Math 570: Mathematical Logic
Fall Semester 2004
Prof. Ward Henson
Monday, September 27, 2004
Problem Set 2 – Due in class Monday, October 4, 2004.
(Problem 5(b) revised 10-1-04)
There are five problems (25 points each) and you should do four of them. To
earn full credit requires a careful writeup of each problem, taking care to justify
everything you claim and to explain your ideas clearly.
2.1. Problem. Let Lbe the first-order language whose non-logical symbols
consist of one binary function symbol fand one constant symbol c. Let Abe the
L-structure (N,+,0). Show that there is no L-term t(x) such that tA(n)=2n
holds for every n∈N.
2.2. Problem. Let Lbe the first-order language with a binary relation sym-
bol Pand no other non-logical symbol. Consider the following three L-structures:
A= (R, <); B= (Z, <); C= ([0,1] <). (Here [0,1] ⊆Ris the unit interval.)
For each of these three structures, give an L-sentence that is true in the given
structure and false in the other two. (Your proofs that the exhibited sentences
have the required properties can be informal and sketchy; explain carefully what
mathematical property you intend for each of your sentences to express.)
2.3. Problem. Let Lbe the first-order language with a binary relation sym-
bol Eand no other non-logical symbol.
(a) Exhibit an L-sentence σwhose models are exactly the L-structures Asuch
that EAis an equivalence relation on A.
(b) Let Abe a model of σsuch that Ais finite. Show that there is an L-sentence
τsuch that for any L-structure B, one has B |=τif and only if B∼
=A.
(c) Show that if Ais a model of σand Ais infinite, then no sentence τwith the
property given in part (b) can exist.
2.4. Problem. Let Lbe the first-order language whose only non-logical sym-
bol is the binary relation symbol <. Let Σ be the set of all L-sentences σsuch
that σis true in every finite linear ordering.
(a) Show that Σ has an infinite model.
(b) Show that not every linear ordering is a model of Σ.
2.5. Problem. In this problem you may not use the Completeness Theorem;
your proofs should be given using basic properties of our proof system for first-
order logic. You may use the facts proved in Section 3.1 of the classnotes. Let
Lbe any first-order language with a constant symbol cand a unary relation
symbol P; let ϕ(x), ψ(x) be L-formulas. Show that:
(a) ` ∀x(ϕ∧ψ)→(∀xϕ ∧ ∀xψ);
(b) `P c → ∀x(x=c→P x);
(c) ` ∀xϕ(x)→ ∃xϕ(x).