Class A is empty if no class belongs to A, so A is empty x (x / A) x (x A). Theorem 3. If two classes are empty, then they are equal.

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1 Everything is a class. The relation = is the equality relation. The relation is the membership relation. A set is a class that is a member of some class. A class that is not a set is called a proper class. The Axiom of Extensionality is A B (A = B x (x A x B)). Theorem 1. A B (A = B x (x A x B)). Let the expression {A : ϕ(a)} denote the class of all sets A having property ϕ. Suppose the letter C is not used as a variable in ϕ. Then we regard the phrase {A : ϕ(a)} exists as an abbreviation of C x (x C ϕ(x) x is a set), so that {A : ϕ(a)} exists C x (x C ϕ(x) x is a set). Theorem 2. {x : x / x} is not a set. More formally, A ( x (x A x / x x is a set) A is not a set). Proof. Let A be a class. Assume x (x A x / x x is a set). Specialize to A and conclude that A A A / A A is a set. It follows from this last statement that if A is a set, then A A A / A, a contradiction. Therefore, A is not a set. Class A is empty if no class belongs to A, so A is empty x (x / A) x (x A). Theorem 3. If two classes are empty, then they are equal. The Axiom of the Empty Set is A ( x (x / A) x (A x)). The empty set is denoted by the special symbol usually reserved for this purpose, namely. The Axiom of the Empty Set can be restated less formally: is a set. Class B is a complement of class A iff the elements of B are the sets that are not elements of A. B is a complement of A x (x B x / A x is a set). By the Axiom of Extensionality, Theorem 4. Every class has at most one complement. The Axiom of Complementation is A B ( x (x B x is a set x / A)), or, more briefly, A ({x : x / A} exists). For every class A, define A by (1) A := {x : x / A}. 1

2 2 Theorem 5. V x (x V x is a set). By this theorem and the Axiom of Extensionality, we may define the universal class V by (2) V := {x : x is a set}. Theorem 6. x (x V x is a set), V =, and V =. Class C is an intersection of classes A and B if x (x C x A x B). The Axiom of Intersection asserts that classes A and B always have an intersection: A B C x (x C x A x B). By Extensionality, we may define the (unique) intersection A B by (3) A B := {x : x A x B}. therefore, For all classes A and B, A B exists. Theorem 7. If A is any class, then V = A A and = A A. For any classes A and B, the Axioms of Intersection and Complementation imply the existence and uniqueness of the binary union A B, the difference A B, and the symmetric difference A B, where (4) (5) (6) Theorem 8. (7) (8) (9) A B := A B, A B := A B, A B := (A B) (A B). A B = {x : x A x B}, A B = {x : x A x / B}, A B = {x : (x A x / B) (x / A x B)}. The Axioms of Intersection and Complementation can be used to prove the existence of an empty class, under the assumption that some class exists. They do not, however, entail that the empty class is a set Calculus of classes. The class A is a subclass of the class B, in symbols, A B, and B is a superclass of A, in symbols B A, if every element of A is a member of B. A is a subset of B if A is both a set and a subclass of B. A is a proper subclass of B, in symbols, A B, iff A B A B. B is a proper superclass of A, in symbols, B A, iff B A and A B. The following theorem gives some common laws involving and. Much of it can be summarized by saying that and are associative, commutative, idempotent, distribute over each other from both sides, and are dual to each other by way of the involution. Its proof uses only the Axioms of Extensionality, Empty Set, Complementation, and Intersection. The

3 consequences of these axioms, especially those exemplified by the statements in the following theorem, constitute the calculus of classes. Theorem 9. Let A, B, C, and D be any classes. Basic laws: 3 (10) (11) (12) (13) (14) (15) A V, A B A, A A B, A B A, A B A C A B C, A C B C A B C. Inclusion and containment are reflexive on classes: (16) (17) A A, A A. Ways to express an inclusion: (18) (19) (20) (21) (22) (23) A B B A, A B B A, A B A B =, A B A B = V, A B A B = B, A B A B = A. Inclusion and containment are antisymmetric: (24) (25) A B B A A = B, A B B A A = B. Transitivity laws: (26) (27) (28) A B B C A C, A B B C A C, A B B C A C. Union and intersection are monotonic: (29) (30) (31) A B A C B C, A B A C B C. Union and intersection are associative: (32) A (B C) = (A B) C,

4 4 (33) A (B C) = (A B) C. Union and intersection are commutative: (34) (35) A B = B A, A B = B A. Union and intersection are idempotent: (36) (37) A A = A, A A = A. Union and intersection distribute over each other: (38) (39) A (B C) = (A B) (A C), A (B C) = (A B) (A C). Absorption laws: (40) (41) (A B) B = B, (A B) B = B. Duality laws and De Morgan s laws : (42) (43) (44) (45) A B = A B, A B = A B, A B = A B, A B = A B. Complementation is idempotent: (46) A = A. Laws involving difference: (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) A B = A B, A B = A (B A), (A B) C = (A C) (B C), (A B) C = (A C) (B C), (A B) C = A (B C), A (B C) = (A B) (A C), A (B C) = (A B) C, A (B C) = (A C) B, A (B C) = (A B) (A C), A (B C) = (A B) (A C), A (B A) = A.

5 5 Laws involving the universal and empty classes: (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) A (B A) =, A A = A A =, A A = V, A = A, A =, A = A, A =, A V = V, A V = A, A V =, V A = A. Laws for the analysis of cases: (69) (70) A = (A B) (A B), A = (A B) (A B). Laws involving symmetric difference: (71) (72) (73) (74) (75) (76) (77) (78) (79) (80) (81) (82) (83) (84) A B = (A B) (A B) A (B C) = (A B) C, A B = B A, A A =, A (B C) = (A B) (A C), A = A, A V = A, A A = V, A B = A (A B), A B = A B (A B), A B = A (A B) A B = A B, A B (A C) (C B), (A B) (C D) (A C) (B D). The Axiom of Unordered Pairs states that, given any (possibly identical) sets a and b, there exists a set c whose only elements are a and b. (85) a b (a is a set b is a set c (c is a set x (x c x = a x = b)))

