These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students.

In high school mathematics, one encounters about 11 basic laws for addition, multiplication and exponentiation of the natural numbers 1,2,3,... .

For example, for any number x, we have x1=x=1x, while for numbers x,y,z we have x^{y+z}=x^yx^z.

In the 1960's, the famous logician Alfred Tarski asked whether or not there are any laws in addition, multiplication and exponentiation that are true for the natural numbers but do not follow from applications of the familiar `high school' ones.

Surprisingly, the answer to Tarski's problem is `yes'. In fact there is no finite collection of laws of this kind that are sufficient to derive all such laws.

In this talk we examine Tarski's problem and how to prove it. While we will encounter some abstract ideas along the way, the talk will assume little more than, well, some high school algebra!