## 80-110 The Nature of Mathematical Reasoning

**Spring 2001**

*Dirk Schlimm*
# Homework No.6

Thursday, February 22, 2001

Due Tuesday, February 27, 2001

**1. Russell on axioms** (2 points)
Russell (Handout #15) accepts Dedekind and
Peano's analysis of the natural numbers, but wants to go further than
them. At the end of the Chapter he raises two objections against
considering Peano's axioms "as an adequate basis for
arithmetic". State Russell's objections in your own words.

(If you're curious about Russell's own definition of number, here it
is: "A number is anything which is the number of some class", where
"the number of a class is the class of all those classes that are
similar to it".)

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

Read *The Language of First-Order Logic* (FOL): Sections 2.4-2.7
(p. 15-22).

On page 20, solve problems 11 and 12.

On page 23, solve problems 15 and 16.

Hand in your answers on
a disk (in PC/IBM format). Name your files exactly as it is specified
in the problem.

**3. Let's Make a Deal!** (3 points)

Suppose you are on a game show in which there are 3 doors, exactly one
of which will contain a terrific prize. The door that
contains the prize will be decided randomly before the show, and each
door has an equal chance of containing the prize. You
are asked to pick a door, but you are not shown what is behind your
door.

To be concrete, suppose you choose door 1. Your
host then shows you one of the doors you *did not* pick, the only
restriction being that the door you are shown must be *empty*.
Say you are shown that door 2 does not have the prize.

Assuming you
want to maximize the chances for getting a great
prize, the question is:

Do you want to *stay* with your original pick
(door 1) or *switch* (to door 3)?

Pick one of the following
answers, and justify your pick.

A) Stay with original pick (door 1 has a better chance of having
the prize than door 3).

B) Switch (door 3 has a better chance of having the prize than door 1).

C) It doesn't matter (door 1 and door 3 have the same chance of
having the prize)

**4. FOL or Models** (2 points)

Do *either* problem 14 on page 22 of FOL, *or* the problem
on the extra sheet.