6 6 By the Axiom of Extensionality, the Axiom of Unordered Pairs can be written as follows. For all sets a and b, {a, b} exists and is a set A class C is an ordered pair (or, simply, pair) with left side A and right side B iff C is a set (86) D E ( x (x C x = D x = E) x (x D x = A) x (x E x = A x = B)) By the Axiom of Extensionality, the definition of ordered pair can be restated as follows: c is an ordered pair with left side a and right side b c is a set c = {{a}, {a, b}} The Axiom of Extensionality justifies standard notation for ordered pairs. For any sets a and b, let (87) a, b := {{a}, {a, b}} Theorem 10. For all sets a, b, c, d V, a, b = c, d a = c b = d The notation { x, y : ϕ(x, y)} should be understood as a name for the (possibly nonexistent) class of ordered pairs of sets x and y such that ϕ(x, y). If { x, y : ϕ(x, y)} exists, then (88) c { x, y : ϕ(x, y)} a b (c = a, b ϕ(a, b)) A class C is a relative product of classes A and B if and C is a relative sum of A and B if C = { x, z : y ( x, y A y, z B)} C = { x, z : y (y is a set ( x, y A y, z B))} If the product and sum exist, they are unique by the Axiom of Extensionality, so define (89) (90) A B := { x, z : y ( x, y A y, z B)} A B := { x, z : y (y V ( x, y A y, z B))} The Axiom of Relative Product guarantees the existence of the relative product of any two classes A and B A B C (C = A B) or, simply, A B (A B exists) By the Axioms of Complementation and Relative Product, the relative sum exists, since A B = V 2 A B, so A B (A B exists) The operation of composition is defined by (91) A B := B A

7 We may now define the unit relation V 2 by (92) V 2 := V V Then V 2 = { x, y : x, y V} A class R is a binary relation iff it is a class of ordered pairs iff R V 2, and R is a binary relation on the class A iff R is a binary relation and the left and right sides of every ordered pair in R are elements of A, that is, c (c R x (x A y (y A c = x, y ))). Theorem 11. Let R, S, T, X, Y, Z be arbitrary classes. Then (93) (94) (95) (96) (97) (98) (99) (100) R S V 2 R S V 2 R S = (R V 2 ) (S V 2 ) = (R V 2 ) S = R (S V 2 ) R V = R V 2 V R = V 2 R R S = (R V 2 ) (S V 2 ) = (R V 2 ) S = R (S V 2 ) R V = R V 2 V R = V 2 R Relative multiplication is normal, and relative addition is dual-normal (101) (102) Laws of properties (103) (104) (105) Schröder s Law [6, p. 150,152] R = = R V R = V 2 = R V S T V R = S (V R T ) R V S T = (S R V) T R V V S = R V S R V R V = V 2 R = Tarski s Law [7] R V = V 2 V R = V 2 Peirce s Remarkable Property [5] R is either or V 2 ( R) V is either or V 2 R V is either or V 2 V R V is either or V 2 ( -assoc, -assoc) Relative multiplication and relative addition are associative (106) (107) R (S T ) = (R S) T R (S T ) = (R S) T 7

8 8 ( -dist) Relative multiplication distributes over union R (S T ) = R S R T (S T ) R = S R T R ( -dist) Relative addition distributes over intersection (108) (109) R (S T ) = (R S) (R T ) (S T ) R = (S R) (T R) (duality) Relative multiplication and relative addition are dual (110) (111) R S = V 2 R S R S = V 2 (R S) ( -mon, -mon) Relative multiplication and relative addition are monotonic (112) (113) (114) (115) R S R T S T R S T R T S R S R T S T R S T R T S (Peirce s Law) Two formulæ so constantly used that hardly anything can be done without them Peirce [5] (116) (117) Miscellaneous interesting laws (118) (119) (120) (121) R (S T ) R S T (R S) T R S T R S (R T ) = R (S T ) (R T ) R S (R T ) = R (S T ) (R T ) R S (X Y ) (R X) S R (S Y ) R S T (X Y Z) (R X) S T R (S Y ) T R S (T Z) The Axiom of Converse says that for every class A there is a binary relation whose elements are all the ordered pairs obtainable from ordered pairs in A by interchanging left and right sides: A B x ( x B x is a set y z (x = y, z z, y A) ) The uniqueness of the converse of a class follows from the Axiom of Extensionality, so define the converse of A to be the class (122) A 1 := { y, x : x, y A} Note that A 1 V 2 whenever A 1 exists. With this notation the Axiom of Converse can be shortened to A B (B = A 1 ) or simply A (A 1 exists)

9 Theorem 12. Let R, S, T, P, Q be arbitrary classes. The converse of a class is a binary relation (123) R 1 V 2 ( 1 1 ) Conversion is an involution on binary relations (124) (R 1 ) 1 = R V 2, R 1 = (R V 2 ) 1 Conversion preserves the universal relation and the empty relation 9 (125) (126) V 1 = ( V 2 ) 1 = V 2 1 = ( 1, 1 ) Laws connecting conversion with relative multiplication and relative addition (127) (128) (R S) 1 = S 1 R 1 (R S) 1 = S 1 R 1 ( 1 -dist) Conversion distributes over union, intersection, difference, and symmetric difference (129) (130) (131) (132) (R S) 1 = R 1 S 1 (R S) 1 = R 1 S 1 (R S) 1 = R 1 S 1 (R S) 1 = R 1 S 1 ( 1 -mon) Conversion is monotonic. (133) R S R 1 S 1 Conversion commutes with relational complementation (134) Laws of properties (135) (136) R 1 = (V 2 R) 1 = V 2 R 1 (R V S) T = R (S 1 V T ) (R V S) T = R (S 1 V T ) (Th. K) DeMorgan s Theorem K [1] (137) R S T R 1 T S T S 1 R (R 10 ) Forms of Tarski s axiom (138) (139) (140) R 1 R S S S R R 1 R R 1 S S S R 1 R R 1 R S S S R R 1

