set with no elements). The set ; has 0 members. We can define 1 = {0} = {;}. The set {;} has 1 member. We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2 members.... In general, we can define n + 1 = n [{n}. With this definition n + 1 is a set with exactly n + 1 many members and these members are all natural numbers m such that m apple n.

15 What about natural numbers, integers, rational, reals and so on? We can define 0 = ; (the empty set, the unique set with no elements). The set ; has 0 members. We can define 1 = {0} = {;}. The set {;} has 1 member. We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2 members.... In general, we can define n + 1 = n [{n}. With this definition n + 1 is a set with exactly n + 1 many members and these members are all natural numbers m such that m apple n.

16 With this definition, each n is an ordinal which is either ; or of the form [{ } for some ordinal and all of whose members are either the empty set or of the form [{ } for some ordinal, and every ordinal which is either ; or of the form [{ } and all of whose members are either the empty set or of the form [{ } for some ordinal is a natural number (the notion of ordinal, which we will see later on, is defined only in terms of sets). What is nice about this is that this gives a definition of the set N of natural numbers involving only the notion of set: N is the set of all those ordinals such that every member of is either the empty set or of the form [{ } for some ordinal ).

17 In particular: We may want to say that a set x is finite iff there is a bijection between x and some member of N. By the above definition of N we thus have a definition of finiteness purely in terms of sets and the membership relation.

18 + and on N can be defined also in a satisfactory way using the notion of set. Then we can define Z in the usual way as the set of equivalence classes of the equivalence relation on N N defined by (a, b) (a 0, b 0 ) if and only if a + b 0 = a 0 + b, and we can define also Q and the corresponding operations from Z in the usual way. We can define R as the set of equivalence classes of the equivalence relation on the set of Cauchy sequences f : N! Q where f g if and only if lim n!1 h = 0, where h(n) =f (n) g(n). Etc. All these constructions involve only notions previously defined together with the notion of set and the membership relation. So they ultimately involve only the notion of set and the membership relation.

19 + and on N can be defined also in a satisfactory way using the notion of set. Then we can define Z in the usual way as the set of equivalence classes of the equivalence relation on N N defined by (a, b) (a 0, b 0 ) if and only if a + b 0 = a 0 + b, and we can define also Q and the corresponding operations from Z in the usual way. We can define R as the set of equivalence classes of the equivalence relation on the set of Cauchy sequences f : N! Q where f g if and only if lim n!1 h = 0, where h(n) =f (n) g(n). Etc. All these constructions involve only notions previously defined together with the notion of set and the membership relation. So they ultimately involve only the notion of set and the membership relation.

20 + and on N can be defined also in a satisfactory way using the notion of set. Then we can define Z in the usual way as the set of equivalence classes of the equivalence relation on N N defined by (a, b) (a 0, b 0 ) if and only if a + b 0 = a 0 + b, and we can define also Q and the corresponding operations from Z in the usual way. We can define R as the set of equivalence classes of the equivalence relation on the set of Cauchy sequences f : N! Q where f g if and only if lim n!1 h = 0, where h(n) =f (n) g(n). Etc. All these constructions involve only notions previously defined together with the notion of set and the membership relation. So they ultimately involve only the notion of set and the membership relation.

21 If there is nothing fishy with the notion of set and the operations we have used to build more complicated sets out of simpler ones, then there cannot be anything fishy with these higher level objects. Similarly: We feel confident with the existence of C (which, by the way, contains imaginary numbers like i) once we become confident with the existence of R and know how to build C from R in a very simple set theoretic way.

22 Also: We can derive everything we know about the higher level objects (like, say, the fact that is transcendental) from elementary facts about sets. And, presumably, we would expect that the combination of elementary facts about sets can ultimately answer every question we re interested in (is e + transcendental?, Goldbach s conjecture,...). This would reduce mathematics to considerations of sets and their (elementary) properties.

23 Also: We can derive everything we know about the higher level objects (like, say, the fact that is transcendental) from elementary facts about sets. And, presumably, we would expect that the combination of elementary facts about sets can ultimately answer every question we re interested in (is e + transcendental?, Goldbach s conjecture,...). This would reduce mathematics to considerations of sets and their (elementary) properties.

