A retired mathematician and his artist wife are joined by two alien mathematicians (one looking like a pencil and the other looking like a toad) to investigate questions about infinity. Where the aliens are from, the abstractions that the mathematician has studied have a physical reality, manifested for example in infinitely branching plants. When the wife is carried up to higher levels of infinity by a lightning storm, the three mathematicians test the husband's theory of the divisibility of space in order to rescue her.

As the story explains well (but very briefly), although people previously naively believed that "infinity is infinity", Georg Cantor proved rather conclusively that there are actually different sizes of infinity. In particular, if we define two sets to be "the same size" if the elements of one can be paired up with the elements of the other (which makes sense if you imagine pairing up the apples in one bag with the oranges in another to see whether the bags contain the same number of fruits), you get some surprising results. It turns out, for example, that the set of integers and the set of rational numbers are the same size, while the set of real numbers is demonstrably larger . And, there are not just two sizes of infinity, there are infinitely many sizes. What was not clear to Cantor was whether there were sets of intermediate size. For example, he pondered the question whether there is a set whose size is smaller than the size of the reals but larger than the size of the integers. The "Continuum Hypothesis" is the name attached to the claim that there is no such intermediately sized set, and many brilliant mathematicians worked to prove either that it was true or false. From my perspective, this question was long ago answered in a strange way when the question was shown to be formally undecideable in the sense of Gödel. To me, this means that it is not a question with a mathematical answer, and more specifically that one gets a consistent set of mathematical axioms regardless of whether one assumes the Continuum Hypothesis is true or false (assuming, of course, that math is consistent to begin with). However, many with a different philosophy of math still seek some way to decide the question by reference to the physical world since it cannot be decided using only mathematics.

Whatever one's philosophy, if you can understand the question and deal with the bizarre fantastical aspects, this is a fun ride that allows you to feel (at least temporarily) as if these extremely abstract mathematical concepts can be seen, felt and heard.

Thanks to Rich Schroeppel for suggesting that this story be added to the database! I am not sure of its publication history. If you have any information, please let me know.