10 10 (141) R R 1 S S S R 1 R (142) R 1 (R S) S (S R) R 1 (143) R (R 1 S) S (S R 1 ) R (144) R 1 (R S) S (S R) R 1 (145) R (R 1 S) S (S R 1 ) R Laws related to Peirce s Law and transitivity (146) R T (R S 1 ) (S T ) (147) R T R S S 1 T (148) R T (R S) (S 1 T ) (149) R T R S 1 S T (150) R T (R S 1 ) (S T ) (151) R T R S 1 S T (152) R T (R S) (S 1 T ) (153) R T R S S 1 T Formulas for equation-solving. If R, S, T V 2, then (154) R S T S R 1 T R T S 1 (155) R S T S 1 R T R T 1 S Tarski s equivalences (156) R S T 1 = T R S 1 = S T R 1 = Cycle law = R S T = T S 1 R (157) = T 1 R S 1 = S 1 R 1 T 1 = S T 1 R 1 = R 1 T S (rot) Modular laws and rotation (158) (159) (160) (161) (162) R S T = (R T S 1 ) (S R 1 T ) T R S T = R (S R 1 T ) T R S T = (R T S 1 ) S T R 1 S T = R 1 (S R T ) T R S 1 T = (R T S) S 1 T

11 11 Zigzag laws (163) (164) (165) (166) R S T R S 1 T R R R 1 R P Q R S T P Q 1 R S 1 T R R R 1 R R 1 R These laws can be proved by appealing to the definitions of the operations involved, or by equational proofs that rely on earlier laws. For example by various distributivity laws we have However, R S (X Y ) = (R (X X)) S (X Y ) = ((R X) S (X Y )) ((R X) S (X Y )) ((R X) S (X Y )) (R X) S. (R X) S (X Y ) (R X) (S (R X) 1 (X Y )) rot (R X) (S X 1 X ;Y ) (R X) (S Y ) R 10 R (S Y ) -mon so, combining this with the previous equation, we get R S (X Y ) (R X) S R (S Y ) 1 -mon, -mon The Axiom of the ɛ-relation asserts the existence of a binary relation consisting of all ordered pairs a, b for which the left side a is an element of the right side b, E x (x E a b (x = a, b a b)), or, briefly, { a, b : a b} exists. This class is unique by the Axiom of Extensionality, so let (167) E := { a, b : a b}, and also define (168) (169) Id := (E 1 E) (E 1 E) Di := E 1 E E 1 E E is the ɛ-relation, Id is the identity relation, and Di is the diversity relation. Theorem 13. (170) (171) (172) (173) Id = { a, a : a V} Di = { a, b : a b} E 1 E = { a, b : a b} E 1 E = { a, b : a b}

12 12 Id and Di partition the universal relation (174) (175) Id Di = V 2 Id Di = Id and Di are symmetric (176) (177) Id 1 = Id Di 1 = Di Id is an identity for relative multiplication, and Di is an identity for relative addition (178) (179) Id R = R V 2 = R Id Di R = R V 2 = R Di Subclasses of Id are symmetric relations (180) R Id R 1 = R (Id-laws) Identity laws (181) (182) (183) (184) Id R 1 R Id R 1 R Id R R 1 Id R R 1 (Di-laws) Diversity laws (185) (186) (187) (188) R R 1 Di R R 1 Di R 1 R Di R 1 R Di The axioms presented up to now (listed below) are enough for the Peirce-Schröder calculus of relations of the nineteenth century, which uses the empty set, complementation, intersection, union, ordered pairs, relative product, relative sum, converse, the unit relation V 2, the identity relation Id, and the diversity relation Di. The Axioms of Extensionality, Unordered Pairs, and the ɛ-relation are products of the twentieth-century set-theoretical setting. Extensionality: membership determines identity, Empty set: exists and is a set, Complementation: R exists, Intersection: R S, R S, and the universal class V exist, Pairs: the sets {x, y} and x, y exist for all sets x and y, Relative product: R S and the relative sum R S exist, Converse: R 1 exists,

13 13 ɛ-relation: the class E exists, so the classes Id and Di exist, The class R is said to be symmetric iff R 1 R, i.e., x y ( x, y R 1 y, x R). Theorem 14., Id, Di, V 2, R 1 R, R 1 R, R 1 R, R R 1, R 1 R, and R R 1 are symmetric. If R and S are symmetric, then R S and R S are symmetric. Let R V 2. R is transitive iff R R R, R is dense iff R R R. Theorem 15., Id, Di, and V 2 are transitive dense relations, and the following statements are equivalent R is a transitive binary relation, R R R V 2, R R R 1, R R 1 R. Theorem 16 (Peirce). R R 1 is transitive, Id R R 1, and the following statements are equivalent R is a transitive binary relation and Id R, R = R R 1. Theorem 17. For any class R, the following statements are equivalent: R R Id R V 2, R = R R 1, R = R 1 R, R = R 1 R, R = R R 1. R is an equivalence relation if R V 2, R is transitive, and R is symmetric. Theorem 18., Id, and V 2 are equivalence relations. If R and S are equivalence relations, then R S is an equivalence relation. Theorem 19. The following statements are equivalent: R is an equivalence relation, R R R R 1, R R = R = R 1, R R 1 = R, R 1 R = R, R V 2, R R 1 R, and R 1 R R, R V 2, R R R, and R R R.

14 14 For all classes R and S, let (189) R S := (R S) (R S). An ordered pair of sets a, b is in the relation R S iff the R-image of the left side a coincides with the S 1 -image (or S-preimage) of the right side b. Theorem 20. For all classes E, R, S, and T, (190) (191) (192) (193) (194) (195) (196) (197) (198) (199) (200) (201) (202) (203) (204) (205) (206) (207) (208) (209) (210) (211) (212) (213) (214) (215) (216) (217) R S = { x, y : z ( x, z R z, y S)}, Id = Id Id, Id = E 1 E, R S V 2, V 2 (R S) = R S R S, R S = R S, (R S) 1 = S 1 R 1, Id R R 1, Id R 1 R, R R 1 Di, R 1 R Di, R R 1 Di, R 1 R Di, R 1 (R S) S, (R S) S 1 R, R 1 (R S) S, (R S) S 1 R, R R 1 and R 1 R are transitive and symmetric, if R is transitive and symmetric, then R (R S) R S, if R is a transitive relation then (E 1 R) R (R 1 E) min E 1 E. (R E) (E 1 S T ) R S T, (R S E) (E 1 T ) R S T, (R E) ((E 1 S) T ) (R S) T, (R (S E)) (E 1 T ) R (S T ), (R E) ((E 1 S) T ) (R S) T, (R (S E)) (E 1 T ) R (S T ), (R S 1 ) (S T ) R T, (R S 1 ) (S T ) R T = R T.

