## Professor Andre Scedrov

Office: Room 4E6 in David Rittenhouse Laboratory
Telephone: eight five nine eight three ( Math. Dept. Office: eight eight one seven eight )
Fax: three four zero six three
E-mail: lastname at math
Office Hours: By appointment

From Encyclopaedia Britannica: Gödel's proof, which states that within any rigidly logical mathematical system there are propositions (or questions) that cannot be proved or disproved on the basis of the axioms within that system and that, therefore, it is uncertain that the basic axioms of arithmetic will not give rise to contradictions. This proof has become a hallmark of 20th-century mathematics, and its repercussions continue to be felt and debated.

## Textbooks

• D. van Dalen. "Logic and Structures", Third Edition. Springer-Verlag, 1994. ISBN 0-387-57839-0 paperback.
• G.S. Boolos and R.C. Jeffrey. "Computability and Logic", Third Edition. Cambridge University Press, 1989. ISBN 0-521-38923-2 paperback.

## Topics

• Propositional Logic: Propositions and Connectives, Semantics, Natural Deduction, Completeness.
• Predicate Logic: Quantifiers, Structures, Semantics, Natural Deduction, The Completeness Theorem, Compactness and Skolem-Löwenheim Theorems, Skolem Functions.
• Undecidability and Incompleteness: Turing Machines, Undecidability of Predicate Logic, Gödel's First and Second Incompleteness Theorems.

## Homework

• ### Homework Due in Class Wednesday, February 2

• Let n and k be two distinct natural numbers. Prove that there is no 1-1 correspondence between a set with n elements and the set with k elements.

• Prove that there is no 1-1 correspondence between a finite set and an infinite set.

• Relying on the 1-1 correspondence between N and the set of non-negative rationals shown in class, describe a 1-1 correspondence between N and Q.

• van Dalen, Exercises 2.2, p. 58: # 1 all, 2.

• van Dalen, Exercises 2.3, pp. 65-66: # 1, 2, 3, and 4.

• ### Homework Due in Class Monday, February 28

• van Dalen, Exercises 2.5, pp. 78-79: # 9, 10, 12, 14, and 15

• van Dalen, Exercises 2.6, p. 80: # 2, 3, and 4.

• van Dalen, Exercises 2.7, pp. 88-89: # 6, 11, and 13.

• van Dalen, Exercises 1.4, pp. 38-39: # 1e, 2a, 3e, 4 both, and 5.

• van Dalen, Exercises 2.8, pp. 94-95: # 1 (ii) and 1 (vii) where x is not free in phi.

## Take-Home Midterm Due in Class Monday, March 27

Please start working on the midterm immediately after turning in the homework on February 28.

• Prove that the square root of 2 is not rational.

• van Dalen, Exercises 2.7, pp. 88-89: # 10 and 12.

• van Dalen, Exercises 3.2, pp. 117-118: # 6, 7, 8, 9, 10, 12, 13, 16, and 17.

This is a complete list of midterm assignments due March 27.

## Take-Home Final Exam Due in DRL 4E6 Monday, May 8 at 4 p.m.

Please start working on the final exam immediately.

• van Dalen, Exercises 3.3, pp. 132-136: # 5, 10, 11, 12, 15, 17, 18, 19, 20, 26, 27, 28(iii).

• van Dalen, Exercises 3.4, pp. 142-143: # 3, 4.

• Boolos and Jeffrey, Exercise 6.1, p. 66.

• Boolos and Jeffrey, Exercise 14.1. p. 168.

• Boolos and Jeffrey, Exercise 15.1, p. 180.

• Boolos and Jeffrey, Exercises 16.4 and 16.6, p. 190.

This is a complete list of midterm assignments due Monday, May 8.