Bop-counting function

The bop-counting function is a function defined by Harvey Friedman, arising in study of binary operations on structures.

Definition
Let \(D\) be an arbitrary set. Any subset \(R\subseteq D^3\) is called a ternary relation on \(D\). We call two ternary relations defined on sets \(D,D'\) respectively equivalent if there is a bijection \(f:D\rightarrow D'\) such that \((a,b,c)\in D\Leftrightarrow (f(a),f(b),f(c))\in D'\).

\(R\) is called a binary operator (bop for short) if every \((a,b)\in D^2\) uniquely determines \(c\in D\) such that \((a,b,c)\in R\) (so that \(R\) is a function \(D^2\rightarrow D\)). Now, for \(k\in\Bbb N\) a \(k\)-restriction of \(R\) is a restriction of \(R\) to a set \(E^2\) such that \(|E|=k\). Let's call two bops \(k\)-similar if their possible \(k\)-restrictions are, up to isomorphism, the same. It can be easily shown that \(k\)-similarity is equivalence relation.

Finally, define bop-counting function \(\theta(k)\) for \(k\in\Bbb N\) to be the number of equivalence classes under \(k\)-similarity.

Properties
By counting the number of possible (up to isomorphism) \(k\)-restrictions of a set, it can be show that \(\theta(k)\) is a function which is at most triply exponential in \(k\) (i.e. it can be bounded by function of the form \(2^{2^{2^{cn}}}\) for some constant \(c\)). Despite that, Friedman has shown that this function is not recursive. This presents algorithmic difficulty in finding values of this function.

A more striking result regards a specific value \(\theta(12)\). Indeed, Friedman has shown that not even very powerful formal theories, including ZFC plus any known large cardinal axioms, is capable of proving what the exact value of \(theta(12)\) is, meaning that for no natural number \(N\) such systems can prove "\(theta(12)=N\)".