Axiomatic Set Theory. Alexandru Baltag (ILLC, University of Amsterdam) Go to. website:

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1 Axiomatic Set Theory Alexandru Baltag (ILLC, University of Amsterdam) Go to website: then click on Teaching, then click on Axiomatic Set Theory 1

2 Course Administration Instructor: ALEXANDRU BALTAG Teaching Assistants (and Markers): ZHENHAO LI and SHENGYANG ZHONG. Office Hours and Inquiries: your questions to BOTH my assistants: Zhenhao Li, at AND Shengyang Zhong, at with CC to me. Or come to see them during office hours. (Time and Place TBA.) In case they cannot answer a question, please me at TheAlexandruBaltag@gmail.com. 2

3 The main textbook for this course is Course Materials K. Devlin, The Joy of Sets, Springer-Verlag, 1993 (Second Edition). In addition, I will use J. Barwise and L. Moss, Vicious Circles, CSLI Lecture Notes no. 60, CSLI Publications, 1996 for the last few lectures of the course (on non-wellfounded sets). Other relevant materials (book, links, slides): on the website.

4 Assessment, Course Plan, Important Announcements Assessment: Four or Five Homework Exercises (worth in total 50% of the final grade), plus a Final TAKE-HOME EXAM (also worth 50% of the final grade). Both are individual (i.e. no collaboration)! Course Plan:See website. It s tentative, and it will be periodically updated. Important Announcements: February 10: No class! BUT you are all invited to attend Yurii Khomskii s PhD defense (in whose committee I am), as a set theory lesson. Look on ILLC news for time and place. April 3: NO CLASS! 4

5 1.1. Naive Set Theory Sets, Objects, Elements, Membership: a x Sets as Properties or Classes : {a P (a)}, or {a : P (a)}. Sets as well-formed, finished collections: totalities that can be thought of as a single unit. Sets of Sets Identity of objects or sets: a = b, x = y Properties of Identity: a = a a = b b = a a = b b = c a = c a = b x(a x b x) 5

6 Leibniz s principle: x(a x b x) a = b Criterion for identity: the Extensionality Axiom x = y a(a x a y) Inclusion: x y Operations with Sets: x y, x y, x y, {a}, {a, b}, (a, b), x y, x, i I y i, x, i I y i, P(x) Bynary Relations R x y: reflexive, transitive, symmetric, antisymmetric, connected, transitive, equivalence relations, equivalence classes, preorders, partial orders, posets, minimal elements, well-foundedness, total orders, well-orders. 6

7 N-ary Relations. Domain and range of a relation. Functions: f : x y. N-ary Functions. Function composition f g. Identity function id x. Constant function. Image f[z]. Pre-image f 1 [z]. Injections, Surjections, Bijections. Inverse f 1. Arbitrary Cartesian Product Π i I x i := {f f : I i I x i ( i I)f(i) x i } 7

8 The Mathematics of Infinity Pre-history: the Greeks, Aristotle, the horror of infinity; Indians and the love for infinity; Christian theologians, scholastics etc; Bolzano. GEORG CANTOR s Work (1870 s): Cantor s background. Trigonometric Series. Dealing with infinity: mathematical and theological arguments. Bolzano-Weierstrass Theorem: every bounded but infinite set of real numbers has a limit point. Cantor-Bendixson derivative A. Iterated derivatives: A (n), A (ω) etc. Perfect sets. Transfinite ordinals ω, ω + 1, 2ω, ω ω etc: infinite processes. Comparing orders: order types. Ordinals as well-ordered types. 8

9 Comparing Infinities: equipotence, cardinality. Alephs: infinite cardinals ( powers ) ℵ 0, ℵ 1 etc. Countable Infinities: Hilbert s Hotel. Rationals versus natural numbers! N = Z = Q = 2N = N N = ℵ 0 Lines versus Planes versus Cubes etc. I see it, but I don t believe it. R = [0, 1] = R 2 = C = [0, 1] 2 = R n = [0, 1] n Reals versus natural (rational, algebraic) numbers: R > N Proving Existence of Transcendental (=non-algebraic) Numbers. 9