15 R is functional iff no distinct ordered pairs in R with the same left sides have distinct right sides: (218) R is functional x y z ( x, y R x, z R y = z). R is a function if R V 2 and R is functional. Theorem 21. R is functional iff R R Di = iff R 1 R Id. R is injective or one-to-one if R 1 is functional. Injectivity can be characterized in a manner similar to the way functionality is characterized. Theorem 22. R is injective iff R R Di = iff R R 1 Id. Next we have two expected theorems, followed by a less obvious one. Theorem 23. If R is a functional class, then R R 1 is an equivalence relation. Theorem 24. If R and S are functional classes, then R S is a function. Theorem 25. If R and S are functions then R S is a function. Th. 25 can be proved without referring to more than three sides of ordered pairs in R and S at any one time, but the same cannot be said for Th. 24: it is necessary to simultaneously consider four objects at some stage of the proof. Define (219) R : = R (R Id), (220) R : = R (Id R), R is the functional part of R, and R is the injective part of R. Theorem 26. (221) (222) (223) (224) R R R V 2, R = R Id, R = Id R, The functional part is a function: R 1 = (R 1 ), R 1 = R (225) (226) (227) (228) R R Di =, R 1 R Id, R is a function, R = R R is a function. The injective part is an injection: (229) R Di R =,

16 16 (230) (231) (232) R R 1 Id, R is an injection, R = R R is an injection. The relation E is easy to express verbally. A set is in the relation E to another set if it is the sole element of the other set, i.e., E maps each set x to the singleton set {x}. Theorem 27. E = { x, {x} : x V} The next two theorems contain some other laws governing functional parts and injective parts. Theorem 28. If R is a class, then (233) R R Di =, (234) (235) (236) (237) (238) (239) R R Di =, R 1 R Id, R 1 R Id, R = R, R R = ( R (R V R 1 ) ) R, R R = ( R ( R V R 1 ) ) R. In general, relative multiplication does not distribute over intersection, but certain distributive laws do hold when some of the classes involved are functional or injective. Theorem 29. Let R, S, and T be arbitrary classes. Then (240) (241) If R is functional then (242) (243) If R is injective then R S R T = R (S T ) = R S R T, S R T R = (S T ) R = S R T R, R (S T ) = R S R T, (S T ) R 1 = S R 1 T R 1. R 1 (S T ) = R 1 S R 1 T, (S T ) R = S R T R.

17 17 The next main goal is to prove the existence of these two binary relations: { x, y, x : x, y V}, { x, y, y : x, y V}. It will be shown that these binary relations can be obtained from E and Id by using only,,,, and. Theorem 30 (Tarski [8, 3.9, p. 108]). Suppose C is any class. Let (244) (245) A := ( C C), B := C C (Id (C C A)). Then A and B are injections and the following statements are equivalent: x y (x V y V z ( w ( w, z C w = x w = y))), A B 1 = V 2. Tarski s construction is mentioned and applied several times in his manuscript Tarski [8, pp. 108, 111, 156, 209, 207, ]. Its converse dual, involving the functional part instead of the injective part, is used by Tarski-Givant [9, 4.6(ii)]. One of Tarski s variations on this construction comes close to our goal, which is Th. 33 below. A class C is called extensional iff (C 1 C) C 1 V C Id. For example, E is extensional. Theorem 31 (Tarski [8, p. 207]). Assume (244) and (245). Let (246) (247) A 1 := A V B (V C C) B 1 := B V A (V C C) If C is extensional, then the following statements are equivalent: (248) (249) x, z A 1 y, z B 1 u v ( w ( z, w C w = u w = v) w ( u, w C w = x) w ( v, w C w = x w = y)) Since E is extensional, it follows from Th. 31 that if C = E then (248) and (249) are also equivalent to x = y, z. Th. 31 requires the assumption that C is extensional. That more complicated constructions would succeed even without the assumption of extensionality was stated by Tarski-Givant [9, p. 130]. Such a construction is given below. We begin by defining four special operations, denoted by superscript suits, that may be performed on classes to obtain new classes. For an arbitrary class C, let C, C, C, and C be the classes defined as follows. (250) (251) (252) C := (C V C 1 ), C := C C, C := (C C Di),

18 18 (253) C := C C (C C (C C C ) ). These relations were designed to work properly under the assumption that C = E 1. Here are illustrations of the meanings of the first three relations obtained in case C = E 1. { { the only singleton {}}{ {x},... non-singletons (if any)... } (E 1 ) {x} the only singleton {}}{ {x},... non-singletons (if any)... } (E 1 ) { the only singleton Theorem 32. For every class C, (254) (255) (256) (257) (258) {}}{ the only non-singleton {}}{ {x}, y } (E 1 ) C, C, C, and C are functional binary relations. C C = (C C) C = C C, C is functional, C C = C (C C Di), C 1 (C (C C C ) Di C 1 ) C 1. Parts of Th. 32 were designed for the proof of the following theorem. Theorem 33. For every class C, and all x, y, z V, the following statements are equivalent: (259) (260) x, y C x, z C, u v ( w ( x, w C w = u w = v) w ( u, w C w = y) w ( v, w C w = y w = z)). If C = E 1 then (259) and (260) are also equivalent to x y (261) x = y, z. Proof. Assume that x, y C and x, z C. Since C = C C, there is some u V such that (262) x, u C and u, y C. Now C C and C C, so (263) x, u C and u, y C. Since C = C C C C (C C C ), the assumption that x, z C leads to two cases.

