## 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.