24 Some elementary facts about sets Given sets A, B: We say that A is of cardinality at most that of B, and write A apple B, if there is an injective (or one to one) function f : A (remember, a function is a special kind of set!).! B We say that that A and B have the same cardinality, and write A = B, if and only if there is a bijection f : A! B. We say that A has cardiality strictly less than B, and write A < B, if and only if there is an injective function f : A is no bijection f : A! B.! B but there

25 Clearly A apple B and B apple C together imply A apple C. Also, it is true, but not a trivial fact, that A = B holds if and only if both A apple B and B apple A hold (Cantor Bernstein theorem, we will see this later on). The notion of cardinality captures the notion of size of a set. (Example: 5 < 6 ). Notation: Given a set X, P(X) is the set of all sets Y such that Y X. (P(X) is the power set of X).

26 Clearly A apple B and B apple C together imply A apple C. Also, it is true, but not a trivial fact, that A = B holds if and only if both A apple B and B apple A hold (Cantor Bernstein theorem, we will see this later on). The notion of cardinality captures the notion of size of a set. (Example: 5 < 6 ). Notation: Given a set X, P(X) is the set of all sets Y such that Y X. (P(X) is the power set of X).

27 Clearly A apple B and B apple C together imply A apple C. Also, it is true, but not a trivial fact, that A = B holds if and only if both A apple B and B apple A hold (Cantor Bernstein theorem, we will see this later on). The notion of cardinality captures the notion of size of a set. (Example: 5 < 6 ). Notation: Given a set X, P(X) is the set of all sets Y such that Y X. (P(X) is the power set of X).

28 The following theorem arguably marks the beginning of set theory. Theorem (Cantor, December 1873) Given any set X, X < P(X). Proof: There is clearly an injection f : X the singleton of x, i.e., to {x}.!p(x): f sends x to Now suppose f : X!P(X) is a function. Let us see that f cannot be a surjection: Let Y = {a 2 X : a /2 f (a)} Y 2P(X). But if a 2 X is such that f (a) =Y, then a 2 Y if and only if a /2 f (a) =Y. This is a logical impossibility, so there is no such a.

29 This theorem immediately yields that not all infinite sets are of the same size, and in fact there is a whole hierarchy of infinities! (which was not known at the time): N < P(N) < P(P(N)) = P 2 (N) <......< P n (N) < P n+1 (N) <......< S n2n Pn (N) < P( S n2n Pn (N)) <...

30 More elementary facts Let R be the collection of all those sets X such that X /2 X R is a collection of objects, and so it is therefore a set. R contains many sets. For instance, ;2R, 1 2R, every natural number is in R, N 2R, R 2R, etc. Does R belong to R?

31 Well, R2Rif and only if R /2 R, which is the same kind of contradiction that we obtained at the end of the proof of Cantor s theorem! So R cannot be a set!! (Russell s paradox)

32 So, our naïve theory of sets is inconsistent and maybe it s not so good a foundation of mathematics after all... Is this the end of the story for set theory?

33 So, our naïve theory of sets is inconsistent and maybe it s not so good a foundation of mathematics after all... Is this the end of the story for set theory?

34 Well, we like to think in terms of objects built out of sets and like the simplicity of the foundations set theory was intending to provide. Also, we find the multiplicities of infinities predicted by set theory an exciting possibility, and there was nothing obviously contradictory in Cantor s theorem.

35 A retreat A valid move at this point would be to retreat to a more modest theory T such that 1 T should express true facts about sets. (or should we say plausible, desirable?), 2 T enables us to carry on enough constructions so as to build all usual mathematical objects (real numbers, spaces of functions, etc.), 3 T gives us an interesting theory of the infinite ( N < P(N), etc.), and such that 4 we can prove that T is consistent; or, if we cannot prove that, such that we have good reasons to believe T is consistent.

36 A retreat A valid move at this point would be to retreat to a more modest theory T such that 1 T should express true facts about sets. (or should we say plausible, desirable?), 2 T enables us to carry on enough constructions so as to build all usual mathematical objects (real numbers, spaces of functions, etc.), 3 T gives us an interesting theory of the infinite ( N < P(N), etc.), and such that 4 we can prove that T is consistent; or, if we cannot prove that, such that we have good reasons to believe T is consistent.

37 A retreat A valid move at this point would be to retreat to a more modest theory T such that 1 T should express true facts about sets. (or should we say plausible, desirable?), 2 T enables us to carry on enough constructions so as to build all usual mathematical objects (real numbers, spaces of functions, etc.), 3 T gives us an interesting theory of the infinite ( N < P(N), etc.), and such that 4 we can prove that T is consistent; or, if we cannot prove that, such that we have good reasons to believe T is consistent.