10 Cantor s Theorem and the Diagonal Method: The Power of Continuum. x < P(x) The Continuum Hypothesis (CH). Cantor s Obsession. Open Questions and Paradoxes. The Status of the Axiom of Choice (AC). Kronecker s Criticism. Madness. 10

11 Cardinals: Proofs N = Z : Bijection given by f(n) = n 2, if n is even, f(n) = n N = 2N : Bijection given by f(n) = 2n., if n is odd. N = N N : Look at N N as an infinite bidimensional array. Enumerate its elements linearly, by going on successively longer diagonals. Illustrations: Hilbert s hotel. 11

12 N = Q : It is enough to show that there is a bijection f : N Q +. (Why?) Arrange positive rational, expressed as integer quotients, in an infinite bidimensional array A(i, j) = i j This gives us a surjective map A : N A Q +. Compose this with the previous enumeration, to get a surjective map F : N Q +, i.e. a linear enumeration of Q + with repetions. Eliminate the repetions a bijective map from N to Q +. 12

13 R = (0, 1) : A bijection f : R (0, 1) is given by f(x) = 1 1 2x f(x) = 1 2(2 x), for x 1,, for x 1, R = (0, 1)] = [0, 1) = [0, 1] = (0, ) : Exercise. R = R R = R n : It is enough to show [0, 1] = [0, 1] [0, 1]. (Why?) Exercise! R = 2 N = P(N), where 2 N = {f f : N {0, 1}}. Exercise! (Hint: use the binary representation of reals; for the second part, use the characteristic function f S of any subset S N.) 13

14 Cantor s Theorem: proof Proof of x P(x) : Cantor s Diagonal Method. Let f : x P(x) be a bijection. Put A = {y x y f(y)}. Since A P(x), there exists some a A such that f(a) = A. But then we have: Contradiction! a A a f(a) a A. Consequences: x < P(x) ; N < R ; there exist transcendental numbers; there is no largest cardinal. 14

15 Wosets: what are they good for? Well-orderings are well, because we can do induction on them: Induction Theorem (Th in Devlin) Let (X, ) be a woset. If E X satisfies the smallest element of X is in E: Min X E; Every element of X is in E if all its predecessors are in E: then E = X. x X( y(y < x y E) x E ) ; This generalizes induction on natural numbers. NOTE (exercise): In fact, the first itemized premise of this theorem is redundant: it follows from the second itemized premise! 15

16 When are two (well-)orders the same? An order-isomorphism between two (totally) ordered sets (X, ), (X, ) is a bijection f : X X such that for all x, y X. x < y f(x) f(y) If there exists such an order isomorphism, then we write (X, ) = (X, ). Essentially, this means that the two sets are ordered in the same way : they have the same order type. Ordinals are the order types of wosets. 16

17 Von Neumann Ordinals In order to formally define ordinals, it would be useful to be able to choose a canonical representative for each (well-)order type: i.e. to find a special family On of wosets (to be called ordinals ), such that every woset is order-isomorphic to a unique woset in On. Definition: A von Neumann ordinal (or ordinal, for short) is a set that is well-ordered by the membership relation. Intuition behind definition: an ordinal is simply the set of all its predecessors. We (or God) form ordinals in stages: a newly created ordinals is just the set of all previously formed ordinals. 17

18 Equivalent Definition (Devlin) Devlin gives an equivalent definition: Given a woset (X, ) and an element a X, the segment of X determined by a is the set of all the predecessors of a. X a = {x X x < a} Definition (Devlin): An ordinal is a woset (X, ) such that X a = a for all a X. 18

19 Ordinals form a Well-Ordered Collection The collection of all ordinals will be denoted by On. As we will see, in standard Set Theory, On is NOT a set. For two ordinals α, β On, we define the precedence relation by: α < β iff α β iff α β, As a consequence, we ll indeed always have that any ordinal coincides with the set of its predecessors: α = {β On β < α}. Moreover, the collection On of all ordinals is itself well-ordered by <. 19

20 Examples of von Neumann ordinals The ordinal 0 has no predecessors, so it is just the empty set: 0 :=. The ordinal 1 has only one predecessor, namely 0, so: 1 := {0} = { }. The ordinal 2 has two predecessors, 0 and 1, so: 2 := {0, 1} = {, { }}. More generally, the ordinal corresponding to the natural number n is the set of all smaller (natural numbered) ordinals: n := {0, 1, 2,..., n 1}. 20

