Transfinite Induction

Alex Gurevich

I race
past my destination,
so I can walk backwards
to where I long to be.
Touching the hollow wash,
I learn
how much thirst I slaked.
I must 
crest the hill 
to measure
the yearning of my ascent.
I stand
on the roof of my house
to count its bricks.
The end of my journey 
lies beyond 
any finite thought.
To calculate the dimensions of love,
I start
by tasting Infinity.

The Science

Imagine an infinite staircase. You know two facts: the first step is safe; and if all the preceding steps are safe, then the following step will be safe as well. Transfinite induction then tells you that the whole staircase is safe, even the parts of it that can't be reached by a finite journey. This principle has applications in computation theory.

The poem is inspired by the study of fast-growing hierarchies (FGH) — iterative functions which allow construction of numbers so large they cannot be described by any notation of finite mathematics. The pace of growth of such a function can only be described by an infinite ordinal number and manipulated with ordinal, rather than finite, arithmetics. In fact, the growth becomes so rapid that regular, finite mathematics can no longer prove each value in the sequence remains finite. Curiously, such structures can arise from combinatorial explosions related to patterns of social media interactions.

Transfinite induction works backwards from an infinitely high step of the growth staircase and shows that it is theoretically computable if all the previous steps are computable.

The Poet

Alex Gurevich holds a Ph.D. in Mathematics from the University of Chicago. A hedge fund manager by day, he has published several science fiction and fantasy stories as well as two investment books.

Next poem: two weeks into retirement, sally ride runs errands and contemplates speed as an empty vector by Jordan Muscal