a list compiled by Alex Kasman (College of Charleston)
For those who may not know, "Aleph" is the first letter of the Hebrew alphabet (a close relative of "a" and "alpha"). This was used as a mathematical symbol by Georg Cantor who needed a notation for the different "sizes" of infinity. Specifically, Aleph with a subscript of "zero" is the cardinality (or size) of the set of all whole numbers. As Cantor famously proved, other infinite sets may be "larger" than this in the sense that there is no one-to-one correspondence between the sets. For instance, the set of all fractions has the same size (again Aleph Sub Zero) as the whole numbers because they can be paired up in a surprising but sensible way. On the other hand, the set of all real numbers (decimal expansions) is larger because no matter how you may try to pair it up with the whole numbers, some of real numbers are left unpartnered. (This is the "diagonalization" argument.) So, we need other sizes of infinity, which Cantor denotes by increasing the subscript on the Aleph. Aleph sub one (for instance) is the next largest size of infinity (one size larger than Aleph sub zero). Now, if you're hearing this for the first time, you may be wondering about the relationship between Aleph Sub One and the previously mentioned "size of the set of real numbers". Good question...and it's not one I can adequately address in this location. By definition, Aleph Sub One cannot be larger than the cardinality of the reals. But, it could be smaller, or it could be equal. The assumption that the latter is true goes by the name "the continuum hypothesis", and I will summarize the status here only by saying that the question of which is true (if that question has any meaning) is not resolvable by standard set theory.
Anyway, in this short story by St. Clair, Oona experiments with Aleph Sub One using a machine that "visualizes" mathematics for the user, with disastrous consequences.
Originally published in Startling Stories, Vol. 16, No. 3, January, 1948. (Part of St. Clair's "Oona & Jick" series of humorous short stories about everyday life in the future.) Reprinted in the anthology "New Eves: Science Fiction About the Extraordinary Women of Today and Tomorrow", edited by Janrae Frank, Jean Stine, and Forrest J. Ackerman, Longmeadow Press, 1994.
Thanks to Fred Galvin for bringing this interesting story to my attention!
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(Maintained by Alex Kasman, College of Charleston)