21 More Examples and Notations: transfinite ordinals The next ordinal after all the finite ones (= the smallest infinite ordinal) is defined as the set of all finite ordinals: The next ordinal after that will be ω := {0, 1, 2,...} = N. ω + 1 := {0, 1, 2,..., ω} = ω {ω} = N {ω}. The series of increasing ordinals goes forever, beyond the confines of any given system of notation: 0, 1, 2,..., ω, ω + 1,..., 2ω := ω 2,..., 3ω := ω 3,..., ω n,... ω 2 := ω ω,..., ω 3,..., ω 4,..., ω n,..., ω ω,..., ω ωω,..., ω ωω...,... ω 1,..., ω 2,..., ω n,..., ω ω,..., ω ω2,..., ω ωω,..., ω ωω...,... 21

22 Successor Ordinals versus Limit Ordinals For every ordinal α, there is a unique next highest ordinal α + 1. Ordinals of the form α + 1 are called successor ordinals. For every set A of ordinals, there is a unique next ordinal (=the least ordinal that is higher than all the ordinals in A). If A has NO last element (in the <-order), then we denote this next ordinal by lim A. Ordinals of this form are called limit ordinals. Examples: lim {0, 1,..., n,...} = ω, lim {ω, ω + 1, ω + 2,..., ω + n,...} = ω 2, lim {ω, ω 2, ω 3,..., ω n,...} = ω ω. Every ordinal is either a successor or a limit. 22

23 The Successor Function, set theoretically Recall that every von Neumann ordinal α coincides with the set of its predecessors: α = {β On β < α}. Applying this to a successor ordinal α + 1, we obtain that α + 1 = {β On β < α + 1} = {β On β < α} {α} = α {α}. So we can represent set-theoretically the successor function as the following operation on von Neumann ordinals: α α + 1 := α {α} 23

24 Attempts to Save Math: Logicism, Formalism, Intuitionism Logicism: Gottlob Frege. All Math is Logic. Logic can axiomatize infinity. Formalism: David Hilbert. Infinities are useful fictions. A proper logical analysis should show that they are consistent with, but not necessary for Math: they can be eliminated, replaced by finitistic methods. Once this is proved, infinities can still be safely used, to shorten the proofs. No one will drive us from the paradise which Cantor created for us. (D. Hilbert) Intuitionism, Constructivism etc: Brouwer, Heyting etc. Infinities are dangerous. They break the laws of classical logic. Only constructive, finitistic methods should be employed. 24

25 Foundations of Math, Logicism and Paradoxes Gottlob Frege versus Bertrand Russell (1901): invented axiomatic predicate logic Logicism: all Math is Logic. Deriving all math from logical axioms. Naive Comprehension Principle: for any logically definable predicate P, we can form the set {a P (a)} Russell s Paradox: {x x x} 25

26 More Paradoxes Well-founded sets: no infinite descending -chains. Mirimanoff s Paradox: {x x is well-founded} Burali-Forti Paradox: the order type of the set of all ordinals (with theor natural order). {x x is an ordinal} Cantor s Paradox: the cardinal of the universe U = {x x is a set} is by definition the largest cardinal, which contradicts Cantor s Theorem. 26

27 Towards a Solution: The Set/Class Distinction Paradoxes show that we have to give up the Naive Comprehension Principle: not every definable collection of objects (=definable by some first-order predicate P (x)) forms a set. Definable collections {x P (x)} are in general called classes in Set Theory. So the lesson from the paradoxes is that not every class is a set. But what does that mean? What distinguishes sets from classes? 27

28 Classes versus Objects The important distinction is that classes are collections of objects, but they are not necessarily objects themselves! Moreover, only objects can be collected to form classes. Syntactically, the variable x in the expression or in the expression {x P (x)} x Y must denote an object; while the variable Y in x Y, or in Y = Z might stand for a class (collection), such as {x P (x)}, which might happen to NOT be an object. 28

29 Classes versus Sets So any definable collection of objects is a class. But not all classes are objects, and hence not all classes can be collected as members of other classes. The classes that happen to also be objects will be called sets. The classes which are not objects (and hence are not sets) are called proper classes. 29

30 Example Russell s class, defined as the class (collection) of all the sets that do not belong to themselves {x x x} cannot be a set (since this would lead to Russel s paradox), so it is a proper class. 30

31 All these are only intuitions. Axiomatic Systems of Set Theory The formal (axiomatic) systems we ll consider will make some of this precise. There are a number of axiomatic proof systems for Set Theory. The system ZF (from Zermelo-Fraenkel), augmented with the Axiom of Choice (AC) to form the Zermelo-Fraenkel system ZF C: the most widely accepted. The von Neumann-Bernays-Godel system N BG is an alternative formulation, which is nevertheless completely equivalent to ZF C. 31

32 The system ZF A of Anti-founded (=non-wellfounded) Set Theory, proposed by P. Aczel, describes a universe of sets that is wider than the universe of ZF C, allowing for sets that disobey one of the axioms (Foundation) of ZF C. However, in a sense ZF A is still equivalent to ZF C! More precisely, ZF A has the same logical strength (=it is equi-consistent) with ZF C: if given any model of ZF C, one can construct a model of ZF A, and vice-versa. Hence, the consistency of ZF C is equivalent to the consistency of ZF A. 32

33 There are also systems that are logically stronger than ZF C: one example is the Morse-Kelley system M K; another examples are the extensions of ZF C with various large cardinal axioms; yet another example is the system ST S of Structural Set Theory. There are also of course many systems that are weaker than ZF C: obtained by dropping some of the axioms. Finally, there are systems of set theory that are not known to be weaker, nor stronger, nor equivalent to ZF C: an example is Quine s system N F of New Foundations (very important from a philosophical perspective and also from a historical and foundational point of view); another example (much more important for current research in set theory) is the system obtained from ZFC by replacing the Axiom of Choice by the Axiom of Determinacy (AD). 33

34 Classes, from a formal-axiomatic perspective In some axiomatic versions of Set Theory (such as ZF C), there are strictly speaking no classes, at least at a formal level: the allowed variables x, y,... range only over objects Indeed, in some versions of ZFC, no other objects are allowed but sets, and so the variables range over sets only. Nevertheless, at a conceptual, informal level, people continue to talk about classes as ways to denote definable collections of objects, which might or not be objects (sets) themselves. The system ZF C interprets such class talk only as a manner of speaking, a metaphor: classes are like imaginary objects, that may be useful as an intuitive way of speaking, but are not to be taken seriously. 34

35 As we ll see, in ZF C there is a systematic way to translate statements that involve classes into statements that do NOT refer to classes in any way. Other systems, such as N BG, formally allow classes: the variables x, y,... now range over classes. However, only sets can be members of other classes. Moreover, N BG still doesn t fully take classes seriously, since it restricts the quantifiers (and free parameters) to sets. In a sense, only definable classes are allowed in NBG. The Morse-Kelley system M K drops this last restriction, allowing reference to undefined, arbitrary classes (as free parameters): it is impredicative. 35

36 When can a class be regarded as an object? The main question is: WHEN does a class (collection of objects) constitute itself an object (hence, a set)? What is special about those classes/collections that are sets/objects? Various answers have been proposed: Sets are well-formed collections: built in stages, in a well-ordered manner, from previously built sets. Sets are small collections (or at least not very big ). Sets are closed, finished collections. Sets are well-defined collections (defined by a predicate that doesn t involve any hidden vicious circles). 36

37 A Solution: The Iterative Conception of Sets Originates in Russell s Type Theory (B. Russell, A. N. Whitehead, Principia Mathematica, ). It is the most accepted intuition behind today s standard setting for Set Theory: A set can be constructed only AFTER its elements are given. Sets are built inductively, in transfinite stages, starting from basic, atomic objects (ur-elements), or even better: starting from nothing! 37

38 Nothing Versus Ur-elements Some versions of Set Theory (e.g. the one presented in the book Vicious Circles ) allow for the existence of some objects that are NOT sets. These are called primitive objects, or atoms, or ur-elements. Ur-elements can be elements of sets, but they themselves havo no elements. On the other hand, they are not equal to the empty set. Hence, ur-elements do NOT respect the Extensionality Axiom! For this reason, in most standard presentations of Set Theory, ur-elements are excluded: every object is a set. 38

