MATH 570: MATHEMATICAL LOGIC
HOMEWORK 5
Due date: Sep 29 (Tue)
Below, problems with are not mandatory.
1. Prove that a graph is 3-colorable1 if and only if so is every nite subgraph of it.
Hint: To be able to express the 3-colorability property, add to th
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 3
Due date: Sep 17 (Wed)
1. Let A B and assume that for any finite S A and b B, there exists an automorphism
f of B that fixes S pointwise (i.e. f (a) = a for all a S) and f (b) A. Show that A B.
2. Show that (Q, <) (
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 2
Due on Sep 10 (Wed)
1. Let be a -sentence. The finite spectrum of is the set
cfw_n N+ there is M with M = n,
where N+ is the set of positive integers.
(a) Let = (E), where E is a binary relation symbol, and let be t
MATH 570: MATHEMATICAL LOGIC, FALL 2015
HOMEWORK 1
Due date: not sure yet, no earlier than next Tuesday (Sep 1)
1. Dene appropriate signatures for vector spaces over Q.
2. If h A B is a -homomorphism then the image h(A) is a universe of a substructure
of
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 2
Due on Sep 8 (Tue)
Below, problems with are not mandatory (for fun).
1. (a) A formula is called universal (resp. existential ) if it is of the form x1 x2 .xn
(resp. x1 x2 .xn ), where is quantier free. Let A, B be
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 3
Due date: Sep 15 (Tue)
Below, problems with are not mandatory.
1. (Chasing denitions) A -theory T is said to be semantically complete if for any -sentence
, T or T .
(a) Show that any satisable -theory T has a satis
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 4
Due date: Sep 22 (Tue)
Below, problems with are not mandatory. You may use the Completeness and Compactness
theorems starting from Problem 4.
1. Carefully prove the Constant Substitution lemma.
2. Pick two of the fo
Math 570: MATHEMATICAL LOGIC
Fall 2015, MWF 12:00pm-12:50pm in 445 Altgeld Hall
ANUSH TSERUNYAN1
The course will introduce the main ideas and basic results of mathematical logic from a fairly
modern prospective, providing a number of applications to other
FIRST ORDER LOGIC AND GODEL INCOMPLETENESS
ANUSH TSERUNYAN
Contents
1. Introduction
2. First order logic: the semantic aspect
2.A. Structures
2.B. Language and interpretation
2.C. Denability
2.D. Theories, models, and axiomatization
2.E. Semantic version
MATH 570: MATHEMATICAL LOGIC
HOMEWORK 1
Due on Wednesday, Sep 3
1. Define appropriate signatures for
(a) vector spaces over Q;
(b) metric spaces.
2. (a) Show by an example that in the signature group = (1, ), a substructure of a group
need not be a subgro