19 Case A. Assume (264) x, z C C (C C C ). Then x, z C C, and C C = C (C C Di) by (257), so there is some v V such that (265) x, v C and v, z C C Di. But C C, so (266) x, v C and v, z C. Now x, z (C C C ) by (264), z, v Di C 1 by (265), and x, v C by (266), so we have x, v C (C C C ) Di C 1. Also, y, x C 1, so y, v C 1 (C (C C C ) Di C 1 ). By (258), C 1 (C (C C C ) Di C 1 ) C 1, so (267) v, y C. Next we prove w ( v, w C w = y w = z). Consider an arbitrary w V. If w = y or w = z, then v, w C by (267) or (266), respectively. For the converse, suppose (268) v, w C. By (266) and (268), (269) x, w C C. If w = y then we are done, so suppose (270) y, w Di. Then x, w C Di by (270) and the hypothesis that x, y C. We have C Di V 2 C since C is functional, so x, w / C. From this and (269) we get (271) w, x (C C C ) 1. Now x, z (C C C ) by (264), so w, z (C C C ) 1 (C C C ) by (271). But (C C C ) 1 (C C C ) Id by (235), so w, z Id, i.e., w = z, as desired. Next we prove w ( u, w C w = y). Consider an arbitrary w V. First note that if w = y then u, w C by (263). For the converse, assume u, w C. From this assumption, (262), and (235), we get 19 so w = y, as desired. w, y C 1 C Id,

20 20 Finally, we show that w ( x, w C w = u w = v). Let w V. If w = u or w = v, then x, w C by (263) or (266), respectively. For the converse, assume (272) x, w C. If w = u we are done, so assume (273) u, w Di. From (262), (272), and (273) we get x, w C C Di. But x, v (C C Di) by (265), so by (235). Thus w = v, as desired. w, v (C C Di) 1 (C C Di) Id, Case B. Assume x, z C C. By (255), C C = (C C) C = C C, so (274) x, z C and x, z (C C) C. But C is functional and x, y C by hypothesis, so it follows from (274) that y = z and (275) x, y (C C) C. By (275), there is some v V such that (276) x, v C, x, v C, and v, y C. However, we have x, u C by (262), and C is functional, so (276) implies that u = v. Hence (277) x, u C and u, y C. Since u = v and y = z, all we need to show is that w (( x, w C w = u) ( u, w C w = y)). Let w V. We have x, u C and u, y C by (263), and also by (277). If x, w C then, by (277), w, u C 1 C Id, so w = u, and if u, w C then, by (277), w, y C 1 C Id, so w = y, as desired. This completes the proof of (260) from (259). To complete the proof, we must assume there are u, v V such that (260), i.e., w ( x, w C w = u w = v), w ( u, w C w = y), w ( v, w C w = y w = z)],

21 and show that (259) holds, i.e., x, y C and x, z C. This is easy but tedious, and we only do part of it. Suppose u = v. Then {y} = {y, z}, hence y = z. In this case, the assumptions are equivalent to w ( x, w C w = u), w ( u, w C w = y), hence also equivalent to (277). From (277) and (255) we get x, y = x, z C C = C C, as desired. Define binary relations P, Q, and V 3 as follows. (278) (279) (280) Theorem 34. (281) (282) (283) (284) (285) P := (E 1 ) (E 1 ) V, Q := (E 1 ) (E 1 ) V, V 3 := (E 1 ) V (E 1 ) V. P = { x, y, x : x, y V}, Q = { x, y, y : x, y V}, V 3 = { x, y, z : x, y, z V} = P V = Q V, V 2 = P 1 Q = Q 1 P = V P = V Q, Id = P P 1 Q Q 1, Theorem 35. For all classes A, B, X, and Y, A B X Y = (A P 1 X Q 1 ) (P B Q Y ). Next we prove an existence theorem for every relation that can be defined by a setbounded formula. We restrict our attention to those formulas whose variables all occur in some predetermined fixed set of 16 variables. The choice of this number of variables is merely an example. Any other small finite number of variables could be used instead, but powers of 2 are natural choices. The definitions in this section mimic the syntactic constructions of the Translation Mapping of Tarski-Givant [9, 4.4]. The Class Existence Theorem presented here (Th. 45) is essentially due to Bernays, Gödel, and Tarski. This theorem have been obtained by modifying the class existence theorem of Gödel [3, M2, p. 13] according to the finite axiomatizability results of Tarski-Givant [9, 6.4(iv)(v)(vi)]. 21 Define 16 relations as follows. (286) R 0 := P P P P, (287) R 1 := P P P Q, (288) R 2 := P P Q P, (289) R 3 := P P Q Q, (290) R 4 := P Q P P, (291) R 5 := P Q P Q, R 8 := Q P P P, R 9 := Q P P Q, R 10 := Q P Q P, R 11 := Q P Q Q, R 12 := Q Q P P, R 13 := Q Q P Q,

22 22 (292) (293) R 6 := P Q Q P, R 7 := P Q Q Q, R 14 := Q Q Q P, R 15 := Q Q Q Q. From these 16 relations we define 16 more relations A i with 0 i 15 by (294) A i := R j R 1 j. j i, 0 j 15 Define W to be the class of ordered pairs whose left sides are ordered pairs of ordered pairs of ordered pairs of ordered pairs: (295) W := R i V. By unwinding definitions we see that 0 i 15 (296) W = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q : a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q V}. If we refer to sets in the domain of W as 16-tuples and the image of a 16-tuple under R i as its i th place whenever 0 i 15, then we may describe A i as the set of pairs of 16-tuples that may differ only in the i th place. Theorem 36. Let 0 i, j 15 with i j. Then W = W V, Id = R i 1 (R i W ) = R i 1 R i, V 2 = R i 1 (R j W) = R i 1 R j, A i A i = A i = A i 1, A i A j = Id W, A i R j R j. By a set-bounded formula we mean any formula that can be obtained according to the following rules of construction. - Basic equality and membership statements are set-bounded if their variables are in V If ϕ and ψ are previously constructed set-bounded formulas and 0 i 15, then the following formulas are also set-bounded formulas: ϕ, ϕ ψ, ϕ ψ, ϕ ψ, ϕ ψ, ϕ ψ, vi (v i V ϕ), and vi (v i V ϕ). Let Φ 16 be the collection of set-bounded formulas. Set-bounded formulas are known from the literature as predicative. Associate a binary relation M(ϕ) with each setbounded formula ϕ in Φ 16 according to the following rules. If ϕ and ψ are predicative formulas in Φ 16 and 0 i, j 15, then (297) (298) (299) (300) M(v i = v j ) = (R i R j ) V W, M(v i v j ) = (R i E R j ) V W, M( ϕ) = W M(ϕ), M(ϕ ψ) = M(ϕ) M(ψ),

23 (301) (302) (303) (304) (305) (306) M(ϕ ψ) = M(ϕ) M(ψ), M(ϕ ψ) = M(ψ) M(ϕ), M(ϕ ψ) = M(ϕ) M(ψ), M(ϕ ψ) = ( M(ψ) M(ϕ) ) ( M(ϕ) M(ψ) ), M( vi (v i V ϕ)) = W A i M(ϕ), M( vi (v i V ϕ)) = W A i (W M(ϕ)). This definition of M is essentially the same as the one used by Tarski-Givant [9, 4.4(vii)]. For historical remarks concerning the definition of M, see Tarski-Givant [9, 4.3]. The next two theorems are essential preliminary theorems for the Class Existence Theorem, and could be called existence theorems for definable relations. Their proofs require neither the Axiom of Singletons nor the Class Union Axiom. (A comparable observation is made by Tarski-Givant [9, p. 182], namely, that S 5 has not yet been used at a certain stage in the proof of Tarski-Givant [9, 6.4(iv)].) The first one says that M(ϕ) is the class of ordered pairs x, y in which the second set y is arbitrary, while the first one, x, is a 16-tuple such that the formula ϕ holds when its variables are appropriately interpreted as the images of x under the 16 projection functions R 0,, R 15. Theorem 37. If ϕ is a 16-variable set-bounded formula in Φ 16 then (307) (308) M(ϕ) W, M(ϕ) = { v 0, v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8, v 9, v 10, v 11, v 12, v 13, v 14, v 15, y : ϕ} Proof. The proofs of both parts proceed by induction on the complexity of setbounded formulas. The proof of (307) is extremely simple. The right-hand sides of (297) (299), (305), and (306) are subclasses of W, while in (300) (304) this conclusion follows quite readily from the appropriate inductive hypothesis. We will deal with a few of the cases in the proof of (308). First suppose ϕ is the equation v i = v j, where 0 i, j 15. To prove (308) in one direction, assume that x, y M(v i = v j ). By definition (297) this means that x, y (R i R j ) V W, so (309) x, y (R i R j ) V and (310) x, y W. From (310) and (296) we conclude that there are sets v 0,..., v 15 V such that (311) R k (x) = v k whenever 0 k 15. By (309) there is some z V such that x, z R i R j, hence x, z R i and x, z R j. But R i and R i are functional, so we rewrite these last two statements using functional notation, obtaining (312) R i (x) = z, R j (x) = z. 23

24 24 From (311) and (312) we conclude that hence v i = R i (x) = R j (x) = v j, x, y { v 0, v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8, v 9, v 10, v 11, v 12, v 13, v 14, v 15, y : v i = v j }, as desired. For the other direction, suppose that x, y is in the set denoted by the expression on the right-hand side of the equation (308), i.e., there are sets v 0,..., v 15 V such that (311) and v i = v j. These hypotheses imply that x, y W and R i (x) = R j (x). Let z = R i (x) = R j (x). Then x, z R i, x, z R j, and z, y V, so x, y (R i R j ) V W = M(v i = v j ). Suppose ϕ is a 16-variable set-bounded formula and ϕ is the existential statement vi (v i V ψ) where 0 i 15. Assume, as inductive hypothesis, that (313) We wish to prove (314) M(ψ) = { v 0, v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8, v 9, v 10, v 11, v 12, v 13, v 14, v 15, y : ψ}. M( vi (v i V ψ)) = { v 0, v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8, v 9, v 10, v 11, v 12, v 13, v 14, v 15, y : vi (v i V ψ)}. Let us first prove the inclusion from left to right in (314). Suppose By (305) this gives us x, y M( vi (v i V ψ)). (315) x, y W and (316) x, y A i M(ψ). By (316) there is some z V such that (317) x, z A i and (318) z, y M(ψ). From (318) and (313) we conclude that there are sets v 0,..., v 15 V such that ψ and (319) R k (z) = v k whenever 0 k 15. From (317) and (294) we get (320) x, z R k R k 1 whenever i k and 0 k 15. We know from (315) that x is in the domain of R k whenever 0 k 15, so it follows from (319), and (320) that R k (x) = v k whenever i k and 0 k 15. Thus x is the same 16-tuple as z except that possibly R i (x) R i (z), and there does exist some set v i = R i (z) such that ψ, i.e., the conditions required for membership of x, y in the right-hand side of (314) do hold. The existential quantification occurring

25 implicitly in the relative multiplication in (316) asserts that a 16-tuple exists, hence 16 sets exist, except that 15 of them are constrained to be equal to previously given sets, so the quantification is, in effect, concentrated on the existence of the set v i V. The argument for inclusion in the opposite direction is equally elementary, and the argument for the case of universal quantification can be done similarly or reduced to the existential case. The only axioms required for the following Relation Existence Theorem are the Axioms of Extensionality, Empty Set, Complementation, Intersection, Unordered Pairs, Relative Product, Converse, and the ɛ-relation. Theorem 38 (Relation Existence Theorem). If ϕ is a set-bounded formula in Φ 16 in which the only free variables are v i and v j, 0 i, j 15, then the binary relation { v i, v j : ϕ} exists. In fact, (321) { v i, v j : ϕ} = R i 1 (R j M(ϕ)). Proof. This theorem follows in a straightforward manner from the previous one. Suppose we have a pair v i, v j such that ϕ. We may arbitrarily choose 14 more sets and construct a 16-tuple x V whose ith and jth components are v i and v j. By pairing this 16-tuple with another arbitrary set we get a pair which, by Th. 36, belongs to M(ϕ). It is then easy to deduce, from R i (x) = v i and R j (x) = v j, that this pair belongs to R 1 i (R j M(ϕ)). Conversely, if a pair u, w belongs to R 1 i (R j M(ϕ)), then there must be some x in the domain of M(ϕ) such that R i (x) = u, and R j (x) = w. Since x is in the domain of M(ϕ) it must also be in the domain of W and hence is a 16-tuple, and furthermore, by Th. 36, ϕ. The Axiom of Singletons says that for every class A there is a class C whose elements are the singletons whose elements belong to A: A C x (x C x V y (y A u (u x u = y))). Note that a, a = {{a}} for every set a V, so the Axiom of Singletons implies the existence of the identity relation on any class. Theorem 39. For every class A, { a, a : a A} exists. On the basis of the preceding theorem, we may, for every class A, let (322) A 1 := {{{a}} : a A} = { a, a : a A}, and say that A 1 is the identity relation on A, and A is the domain of A 1. The range of A 1 is the same as its domain. The Axiom of Singletons guarantees that every class is the domain of some binary relation. It also allows us to define the direct product of the classes A and B by (323) A B := A 1 V B 1. Theorem 40. For any two classes A and B, A B exists and (324) A B = { x, y : x A y B}. 25

