80-110 Nature of Mathematical Reasoning

Russell (Handout #8) 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".)

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.

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)?

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)

Please do not spend too much time trying to figure out the "correct" answer, but write what you think about it in a paragraph or two.

Do either problem 14 on page 23 of FOL, or the problem on the extra sheet.