UNIVERSITY OF LONDON BA EXAMINATION for Internal Students This paper is also taken by Combined Studies Students PHILOSOPHY Optional Subject (i): Set Theory and Further Logic Answer THREE questions, at least ONE from EACH section SECTION A 1. (i) Show that not all instances of the following schema are true, where F (x) is schematic for a predicate:. B x[x B F (x)]. (ii) Prove that there is no universal set. (iii) Prove that for any set A there is no set of everything not in A. (iv) Prove that if there is a set, there is an empty set. (v) What is it for a two-place predicate Ψ(x, y) to be univalent on a set B? Prove that the predicate x = {y} is univalent on any non-empty set B. State the Axiom Schema of Replacement and use it to show that there is no set of all unit sets. 1
2. (i) Let x y abbreviate there is a bijection from x to y. Let A be a non-empty set and let R be the relation { x, y : x, y PA & x y}. Prove that R is an equivalence relation. You may assume that the empty relation is a bijection from to itself. (ii) For any subset x of A, let [x] R = {y : x, y R}, where R is as given in (i). Let C = {[x] R : x PA}. If A has exactly three members, how many members does C have? Give your reasons. (iii) Let S be a transitive relation. Prove that S is irreflexive if and only if it is antisymmetric: For all x and y, x, y S y, x S. (iv) Let T be a (strict) partial ordering that is connected, in the sense that for all x, y in the field of T, x y x, y T y, x T. Prove that T is a (strict) total ordering. 3. (i) What is a function? Define the inverse F 1 of a function F. Define the composition G F of functions G and F. What is it for a function to be (a) an injection and (b) a surjection from A to B? (ii) Prove that for any function F, F 1 is a function if and only if F is an injection. (iii) Prove that if G and F are functions such that ran(f ) dom(g), (a) G F is a function and (b) dom(g F )=dom(f ). (iv) Let I A be the identity function on A. Prove that F 1 F = I dom(f ). (v) Prove that for any sets A, B and C, if A B and B C, then A C, where x y is used as in 2(i). 4. (i) Let x y be used as in 2(i) and let S be the cardinality of S. Assuming that A = B if and only A B, show the following: If A B = and Y Z = and A = Y and B = Z, then A B = Y Z. (ii) Define the operations of cardinal addition and cardinal multiplication and prove that cardinal multiplication is associative: ( A B ) C = A ( B C ). 2
(iii) Prove that cardinal addition distributes over cardinal multiplication: For any sets A, B and C, ( A + B ) C =( A C ) + ( B C ). 5. (i) What is the von Neumann successor A + of a set A? What is an inductive set? Define the predicate NN(x) (read x is a natural number ); define ω. (ii) Given that ω is a set, prove that it is an inductive set. (iii) Prove that ω is a subset of every inductive set. (iv) State and prove the principle of proof by induction on ω. You may assume that ω is a set. 6. (i) What is a total ordering? Show that for any set A with at least two members, the relation of inclusion on the power set of A, { x, y : x, y PA & x y}, is not a total ordering. (ii) What is it for a set to be well-ordered by a relation? Give an example of a set B and a relation R on B such that B is totally ordered by R but not well-ordered by R, and explain why it is not well-ordered by R. (iii) What is a transitive set? What is an ordinal? Prove that every member of an ordinal is an ordinal. (You may assume that if B A and membership on A, A, is a well-ordering, then membership on B, B, is a well-ordering.) (iv) Assuming that a transitive set of ordinals is an ordinal, prove that if A is an ordinal so is its von Neumann successor, A +. 7. (i) Prove that there is an injection from a set A to its power set PA. (ii) Prove that there is no bijection from a set A to its power set PA. (iii) Let C D. What is the characteristic function of C in D? For any set A, let H be the function with domain PA such that for any B PA, H(B) = the characteristic function of B in A. Show that H is a bijection from PA to A 2. (iv) Define cardinal exponentiation: C D. Prove that A < 2 A. (You may assume that for any cardinal number κ, κ = κ.) 3
8. (i) Prove that any infinite ordinal α is equinumerous to its successor α +. (ii) Define the cardinal A of a set A. What is it for an ordinal to be a limit ordinal? Prove that every infinite cardinal is a limit ordinal. (iii) State the theorem that legitimates definition by transfinite recursion on the ordinals. Define the aleph operator ℵ by transfinite recursion on the ordinals. (You may assume that for any set B of cardinals there is an infinite cardinal not in B.) (iv) Prove that for any ordinals α and β, if α β then ℵ α < ℵ β. SECTION B 9. Let three frames (G 1,R 1 ), (G 2,R 2 ).(G 3,R 3 ), be defined by G 1 = {a}, R 1 = { a, a, b, b } G 2 = {a, b}, R 2 = { a, b, b, a, b, b } G 3 = {a, b}, R 3 = { a, a, a, b, b, b } ( x, y in R i means xr i y). For each of the three fames determine which of the following formulas is/are valid on the frame. (a) P P (b) P P (c) P P (d) P P Moreover, if one of the formulas φ is not valid on frame (G i,r i ), give a world x in G i and a forcing relation between G i and {P } such that x φ. 4
10. Use the propositional tableau proof systems to test the following three formulas for validity (a) ( φ φ) in the T system (b) ( φ ψ) ( ψ φ) in the system S4, (c) ( φ ψ) ( ψ φ) in the system S5, Use the constant domain tableau system to determine whether (d) is valid on all K models with constant domain (d) ( ( x)a(x) ( x)b(x)) ( x) (A(x) B(x)). Use the variable domain tableau system to determine whether (e) is valid on all K models with varying domain (e) ( ( x)a(x) ( x)(b(x)) ( x)(a(x) B(x)). 11. (i) What is a model for quantified modal logic that allows domains to vary over worlds? With respect to such models specify the semantic rules (i.e. clauses in the truth-definition) for atomic sentences, quantified sentences, and modal sentences. Let a one-place predicate E (x) be defined thus: E (x) y(y = x). Describe a model in which x E (x) is true at some world. (ii) Show that the Barcan formula xp (x) x P (x) (a) is not valid in some model of S4 satisfying the inclusion requirement that if wrw,d(w) D(w ) but (b) is valid in every model of S5 satisfying the inclusion requirement. (iii) Show that the converse of the Barcan formula, namely x P (x) xp (x), is valid in every model for quantified modal logic satisfying the inclusion requirement. (iv) When the operator is interpreted as necessarily in the metaphysical sense, should we accept (a) the Barcan formula, and (b) its converse? Justify your answers, making clear what they imply for constraints on variation of domains over possible worlds. 5
12. Let (G, R) be the following frame: G = {0, 1, 2, 3...}, the set of natural numbers, and nrm holds if n is a smaller number than m. Let (G, R, ) be a model over (G, R). In such a model, n P, means that natural number n has some property φ P, that is, the model interprets P as a specific property of natural numbers (eg., n is smaller than a given number a, n is a multiple of 3, n is 4, 7 or 18, etc.). (i) For each of the following formulas give a property Φ P of natural numbers such that the formula is valid on the model (G, R, ) (a) P P (b) P P (c) (P P ) (ii) Argue that ( P Q) (( P Q) (P Q) (P Q)) is valid on the frame (G, R). (iii) Let nqm mean that number n is larger than number m. Give a formula that is valid on the frame (G, Q) but not on (G, R). 13. Considering, as temporal operators, where P is interpreted as P is and will always be the case give a formalization using predicate abstraction of the sentence Someday the prime minister won t be the prime minister (anymore) and discuss how a logical contradiction is avoided. 6 END OF PAPER