UNIVERSITY OF EAST ANGLIA. School of Mathematics UG End of Year Examination MATHEMATICAL LOGIC WITH ADVANCED TOPICS MTH-4D23

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics UG End of Year Examination MATHEMATICAL LOGIC WITH ADVANCED TOPICS Time allowed: 3 hours Attempt Question ONE and FOUR other questions. Candidates must show on each answer book the type of calculator used. Do not turn over until you are told to do so by the Invigilator Copyright of the University of East Anglia

2 (a) What does it mean to say that a ring A ;,,0,1 is a Boolean algebra? Say how to define the ordering and the operations of least upper bound x y and greatest lower bound x y on a Boolean algebra A (with x,y A). Suppose X is any set. Describe the usual operations on P(X), the power set of X, which make it into a Boolean algebra. What do,, correspond to in this case? (c) Let F be a finite set with at least two elements and let X F be the compact space F N consisting of all functions from N to F, with basic open sets of the form O a 1, K, a n ; b 1, K,b n ={ fîx F : f a i =b i for i=1, K, n} where a 1, K, a n ÎN, b 1, K,b n ÎF. Describe the subsets of X F which are both closed and open, and show that the set B of these forms a Boolean subalgebra of P(X F ). (iii) Show that B is countable and atomless. Explain how the fact that any two countable atomless Boolean algebras are isomorphic allows you to deduce that X F is homeomorphic to X F, where F is any other finite set with at least two elements.

3 The formal system L for propositional logic has axioms: A1 f y f A2 f y c f y f c A3 Øf Øy y f (for propositional formulas f, y, c ), and deduction rule Modus Ponens. (a) Show that the axioms of L are tautologies. Suppose Γ is a set of L-formulas and ψ is an L-formula. Define what is meant by saying that there is a deduction of ψ from Γ (abbreviated as Γ ψ). Show that if Γ ψ and v is a valuation of L with v(φ) = T for all φ Γ, then v(ψ) = T. (c) State the Deduction Theorem for L and show that ((φ ψ) ((ψ χ) (φ χ))) is a theorem of L (whenever f, y, c are propositional formulas). TURN OVER

4 The formal system L for propositional logic is described in Question 2. (a) What is meant by an extension L* of L? What does it mean for an extension of L to be complete, and consistent? You may assume the following result: If L* is a consistent extension of L and φ is a formula which is not a theorem of L*, then the extension of L* obtained by including ( φ) as an extra axiom is consistent. Show that any consistent extension L* of L has a consistent extension L** which is complete. (c) Using, outline a proof that any tautology is a theorem of L.

5 The first-order language L has a 2-ary relation symbol R, a constant symbol c and a 1-ary function symbol f. (a) Describe the terms and atomic formulas of L, and say how to construct the formulas of L. Give L-structures A,B such that the following formula is true in A and not true in B : x rsub { size 8{1} } \) \( \( $x rsub { size 8{2} } \) R \( x rsub { size 8{1} },x rsub { size 8{2} } Explain your answer briefly. (c) In each of the following, two L-structures A i and B i are given (for i = 1,2). In each case, find a closed L-formula φ i which is true in A i but not in B i. Explain your answer. The domains of A 1 and B 1 are both Z, the set of integers. In both A 1 and B 1 the constant symbol c is interpreted as 0 and R(x 1,x 2 ) is interpreted as x 1 < x 2. In A 1, the function symbol f is interpreted as the function x 1 ax 1 4 ; in B 1, it is interpreted as the function x 1 ax 1 3. The domains of A 2 and B 2 are both Z, the set of integers. In both A 2 and B 2 the constant symbol c is interpreted as 0 and R(x 1,x 2 ) is interpreted as 'x 1, x 2 are congruent modulo 5'. In A 2, the function symbol f is interpreted as the function x 1 ax 1 2 ; in B 2, it is interpreted as the function x 1 ax TURN OVER - 6 -

6 5. (a) Suppose L is a first-order language, φ is an L-formula, and A is an L-structure. Define, in terms of valuations satisfying φ, what it means for the valuation v in A to satisfy ( x i )φ (in A). The notation ( x i )φ is shorthand for ( ( x i )( φ)). Show that the valuation v in A satisfies ( x i )φ if and only if there is a valuation v in A which is x i -equivalent to v and which satisfies φ. State Gödel's Completeness Theorem for the formal system K L (you need not give a description of K L ). State the Compactness Theorem. Explain how both these results can be deduced from the following result: If S is a consistent extension of K L, then there is an L-structure A such that A = φ for all closed formulas φ which are theorems of S

7 6. (a) What is meant by a normal model of a set Γ of closed formulas in a firstorder language with equality L =? State the Compactness Theorem for normal models. For n a positive integer, write down a closed L = -formula σ n whose normal models are precisely the normal L = -structures with at least n elements. (iii) Suppose Γ 1 is a set of closed L = -formulas with the property that for every positive integer n, there is a normal model of Γ 1 with at least n elements. Show that Γ 1 has an infinite normal model. Suppose L = is a first-order language with equality which has a 2-ary relation symbol R(x 1,x 2 ) (as well as a 2-ary relation symbol for equality). Let Q be the normal L = -structure with domain Q, the set of rational numbers, and R(x 1,x 2 ) interpreted as x 1 x 2. Write down a closed L = -formula χ which is true in Q and has the property that any countable normal model of χ is isomorphic to Q. Justify your answer briefly. Give a normal model A of χ which is not isomorphic to Q. (iii) Explain briefly why, for every closed L = -formula φ A = φ Q = φ. END OF PAPER