Eventual domination

Eventual domination is a relation describing the asymptotic behavior of two functions. The function \(f\) is said to eventually dominate \(g\) iff \(f(n) > g(n)\) for all sufficiently large \(n\). That is, \(f\) asymptotically outgrows \(g\).

Theorems of the form "\(f\) eventually dominates every function with property \(P\)" are useful in formally giving a sense of scale for fast-growing functions. A common choice for \(P\) is the property of being provably recursive in certain formal theory, like Peano arithmetic or ZFC. In hypercomputable scenarios, \(P\) can be the property of being computable within a system such as Turing machines.

Over \(\mathbb{N} \mathbb \mathbb{N}\), eventual domination is transitive and total. It is not antisymmetric, since we can construct two different functions such that neither dominates the other. Thus it is not a, and is not isomorphic to any "linear" number system.