39 The Cummulative Hierarchy of Sets In Day 0, God takes nothing and forms the empty set 0 :=. In Day 1, God puts the empty set into a set, creating 1 := { } = {0}. In Day 2, God uses the previous creations to create two sets 2 := {0, 1} = {, { }} and {{ }}. In Day ω, God puts together the infinitely many previous creations in various ways, creating among other things the set of natural numbers but also the set N = {0, 1, 2,...} = {, { }, {, { }},...} {, { }, {{ }},...} In Day ω + 1, God creates among other things the real numbers R, i.e. P(N). 39

40 Formalizing the Cummulative Hierarchy of Sets The collection of all the sets created by God before Day α forms the set-theoretic universe V α on (the morning of) day α. Before Day 0 there were no sets, so V 0 = In Day α, God forms sets by collecting in all possible ways his previous creations, i.e. the objects in V α : V α+1 = P(V α ) If λ is a limit ordinal, then V λ consists all sets created before day λ, i.e. created in any of the previous days; i.e. V λ = β<λ 40 V β

41 These three equations define the cummulative hierarchy of sets. As in the Big Bang, the universe keeps expanding every day. Each universe V α becomes just a set in the next universe V α+1. There is no last day, since ordinals never end. Hence, the set-theoretical universe grows forever: we cannot form a set of all sets in any day! The intuitive assumption behind this hierarchy is that all sets are formed in this iterative way: every set is created in a particular day. Hence, there cannot exist any set of all sets. However, we can still in a sense refer to the real universe of all sets as being a (proper) class, namely the class V = {x x is a set }. 41

42 Foundation Axiom Our intuitive assumption ( every set is created in some day ) can now be captured by the so-called Foundation Axiom: V = α V α. An equivalent formulation: Every set is well-founded ( when considered as a structure with membership as its binary relation): i.e. there are NO infinite descending -chains of sets x 0 x 1 x Another equivalent formulation: Every non-empty set has an -minimal element: If x is a non-empty set then there is some a x such that a x =. This last one is the formulation in Devlin. 42

43 Null Set Axiom versus Pairs Axiom Devlin s first ZFC axiom is the Null Set Axiom: There exists a set with no members. This provides the starting point for building the set hierarchy. Justification: God creates in day 0. In more standard formulations of ZF C, this axiom is replaced by the Pairs Axiom: For every two sets a, b, there exists a set {a, b}. Justification: if a and b were created in some days, take the latest of these two days, say α; then {a, b} is created in day α + 1. NOTE: In the context of the other ZF C axioms, both the Null Set Axiom and the Pairs Axiom are redundant (=not independent): they can be deduced from the other axioms! 43

44 Powerset And Union Axioms Powerset Axiom: If x is a set, then there is a set P(x) consisting of all subsets of x. Justification: if x is created on day α, then all its members were created before day α; hence, all collections that can be formed from x s members (=all subsets of x) are also created in day α (or earlier); the set P(x) consisting of all these subsets is thus created in day α + 1. Union Axiom: If x is a set, then there is a set x consisting of all elements of all elements of x. Justification: if x is created in day α, then its elements were created before day α; hence, all the elements of its elements were also created before day α. The set x, collecting all these (elements of elements), will thus be created in day α (unless it was already created earlier). 44

45 Subset Selection Axiom Subset Selection Axiom: If x is a set and P (v) is a unary predicate, then there exists a set {y x P (y)} consisting of all (and only) the members of x which satisfy property P. Justification: If x is created in day α, then all its elements were created before day α; hence, the collection {y x P (y)} is also formed in day α (unless it was already created earlier). 45

46 Axiom of Replacement The Subset Selection Axiom can be strengthened to the Replacement Axiom: If x is a set and F is a functional operation on sets, then there is a set F [x] := {F (y) y x} consisting of all (and only) the F -images of all elements of x. EXERCISE: This axiom is stronger than (i.e. it implies) the Subset Selection Axiom. JUSTIFICATION?? Replacement was added later to ZF C than the other axioms. It is hard, or maybe even impossible, to justify based on the iterative picture. 46