26 26 A consequence of this theorem is Axiom B5, Gödel s axiom of direct product, which says that if C is a class, then V C exists. For every class A, the direct square of A is A A, also known as the Cartesian square of A. Let (325) (326) (327) A 2 := A A = A 1 V A 1, A 3 := A 2 A = (A 1 V A 1 ) 1 V A 1, A 4 := A 3 A = ((A 1 V A 1 ) 1 V A 1 ) 1 V A 1, and so on. According to (325), applied to V, V 2 = V V, but the unit relation V 2 was defined earlier by V 2 = V V. However, the two definitions agree, since V V = V V. Let R and A be arbitrary classes. R is a binary relation on A iff R A 2. The smallest binary relation on A is the empty set (which qualifies as a binary relation on A because none of its elements fails to be an ordered pair with sides in A). The largest binary relation on A is A 2 itself, and V 2 is the largest binary relation. The diversity relation on A is A 2 A 1. We say that R is a ternary relation on A iff R A 3, that R is a ternary relation iff R V 3, that R is a quaternary relation on A iff R A 4, and that R is a quaternary relation iff R V 4. Note that the left side of an ordered pair in a ternary relation is an ordered pair. The next theorem collects a few observations involving direct products. Theorem 41. Let A, B, C, D be classes. Then (328) (329) (330) (331) (332) (333) (334) (335) (336) (337) (338) (339) (340) (341) (342) (343) (344) (345) A 1 B = (A V) B, A V = A 1 V, B A 1 = B (V A), V A = V A 1, (A B) 1 = B A, A = = A, A B = A = B =, A B C D A C B D, B C (A B) (C D) = A D, (A B) (C D) = (A (B C)) ((B C) D), A (B C) = (A B) (A C), A (B C) = (A B) (A C), (A B) (C D) = (A C) (B D), (A B) (C D) = (A C) (B D), A (B C) = (A B) (A C), (A B) C = (A C) (B C), (A B) C = (A C) (B C), (A B) C = (A C) (B C).

27 The Axiom of Singletons allows us an alternative route to the existence of the identity relation Id, since V 1 = Id. The ability to form direct products, guaranteed by the Axiom of Singletons, provides somewhat simpler ways to construct the projection functions P and Q, as shown by the following theorem. Theorem 42. (346) (347) (348) (349) (350) (351) { x, y, x : x y} = (Di V) E 1 (E 1 ), { x, y, y : x y} = (Di V) E 1 E 1 E 1 (E 1 ), { x, x : x V V 2 } = Id V 3, { x, x, x : x V} = (Id V) E 1 E 1, P = V 3 E 1 (E 1 ), Q = (Id V) E 1 E 1 (Di V) E 1 E 1 E 1 (E 1 ), The addition of the Axiom of Singletons allows a parameterized version of the Relation Existence Theorem. Let C be a collection of letters denoting classes. The C-parameterized set-bounded formulas are defined by extending the definition of set-bounded formulas by including - basic statements of the form v i C with 0 i 15 and C in C. Let Φ C 16 be the collection of all C-parameterized set-bounded formulas. Such formulas are also called predicative. Associate a binary relation M(ϕ) with each C-parameterized set-bounded formula ϕ in Φ C 16, by extending the previous definition of M in (297) (306) with, for any parameter C in C and 0 i 15, this clause: (352) M(v i C) = R i (C V) W. Theorem 43. If ϕ is a 16-variable C-parameterized set-bounded formula in Φ C 16 then (307) and (308) hold. Proof. The proof is by induction on the complexity of C-parameterized set-bounded formulas, and is nearly the same as the proof of Th. 37. In the proof of (308) there is an additional base case that arises when ϕ is the statement v i C. To prove (308) in one direction, assume that x, y M(v i C). By definition (352), we have x, y R i (C V) W. Since x, y W, there are sets v 0,..., v 15 V such that R k (x) = v k whenever 0 k 15. Since x, y R i (C V), there is some z V such that x, z R i and z, y C V. But then v i = R i (x) = z C, hence (353) x, y { v 0, v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8, v 9, v 10, v 11, v 12, v 13, v 14, v 15, y : v i C}, as desired. For the other direction, if (353) holds, then there are sets v 0,..., v 15 V such that R k (x) = v k whenever 0 k 15 and v i C. These conditions imply that x, y W and R i (x) C. Let z = R i (x). Then x, z R i and z, y C V, so x, y R i (C V) W = M(v i C). 27

28 28 Theorem 44 (Parameterized Relation Existence Theorem). If 0 i, j 15 and ϕ is a 16-variable C-parameterized set-bounded formula in Φ C 16 whose only free variables are v i and v j then the binary relation { v i, v j : ϕ} exists, and (321) holds. The class A is a union of the class B if the elements of A are the elements of elements of B. The Class Union Axiom asserts that every class A has a union: B A x ( x A y (x y y B) ). By the Axiom of Extensionality, a class can have at most one union. The existence and uniqueness of a union justifies the introduction of notation for the union. For every class B, let (354) B := {x : y (x y y B)}. The class F is a field of B if x (x F y ( x, y B y, x B)). By the Axiom of Extensionality, a class can have at most one field. Existence follows from the Axiom of Union, for if B is a binary relation then its field is B. For an arbitrary class B, define its field to be the field of its relational part. (355) F d (B) := (B V 2 ). The Parameterized Relation Existence Theorem asserts the existence of any relation that is definable by a formula whose quantified variables are relativized to some class parameters. The parametric classes are represented as binary relations via the Axiom of Singletons. To prove the Class Existence Theorem we also need to be able to recover a class from some relation of which it is the domain. The Class Union Axiom does just that. In particular, the field of the identity relation A 1 is just its domain, namely A: A 1 = A. Applying the Class Union Axiom to the Parameterized Relation Existence Theorem gives the Theorem 45 (Class Existence Theorem). Suppose ϕ Φ C 16 is a 16-variable C- parameterized set-bounded formula in which the only free variable is v i. Then the class {v i : ϕ} exists. For every class R, let (356) R := E 1 R E. The symbol is used for this same purpose by Henkin-Monk-Tarski [4]. Theorem 46. For every class R, R is a function. Theorem 47. For all sets X, Y V, X, Y R Y = {y : x (x X x, y R)}.