38 A retreat A valid move at this point would be to retreat to a more modest theory T such that 1 T should express true facts about sets. (or should we say plausible, desirable?), 2 T enables us to carry on enough constructions so as to build all usual mathematical objects (real numbers, spaces of functions, etc.), 3 T gives us an interesting theory of the infinite ( N < P(N), etc.), and such that 4 we can prove that T is consistent; or, if we cannot prove that, such that we have good reasons to believe T is consistent.

39 A retreat A valid move at this point would be to retreat to a more modest theory T such that 1 T should express true facts about sets. (or should we say plausible, desirable?), 2 T enables us to carry on enough constructions so as to build all usual mathematical objects (real numbers, spaces of functions, etc.), 3 T gives us an interesting theory of the infinite ( N < P(N), etc.), and such that 4 we can prove that T is consistent; or, if we cannot prove that, such that we have good reasons to believe T is consistent.

40 First questions: (1): What is a theory? (2): Which should be our guiding principles for designing T? We answer (1) first.

41 First questions: (1): What is a theory? (2): Which should be our guiding principles for designing T? We answer (1) first.

42 The axiomatic method: A crash course in first order logic. For us a theory will be a first order theory or, more accurately, a theory in classical first order logic. A theory T will always be a theory in a given first order language L. It will be a set (!) of L sentences expressing facts about our intended domain of discourse. Talk of sets of L sentences before we have even defined T (which might end up being an intended theory of sets)? Well, those sets of sentences, as well as the sentences, the language L, etc., are objects in our meta theory. Presumably they will obey laws expressible in some meta meta theory (perhaps the same laws the same theory T is, in its intended interpretation, trying to express!).

43 The axiomatic method: A crash course in first order logic. For us a theory will be a first order theory or, more accurately, a theory in classical first order logic. A theory T will always be a theory in a given first order language L. It will be a set (!) of L sentences expressing facts about our intended domain of discourse. Talk of sets of L sentences before we have even defined T (which might end up being an intended theory of sets)? Well, those sets of sentences, as well as the sentences, the language L, etc., are objects in our meta theory. Presumably they will obey laws expressible in some meta meta theory (perhaps the same laws the same theory T is, in its intended interpretation, trying to express!).

44 A language L consists of a (possible empty) set of constant symbols c, d,... a (possibly empty) set of functional symbols f, g,..., together with their arities (this arity is a natural number; if f is meant to express a function f M : M! M it has arity 1, if it is meant to express a function f M : M M! M, then it has arity 2, etc.) a (possibly empty) set of relational symbols R, S,... together with their arities (this arity is a natural number; if R is meant to express a subset R M M, then it has arity 1, if it is meant to express a binary relation R M M M, then it has arity 2, etc.) These are the non-logical symbols and completely determine L.

45 We also have logical symbols, which are independent from L: ^, _,,!, $ 8, 9 (, ), = (quantifiers) (connectives) = is sometimes omitted. Also, many of these symbols are not necessary; we could do with just, _ and 9. Finally, we have a sufficiently large supply of variables: Var = {v 0, v 1,...,v n,...}. For most uses it is enough to take the set of variables to have the same size as the natural numbers.

46 Language of set theory: Only non logical symbol: A relational symbol 2 of arity 2. Let s focus on the language of set theory from now on: 1 every expression of the form (v i 2 v j ) or (v i = v j ), with v i and v j variables, is a formula (an atomic formula). 2 If ' and are formulas, then ( '), (' _ ), (' ^_), (' ^ ), ('! ), (' $ ) are formulas. Also, if v is a variable, then (8v') and (9v') are formulas. 3 Something is a formula if and only if it is an atomic formula or is obtained from formulas as in (2). When referring to a formula, we often omit parentheses to improve readability (these expressions are not actual official formulas but refer to them in an unambiguous way).

47 A sentence is a formula ' without free variables, i.e., such that every variable v occurring in ' occurs in some subformula of the form 8v or of the form 9v.

48 Examples of formulas are the formulas abbreviated as: 8x8y(x = y $8z(z 2 x $ z 2 y)) (The axiom of Extensionality) 8x8y9z8w(w 2 z $ (w = x _ w = y)) or, even more abbreviated, for all x, y, {x, y} exists (Axiom of unordered pairs).

49 Another example: 9a9b8y(y 2 x $ ((8w(w 2 y $ (w = a _ w = b))) _ (8w(w 2 y $ w = a))))) (x is in ordered pair) The first two formulas are sentences. The third one is not.

