ABSTRACT
This chapter provides examples of paradox exercises in Mathematical Imagining. While reasoning exercises seek to make a mathematical idea credible to the students, the purpose of paradox exercises is to confront learners with an (apparently) inescapable contradiction. Each exercise is designed around a conflict between different kinds of knowledge (intuitive, experiential, mathematical), which students then investigate and expand. In order to accommodate the infinitely many new guests, one option is to have the existing guests each move into a room whose number is twice as large. All of the odd-numbered rooms are then free and can accommodate the new guests. The instructions are geared toward the visual mental image of a hotel with infinitely many rooms and the mental images of finitely and infinitely many new guests. With the concept of cardinality, the concept of the number of elements is generalized to infinite sets.
