In the late 1880s Georg Cantor arrived at a theory of infinite sets leading to the notion that the infinite has hierarchical structure. It was somewhat difficult for others to understand at the time. Cantor himself wrote of it in an 1877 letter to Dedekind "Je le vois, mais je ne le crois pas!" Evidently, Kronecker neither saw nor believed — the mathematical giant denounced Cantor at every opportunity. Cantor's work culminated in what became known as the Continuum Hypothesis (CH), which sufficiently impressed Hilbert that he gave it first among his famous 23 problems. The CH matter was settled by latter-day giants Gödel and Cohen — they proved it was undecideable. Despite his fundamental contributions to set theory, Cantor's life sadly ended in a mental institution. In this talk we'll take a tour of Cantor's transfinite world and explore some of its implications. If you're looking for applications, this talk is not likely for you.