50 Satisfaction This takes place of course in the meta theory: A pair M =(M, R), where M is a set and R M M, is called an L structure. Given an assignment ~a : Var! M: M = (v i 2 v j )[~a] if and only if (~a(v i ),~a(v j )) 2 R. M = (v i = v j )[~a] if and only if ~a(v i )=~a(v j ). M = ( ')[~a] if and only if M = '[~a] does not hold. M = (' 0 _ ' 1 )[~a] if and only if M = ' 0 [~a] or =' 1 [~a]; and similarly for the other connectives. M = (9v')[~a] if and only if there is some b 2 M such that M = '[~a(v/b)], where ~a(v/b) is the assignment ~ b such that ~ b(v i )=~a(v i ) if v i 6= v and ~ b(v) =b. M = (8v')[~a] if and only if for every b 2 M, M = '[~a(v/b)].

51 We say that M satisfies ' with the assignment ~a if M = '[~a]. Easy fact: If ' is a sentence, then M = '[~a] for some assignment ~a if and only if M = '[~a] for every assignment ~a. In that case we say that M is a model of '.

52 Given a set T of formulas and a formula ', we write T = ' if and only if for every L structure M =(M, R) and every assignment ~a : Var! M, IF M = [~a] for every 2 T, THEN M = '[~a]. The relation = aims at capturing the notion of logical consequence : ' follows logically from T if and only if ' is true in every world in which T is true. = is often called the relation of logical consequence. This framework is mostly due to the logician A. Tarski (1930 s). First order in first order logic refers to the fact that variables range in the above definition only over the individuals of the universe of the relevant L structures M. In second order logic we can have variables that range over (arbitrary) subsets of the universe of the relevant L structures M. Etc.

53 Given a set T of formulas and a formula ', we write T = ' if and only if for every L structure M =(M, R) and every assignment ~a : Var! M, IF M = [~a] for every 2 T, THEN M = '[~a]. The relation = aims at capturing the notion of logical consequence : ' follows logically from T if and only if ' is true in every world in which T is true. = is often called the relation of logical consequence. This framework is mostly due to the logician A. Tarski (1930 s). First order in first order logic refers to the fact that variables range in the above definition only over the individuals of the universe of the relevant L structures M. In second order logic we can have variables that range over (arbitrary) subsets of the universe of the relevant L structures M. Etc.

54 Given a set T of formulas and a formula ', we write T = ' if and only if for every L structure M =(M, R) and every assignment ~a : Var! M, IF M = [~a] for every 2 T, THEN M = '[~a]. The relation = aims at capturing the notion of logical consequence : ' follows logically from T if and only if ' is true in every world in which T is true. = is often called the relation of logical consequence. This framework is mostly due to the logician A. Tarski (1930 s). First order in first order logic refers to the fact that variables range in the above definition only over the individuals of the universe of the relevant L structures M. In second order logic we can have variables that range over (arbitrary) subsets of the universe of the relevant L structures M. Etc.

55 Syntactical deduction Let T will be a set of formulas. We will view T as a set of axioms and deduce theorems from T : A theorem of T will be the final member n of a deduction =( 0, 1,..., n) from T, where we say that =( 0, 1,..., n) is a deduction from T if it is a finite sequence of L formulas and for every i, i is either in T, or i is a logical axiom of first order logic, or i is obtained form j and k, for some j, k < i, by the rule of Modus Ponens If '! and ', then (for all L formulas ', ). In other word, in this last case, there are j, k < i and an L formula ' such that j = ' and k is '! i.

56 Here, a logical axiom is a member of a certain infinite easily specifiable list of formulas that express logical / completely general truths. Typical members of this list are for example, '! (! ') for all formulas ',, or ' _ ' for all formulas '. Indeed, we see it as a general truth that if ' is true, then it is true that if is true then ' is true. And we see it as a general truth that for every ' either ' is true or ' is true. 1 This list of axioms is not unique: Many different lists of axioms give rise to the same system of logic. 1 If we are classical logicians. There are weakening / versions of classical first order logic in which ' _ ', also known as Law of Excluded Middle, is not true for some choices of '.

57 If ' is a theorem from T, we write T ` ' ` is often called the relation of logical derivability.

58 A priori, = and ` look like quite different relations, aimed at capturing two apparently different notions: The notion of logical (semantical) consequence and the notion of deductibility in a reasonable calculus. However,

59 Theorem (Completeness theorem for first order logic) (K. Gödel, 1930 s) ==`