# Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics":

Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that infinitely many values of $f(n)$ are palindromes when written in base $10$. Show that $\mathcal{P}$ cannot be proved or disproved in Zermelo-Fraenkel set theory.

Stanley writes that it's there just "to point out that some simply stated facts about posets may be forever unknowable". It's also stated that it hasn't been solved.

Is it possible to give some intuition about why this result might be true in terms that can be understood with very basic knowledge of logic and model theory?