Georg Cantor (1845-1918), a German mathematician, first dared to think the counter-intuitive notion that not all infinite sets have the same size--and then he proved it: The set of all real numbers (including all of the decimal numbers representable on the number line) cannot be matched in a one-to-one pairing with the set of counting (or natural) numbers -- 1,2,3,4, . . . . Sets whose elements can be matched one-to-one with the counting numbers are termed "countable" -- and Cantor's result showed that the set of all real numbers is uncountable.
Cantor developed an extensive theory of transfinite numbers -- and poet (as well as philosopher and professor) Emily Grosholz reflects on these in a poem:
Reflections on the Transfinite by Emily Grosholz
Reading about the tower or great-boled tree
of ordinals, I think how Cantor grew
more wise and more insane, trying to save
his tree of Jesse from the pruning shears
and kitchen gardening of Kronecker;
though I must share the latter's feeling for
the natural numbers, those deceptively
well-ordered, step-wise creatures, which appear
transparent as then mount, but all in all
among themselves are most unknowable.
Dreaming about the cardinals, at night,
the alephs flaming like a candelabrum,
I see you in the attic of your house
installed between Diogenes Laertius,
Nachgelassene Schriften, commentary
on Aristotle, Plato, and the latest
fashionable fact-shredders out of Paris.
So cloudy is the place you occupy
in the last hierarchies of my world
that I hardly discern you; yet I know
you are not just a postulate I made.
You are the great collection of desires,
forever incomplete, unsatisfied,
toward which all finite sequences in time
with little steps so trustfully aspire.
Though you outrank them all, see how they run
like atomies of fire toward the sun,
sent over the abyss with no alarm
to make the leap across into your arms.
"Reflections on the Tranfinite" first appeared in The Kenyon Review and later in Grosholz's collection, The River Painter (University of Illinois Press, 1984); it may also be found in the anthology Verse and Universe; Poems about Science and Mathematics (Ed. Kurt Brown, Milkweed Editions, 1998). Grosholz holds positions at Penn State University and at the University of Paris Denis Diderot. Author of five books of poems, her current work-in-progress is Mathematics and Poetry, a collection of essays.
Cantor's connection to infinite sets has been mentioned in earlier blog postings on July 24 and September 14.