Homework #3

• Exercise 8.1 - To show the axiomatic system is consistent, one can construct a "model", that is a concrete realization of a set of "ags" and "trads" with an operation "kow" relating them.

  One can put the ags in one column and trads in another and graphically apply the axioms.

• Exercise 8.4 - Use induction, that is assume it is true for c=1, assume it is true for arbitrary c, then show it is true for c'.

• Example 8.5 - The "Nested Interval Theorem" states that if I₁, I₂,... is a sequence of nested, closed, bounded, nonempty intervals then the intersection is nonempty. If the length of the intervals approaches 0, then the intersection consists of a single point.

  Apply this to an enumeration of the set of all rationals on [0,1].

  Note, this procedure depends on the particular enumeration chosen, and results in one irrational, and is essentially non-constructive. Cantor's diagonalization argument can produce an infinite number of irrationals quite easily!