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This page lists various googological functions arranged roughly by growth rate. They are grouped roughly by what theories are expected to prove them total recursive, and individual functions are also compared to the fast-growing hierarchy.

  • \(\approx\) means that two functions have "comparable" growth rates in some unfixed sense.
  • \(>\) means that one function significantly overgrows the other.
  • \(\geq\) means that it is not known exactly whether one function overgrows the other or not.
  • (limit) means that the function has many arguments, and the growth rate is found by diagonalizing over them.

Be careful that for Computable functions A and B, even if the weakest theory among those which are known to prove the totality of B can prove the totality of A, B does not necessarily eventually dominate A. Similarly, the uncomputability of a function does not mean that it eventually dominates all total computable functions. (It's possible to make an uncomputable function that only outputs 0 or 1.) In other words, the order of this list does not mean the order of the growth rate.

Primitive recursive[]

These functions are all bounded by primitive recursive functions and it is believed that most can be proved total within the theory of primitive recursive arithmetic (PRA).

RCA0[]

The totality of these functions cannot be proved in RCA0 (see second-order arithmetic) and they eventually dominate all primitive recursive functions. The fast-growing hierarchy here is given by Wainer hierarchy.

PA[]

The following functions eventually dominate all multirecursive functions but are still provably recursive within Peano arithmetic (PA). The fast-growing hierarchy here is given by Wainer hierarchy.

ATR0[]

Starting from here, the following functions eventually dominate all hyperrecursive functions and the totality of these functions is not provable in PA. The fast-growing hierarchy here is given by Veblen hierarchy.

ZFC[]

These functions cannot be proved total in arithmetical transfinite induction but are believed to be provably total in \(\textrm{ZFC}\). It does not mean that the totality is actually verified, and actually the list contains functions whose totality or even computability is not known in the current googology community.

The fast-growing hierarchy here below the Small Veblen ordinal is given using Veblen function and beyond that we use Buchholz's function and Extended Buchholz's function. Beyond the Countable limit of Extended Buchholz's function, we use the tautological norm on the ordinal notation associated to Rathjen's psi for expressions including the function. On the other hand, \(\psi\) with inputs larger than \(\varepsilon_{\Omega_{\omega}+1}\) with respect to Buchholz's function (or \(\Omega_{\Omega_{._{._.}}}\) with respect to Extended Buchholz's function) in estimations are unspecified (and possibly ill-defined) OCFs, and hence those estimations are meaningless. Also, these estimations are not necessarily verified, and could be simply expectations, i.e. conjectural one without proofs.

Stronger set theories[]

These functions cannot be proven total in \(\textrm{ZFC}\), but are known to be provably total in stronger set theories.

Uncomputable functions[]

These functions are uncomputable, and cannot be evaluated by computer programs in finite time.

Other[]

The following are functions which are not single maps:

  • Slow-growing hierarchy, Hardy hierarchy, Fast-growing hierarchy. These three hierarchies can be extended indefinitely, as long as ordinals and their fundamental hierarchies can be defined. Although they are literally uncomputable with transfinite indices, their segments can be interpreted into computable functions if ordinals are replaced by terms in an ordinal notation and fundamental sequences are given by an algorithm on expressions.
  • Bowers' Exploding Array Function (BEAF). BEAF is not formalized and well-defined beyond tetrational arrays, so there are multiple interpretations by other googologists than the creator. A conjectural growth rate of a desired interpretation heavily depends on analysts.
  • Strong array notation (SAN). SAN is not formalized, so there are multiple interpretations by other googologists than the creator. A conjectural growth rate of a desired interpretation corresponds to C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),0),0),0)[3] in Taranovsky's ordinal notation using Hyp_cos's fundamental sequences, but there is no actual proof.
  • Username5243's Array Notation (UNAN). The original definition of UNAN is ill-defined due to domain-related issues. The creator's expectation of the limit of the notation (assuming that UNAN is well-defined) is approximately \(f_{\Gamma_0}(n)\) in the website. In addition, the creator also continued to extend the notation beyond dimensional first-order array notation level to beyond nested array subscripts (see introduction and analysis of UNAN from basic array notation level to array subscript notation level here), which its limit is expected to reach the fast-growing hierarchy ordinal level of the countable limit of Extended Buchholz's function (\(>f_{\psi_0(\Omega_{\Omega_{...}})}(n)\)).
  • Lossy channel systems and priority channel systems. The complexity classes of some decision problems are googologically large, but no single fast-growing function or number has been extracted from these.
  • TR function. It is a hierarchy of functions indexed by formal theories, or a function on the pair of a formal theory and a natural number.
  • Revised Pehan Notation

References[]

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