29 The Replacement Axiom says that for every set s and every functional class F there is a set b, called the image of s under F, containing the right side of every ordered pair in F whose left side is an element of s, that is, s (s V F (F is functional b (b V y (y b z (z s z, y F ))))). In the context of the axioms already assumed, the next theorem is just a restatement of the Axiom of Replacement. Theorem 48. For every class R, (357) (358) R R Di = V 2 = R V, V 2 = ( R) V. Theorem 49. Every subclass of a set is a set. In particular, Id s V for every set s V. Proof. Let s be a set and let C be a class. First, note that C s = {x : x C x s} = {y : x s x, y Id C 2 } = (C 1 ) (s). It follows that C s is a set because, applying the Axiom of Replacement to the set s and the functional class C 1, we conclude that (C 1 ) (s) is a set. In case C is a subclass of s, i.e., C s, we have (C 1 ) (s) = C s = C, hence C is a set. Id s is the intersection of the class Id with the set s, and hence is a set. A class B is a set union of the class A if B is a set and B is the union of A, that is, A = B V. The Set Union Axiom asserts that the union of every set is a set: s (s V u (u V x (x u y (x y y s)))). Every class has a union, so the Set Union Axiom is equivalent to s (s V s V). However, for any sets u, s V, we have s = u s, u E 1 E 1 E, 29 so we may use the notation not only as an operation symbol on classes, but also as a class in its own right. Therefore, let (359) := E 1 E 1 E = ((E 1 ) ) 1, The Set Union Axiom is equivalent to the equation V 2 = V. Theorem 50. If x, y, R V then x y, x y, x y V and R V = V 2.

30 30 A class B is a powerset of a class A if B is a set, and the elements of B are the subsets of A, i.e., x (x B x A x is a set). The Powerset Axiom asserts that every set A has a powerset: A (A V B (B is a powerset of A)). The uniqueness of powersets follows from the Axiom of Extensionality, so for any set A V, we may define Sb (A) to be the powerset of A: (360) Sb (A) := {x : x V x A}. The Powerset Axiom is equivalent to A (A V Sb (A) V), and is also equivalent to the following equation in the calculus of relations. V 2 = ((E 1 E) E) V. Theorem 51. If x, y V then x y, x y, x 1 V. Proof. By Th. 50, x y is a set since x and y are sets. By the Powerset Axiom, applied twice, Sb (Sb (x y)) is a set. However, x y Sb (Sb (x y)), so x y is a set because every subclass of a set is a set. By the Union Axiom, applied twice, Sb (Sb (x y)) is a set, hence (Sb (Sb (x y))) (Sb (Sb (x y))) is also a set by what we have proved so far. However, (x y) (x 1 ) (Sb (Sb (x y))) (Sb (Sb (x y))), so x y and x 1 are sets. The Axiom of Infinity is that an infinite set exists. There are many ways to say this. In the version used by Gödel [3], the Axiom of Infinity states that there is a nonempty set C such that every element of C is a proper subset of another element of C: C ( x (x C V) x (x C y (x y y C))). Gödel s axiom of infinity can also be expressed as an equation in the calculus of relations, such as ( (E V 2 = V E V ( 1 E E 1 E ) )) E E E V. The strong form of the Axiom of Choice is that there is universal choice function, i.e., a functional binary relation that picks an element of every nonempty set: C ( C is functional x (x is a nonempty set y (y x x, y C)) ). Let C be a class whose existence is guaranteed by the Axiom of Choice. Recall, for comparison, that the Axiom of the ɛ-relation asserts the existence of the class E = { x, y : x y}. Using E, the conditions satisfied by C can be expressed more succinctly in the calculus of relations. Here is one way: C C Di =, C E 1, C V = E 1 V. The Axiom of Choice is equivalent to the assertion that every binary relation contains a function with the same domain. Here are two well-known consequences of the Axiom of Choice.

31 Theorem 52 (The Well-Ordering Theorem, Zermelo [12, 13]). Every set can be well ordered. 31 Proof. Let s be any set. From the Axiom of Choice first deduce the existence of a function g : Sb (s) {s} s such that g(x) s x for every proper subset x s. To obtain g, just compose complementation relative to s with the universal choice function: (361) g := Q 1 (((P { s, s } E 1 ) Q E 1 E) V 3 ) C. Let K Sb (s) be the intersection of all sets X satisfying the following conditions: x X x {g(x)} X, Y X Y X. K exists by the Class Existence Theorem and the form of its definition by a setbounded formula: (362) K := {k : X (X V ( x (x X x {g(x)} X) Y (Y X Y X)) k X)}. Then s = K and s is well-ordered by R, the relation that holds between a s and b s iff there is some k K such that a k and b / k, that is, (363) R := E K 1 E 1 s 2. Theorem 53 (Zorn s Lemma [14]). If R is a partial ordering of X and every linearly ordered subset of X has an upper R-bound, then there is a maximal element for R. The Axiom of Regularity states that every nonempty class has an element with which it has no common elements: A ( b (b A) b(b A x (x / A x / b))). This axiom is essentially due to J. von Neumann [11, p. 231, Axiom VI 4]; the version used here is due to P. Bernays; see [3, p. 7]. It was proved consistent relative to the other axioms by von Neumann [11]. As Gödel [3, p. 6] said, it is not indispensable, but it simplifies considerably the later work. Using it, we can now prove that the class of all sets is not a set. Theorem 54. For every class X, X X. In particular, V / V. Proof. Suppose X X. Then X V, hence {X} V. Every element of {X} has an element in common with X, namely X itself, contradicting the Axiom of Regularity.

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