CHAPTER 5. THE INCOMPLETENESS THEOREM
1. Introduction
G
odels famous Incompleteness Theorem states that there are sentences
true on N which are not consequences of PA. In fact, this holds for any
reasonable set of axioms in place of PA, as we will explain
Math 445: Exam 3
Fall 2013
Important: You may (and really should!) use any Axioms and/or Lemmas (from the attached
sheet, which you may tear off) you wish unless a problem specifies otherwise. If you use an Axiom
or Lemma please specify which one(s) you u
MATH 445 Exam I 2 October 2009
Work each numbered problem on a separate answer sheet. Show all your
work for each problem clearly on the answer sheet for that problem. Write
your name and the problem number on each answer sheet. Good luck.
1. [15 pts] Use
Math 445: Exam 1
Fall 2013
Directions: You may only use completeness/compactness on question 9. On all other questions you
may use whatever you want unless specified.
1. Find all ways that parentheses can be added to A B C in order to make it a sentence.
APPENDIX: AXIOMS FOR ARITHMETIC ON THE
NATURAL NUMBERS
1. Introduction
In this appendix we give an axiomatic system for arithmetic on the natural numbers, that is, the set N of non-negative integers. The arithmetic
functions we consider are s (successor),
CHAPTER 4. COMPUTABILITY AND DECIDABILITY
1. Introduction
By definition, an n-ary function F on N assigns to every n-tuple k1 , . . . , kn
of elements of N a unique l N which is the value of F at k1 , . . . , kn ,
F (k1 , . . . , kn ). In most common exam
CHAPTER 3. THE COMPLETENESS THEOREM
1. Introduction
In this Chapter we prove G
odels Completeness Theorem for first order
logic.
Theorem 1.1. (Completeness Theorem) Let SnL .
(a) is satisfiable iff is consistent.
(b) For any SnL , iff |= .
By Soundness (T
CHAPTER 2. FIRST ORDER LOGIC
1. Introduction
First order logic is a much richer system than sentential logic. Its interpretations include the usual structures of mathematics, and its sentences enable
us to express many properties of these structures. For
CHAPTER 1. SENTENTIAL LOGIC
1. Introduction
In sentential logic we study how given sentences may be combined to form
more complicated compound sentences. For example, from the sentences 7
is prime, 7 is odd, 2 is prime, 2 is odd, we can obtain the followi