47 The Axiom of Infinity Infinity Axiom: There is a set x such that x and which is closed under taking singletons (i.e. for every a x, we have {a} x). The essential content of this axiom is the claim that there exists an infinite set. Justification: is created in day 0, { } in day 1, {{ }} in day 2 etc; so, in day ω, the set is created. {, { }, {{ }},...} 47

48 The Zermelo-Fraenkel Axioms The axioms of the system ZF C are: 1. Extensionality Axiom: sets with the same elements are the same. 2. Null Set Axiom: is a set. Alternatively, the Pairs Axiom: if a, b are sets, then {a, b} is a set. 3. Axiom of Infinity: {, { }, {{ }},...} is a set. 4. Power Set Axiom: if x is a set then P(x) is a set. 5. Union Axiom: if x is a set then x is a set. 6. Subset Selection Axiom: if x is a set and P (y) is a unary predicate, then {y x P (y)} is a set. 48

49 7. Axiom of Replacement: if x is a set and F is a functional operation on sets, then {F (y) y x} is a set. 8. Foundation Axiom: If x is a non-empty set, then there exists a x such that a x =. 9. Axiom of Choice (AC): The Cartesian product of a set of non-empty sets is non-empty: if {x i i I} is a set s.t. i I : x i, then i I x i. 49

50 The Language of Set Theory: LAST The language LAST presented in Devlin (Ch. 2.1) is one of the various languages that were used to formalize the system ZF C. Vocabulary of LAST: Constants (=names for sets): a countably infinite collection of symbols w 0, w 1,..., w n,... (Strictly speaking, these are not really necessary: they can always be simulated by variables!) Variables (for sets): another countably infinite collection of symbols v 0, v 1,..., v n,... The membership symbol 50

51 The equality symbol = (Again, this is not really necessary: in some presentations of Set Theory, is the only primitive relational symbol, and = is defined in terms of.) Logical connectives (AND), (OR), (NOT) (The OR symbol is not really necessary: it can be defined as an abbreviation via ϕ ψ := ( ϕ ψ.) Quantifier symbols: (for all), (there exists) Brackets: (, ). We use x, y, z,... as metavariables, denoting arbitrary constants w n or variables v n. 51

52 Formulas of LAST: x y, x = y are formulas; if ϕ, ψ are formulas, then (ϕ ψ), (ϕ ψ), ( ϕ) are formulas; if ϕ is a formula, then ( v n ϕ), ( v n ϕ) are formulas. We drop the brackets whenever there is no ambiguity. We make the usual conventions concerning free and bound variables. A sentence is a formula that contains no free variables. We write ϕ(v 0, v 1,..., v n ) for a formula whose free variables are among the ones in the list v 0, v 1,..., v n. We make the usual abbreviations: ϕ ψ := ϕ ψ, ϕ ψ := (ϕ ψ) (ψ ϕ). 52

53 More Abbreviations x y := v n (v n x v n y), where v n x, y ; x = y := v n (v n x v m (v n v m v m y)), where v n v m, v n, v m x, y. x = {y} := v n (v n x v n = y), where v n x, y; x = {y, z} := v n (v n x v n = y v n = z), where v n x, y, z; x = (y, z) := v n (v n x v n = {y} v n = {y, z}), where v n x, y, z; x = y z := (x = {y, z}). 53

54 Formalizing the ZFC axioms Each of the ZF C axioms can be encoded as formula, or an infinite set of formulas (an axiom scheme), in LAST. E.g. the Union Axiom becomes, with the above abbreviations: v i v j (v j = v i ), or more explicitly (unfolding the abbreviation) v i v j v n (v n v j v m (v n v m v m v i )). Some ZF C axioms are axiom schemata, e.g. the Axiom of Subset Selection actually consists of infinitely many instances: v i v m v n (v n v m v n v i ϕ(v n )).

55 Do the following exercises from Homework 1 Devlin (The Joy of Sets, 1993 edition): Exercise (page 34): parts (ii), (iii), (v), (vi), (vii); Exercise (page 39); Exercise (page 39); Exercise (page 39); Exercise (page 40); Exercise (page 44); Exercise (page 45). DUE Friday 24 February before 10:00 am. Please leave it in SHENGYANG ZHONG s mailbox at ILLC, or else him at zhongshengyang@163.com Late submissions will NOT be considered!! 55