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Math 570 – Mathematical Logic Prof. Ward Henson Final Exam – December 18, 2004
Problem: 1 2* 3 4* 5 6* Points 20 20 20 20 20 20 Score:
Problem: 7 8 9* 10* Total Points 25 15 20 20 200 Score:
There are ten (10) problems on this Exam and you should do all of them. They are weighted as indicated.
To receive full credit, each solution must be explained and justified as fully as possible. Write your solutions carefully and precisely.
In justifying your proofs, unless the instructions say otherwise, you may cite any result from the text or proved in class; however, you may not cite homework problems.
See separate sheets for definitions of the proof systems for propositional logic and first order logic, for the axioms of the system N, for the definition of computable function and relation, and for a statement of computability facts related to G¨odel numbering.
- The score you earned on the midterm exam will be replaced by your total score on problems 2, 4, 6, 9, and 10 here, if that is higher.
Problem 1. (20 points) Let L be a first order language with a finite number of nonlogical symbols, and let Σ be a finite set of L-sentences. Assume that Th(Σ) is undecidable. Prove the following:
(a) (8 points) Σ has at least two non-isomorphic countable^1 models. (b) (6 points) For each L-sentence σ, either Th(Σ ∪ {σ}) is undecidable or Th(Σ ∪ {¬σ}) is unde- cidable. (c) (6 points) Σ has infinitely many non-isomorphic countable models.
Problem 2. (20 points) Let L be any first order language that includes the constant symbol c. Suppose Σ is a set of L-sentences and that ϕ(x) is an L-formula. Assume that c does not occur in any member of Σ and that Σ ϕ(c). Without using the Completeness Theorem, show that Σ ∀x ϕ(x).
Problem 3. (20 points) Let L be the first order language whose only nonlogical symbols are the (distinct) constant symbols cn for n ∈ N. Let Σ be the set of all L-sentences of the form cm 6 = cn where m, n ∈ N and m 6 = n.
(a) (8 points) Show that Th(Σ) is complete. (b) (12 points) Let A be any model of Σ. Characterize the subsets of A that are definable in A by an L-formula ϕ(x).
Problem 4. (20 points) Let L be a first-order language and let Σ be a maximal consistent (in L) set of L-sentences. Let σ and τ be L-sentences. Without using the Completeness Theorem, prove that Σ L σ ∨ τ if and only if ΣL σ or Σ `L τ.
Problem 5. (20 points) Let L be the first order language whose only nonlogical symbol is a unary function symbol F. Consider the L-structures A = (R, S) and B = (Q, S|Q), where S is defined for all x ∈ R by S(x) = x + 1.
(a) (7 points) Which sets X ⊆ R are definable in A by an L-formula ϕ(x)? (b) (13 points) Show that A and B are elementarily equivalent.
Problem 6. (20 points) Let L and L′^ be first order languages such that L ⊆ L′. Let Σ be a set of L-sentences and let σ be an L-sentence such that Σ L′^ σ. Prove that ΣL σ.
Problem 7. (25 points) (a) (5 points) Define the concept representability in N for relations R ⊆ Nk. (b) (5 points) Define the concept representability in N for functions f : Nk^ → N. (c) (7 points) Suppose A, B ⊆ Nk^ are representable in N; show that their intersection A ∩ B is representable in N. (d) (8 points) Suppose f, g : N → N are representable in N; show that their composition f ◦ g is representable in N.
(^1) countable means finite or countably infinite
