80-110   The Nature of Mathematical Reasoning

Spring 2001

Dirk Schlimm

Homework No.5

Thursday, February 15, 2001
Due Tuesday, February 20, 2001

1. Axiom systems and Models (4 points)

Read the Handout #11, Axiom systems. Models. Consistency and Independence. (The material covered in sections 3.4-3.5 will be covered later in more detail using Tarski's World).

Make exercises 26-29 on page 89 of the handout. Remember to justify every little step in your proof.

Make exercises 1 and 2 on page 97 of the handout.

2. Reflections on Geometry (3 points)

Think back at the discussion of the development of mathematics in the first lectures and read also Handout #13, Space and the Geometrization of Mathematics. In the past lectures geometry was introduced as a mathematical theory based only on a few axioms. Describe the transition from the science of land measuring to the formulation of axioms for geometry in a few paragraphs. Consider in particular the following questions: Why do you think this transition happend; what are the advantages? Has something been lost in this transition?

Write this exercise in the form of a letter to a friend from high school who is now working as a painter in New York City.

3. The Language of First-Order Logic (3 points)

Read The Language of First-Order Logic (FOL): Sections 1.1-1.3 and 2.1-2.3 (p. 1-13).

Solve problems 1-4 on pages 13-14.
Hand in your answers on a disk (in PC/IBM format). Name your files exactly as it is specified in the problem.

If you don't have the book yet, find somebody in the class who has it and either borrow the book or do this exercise together. If you work together in a group, you need to hand in only one disc, but in addition to the above problems, hand in also exercises 5, 6, and 7 on page 14.