Here's probably the simplest manifestation of the field-with-one-element phenomenon. Define a projective n-space of order q to be a collection of points, lines, planes, etc. satisfying the usual incidence relations with the additional condition that every line has q+1 points on it, every plane has q^2+q+1 points on it, and so forth. For q a prime power, all such spaces come from the usual definition of projective n-space Pⁿ(F_q) over a finite field.

But a projective n-space of order 1 is precisely the Boolean algebra of subsets of a set with n elements!

(This example is due to Henry Cohn, and it has the virtue that any theorems one wants to prove in this abstract setting don't depend on the value of q.)

Here's probably the simplest manifestation of the field-with-one-element phenomenon. Define a projective $n$-space of order $q$ to be a collection of points, lines, planes, etc. satisfying the usual incidence relations with the additional condition that every line has $q+1$ points on it, every plane has $q^2+q+1$ points on it, and so forth. For $q$ a prime power, all such spaces come from the usual definition of projective $n$-space $\Bbb P^n(\Bbb F_q)$ over a finite field.

But a projective $n$-space of order $1$ is precisely the Boolean algebra of subsets of a set with $n$ elements!

(This example is due to Henry Cohn, and it has the virtue that any theorems one wants to prove in this abstract setting don't depend on the value of $q$.)