Logic Comprehensive Examination
Math 570 (formerly Math 410)
May 25, 2005
There are five problems and you should do all of them. In doing a part of any problem, you may
assume the results of any earlier part of the same problem, whether or not you correctly worked it.
Each problem is worth 20 points, for a total of 100 points. To receive credit, each of your solutions
must be justified.
Notation and terminology:Ldenotes a countable first order language (with equality, as a logical
symbol) and Σ denotes a set of L-sentences. For each Σ, Th(Σ) denotes the set of L-sentences σ
such that there is a formal proof of σfrom Σ in L(i.e., Σ `Lσ). The symbols N,Z,Qand R
denote (respectively) the sets of all natural numbers (including 0), all integers (positive, negative,
and zero), all rational numbers, and all real numbers. A set Sis countable if there is a bijection
between Sand a subset of N; in particular, all finite sets are countable.
Problem 1. Recall that an L-formula is existential if it has the form ∃x1. . . ∃xnϕwhere ϕis
quantifier free. Assume that every L-formula of the form ∀xψ, in which ψis quantifier free, is
Σ-equivalent to an existential L-formula.
(a) (10 points) Show that every L-formula is Σ-equivalent to an existential L-formula.
(b) (10 points) Show that if Mand Nare models of Σ with M ⊆ N , then M N ; that is, for
each L(M)-sentence σ, show that M |=σiff N |=σ.
Problem 2. Let Lbe the language of abelian groups; so Lhas binary function symbols + and
−for addition and subtraction, respectively, and a constant symbol 0 for the identity element.
Consider the additive abelian groups Zand Z×Zas L-structures.
(a) (10 points) Show that the set {(a, a)|a∈Z}is not definable in Z×Zby any L-formula ϕ(x).
(b) (10 points) Find an L-sentence that is true in Zbut false in Z×Z. (Hint: think modulo 2.)
Problem 3. Specify a first order language Land L-structures Mand Nsuch that Mand Nare
elementarily equivalent, Mhas exactly one automorphism, and Nhas infinitely many automor-
phisms.
Please turn over for the remaining problems.
