# Can algorithms of arbitrarily worse complexity be systematically created?

We’ve all seen this:

Can we get worse?

Part 1: Can mathematical operations of increasing orders of growth be generated, with or without Knuth’s up-arrow notation?

Part 2: If they can, can algorithms of arbitrary complexities be systematically generated?

Part 3: If such algorithms can be generated, what about programs implementing those algorithms?

Yes, of course. $$2^{f(n)}$$ is asymptotically larger than $$f(n)$$, so you can come up with an unending sequence of larger and larger running times.
On the one hand, every algorithm (by which I will mean a Turing machine which halts on all inputs) has computable runtime. On the other hand, by diagonalization for every computable function $$f$$ there is a computable set $$X_f$$ such that no Turing machine computing $$X_f$$ runs in time $$O(f)$$.