This is a fun talk I will give at Temple University for the mathematics undergraduates in the Senior Problem Solving forum. We’ll be exploring some of the best puzzles and paradoxes I know of that arise with large numbers and infinity. Many of these paradoxes are connected with deep issues surrounding the nature of mathematical truth, and my intention is to convey some of that depth, while still being accessible and entertaining.

**Abstract**: Are there some real numbers that in principle cannot be described? What is the largest natural number that can be written or described in ordinary type on a 3×5 index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every true statement provable? Does every mathematical problem ultimately reduce to a computational procedure? Is every sentence either true or false or neither true nor false? Can one complete a task involving infinitely many steps? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics.