Fast Growing Hierarchy

The extended function of transcendental integers, which is denoted by \(\textrm{TR}\), is a family of computable large functions coined by Googology Wiki user Fish. It extends the computable function which naturally arises from the definition of transcendental integer.

Definition
Let \(T\) be a formal theory with a fixed embedding of an arithmetic, and \(n\) a natural number. Then \(\textrm{TR}(T,n)\) is defined as the least integer \(N\) such that for any Turing machine \(M\), if the termination of \(M\) is provable in \(T\) within \(n\) symbols, then \(M\) actually halts within \(N\) steps.

Explanation
We are working in a base theory such as \(\textrm{ZFC}\) set theory, and considering \(T\) as a formal theory coded in the base theory. For each Turing machine \(M\) in the base theory, there is a known way to code \(M\) in an arithmetic, and hence in \(T\). Therefore the termination of \(M\) in \(T\) naturally makes sense. In this way, \(\textrm{TR}\) function generates a computable function on \(n\) for each formal theory \(T\) with a fixed embedding of an arithmetic.

This function \(\textrm{TR}\) itself is not total, because there are inconsistent formal theories. For example, suppose that the base theory is consistent, \(T\) is Peano arithmetic augmented by the disprovable formula \(0 = S0\), and \(M\) is non-terminating. By the principle of explosion, the termination of \(M\) is provable in \(T\). If \(n\) is greater than or equal to the minimum of the symbols of a proof of the termination of \(M\) in \(T\), then there is no integer \(N\) such that \(M\) halts within \(N\) steps, because \(M\) does not halt. Therefore \(\textrm{TR}(T,n)\) is ill-defined in this case.

Even if \(T\) is consistent in the sense \(\textrm{Con}(T)\) holds in the base theory, then \(T\) might proves the termination of a non-terminating Turing machine. In order to ensure the well-definedness of \(\textrm{TR}(T,n)\) for any \(n\), we need to assume a strong assumption called the \(1\)-soundness of \(T\) in the base theory. If we just want to define \(\textrm{TR}(T,n)\) for a specific \(n\), e.g. \(2^{1000}\), then we just need a weaker assumption that for any Turing machine \(M\), if the termination of \(M\) is provable in \(T\) within \(n\) symbols, then \(M\) actually halts.

For example, if \(T\) is \(\textrm{ZFC}\) set theory, then \(\textrm{TR}(T,n)\) is total under the assumption of the \(1\)-soundness of \(\textrm{ZFC}\) set theory in the base theory, and \(\textrm{TR}(T,2^{1000})\) coincides with the least transcendental integer. That is why \(\textrm{TR}\) is called the extended function of transcendental integers.

Specialisation
Fish coined a specific function called \(\textrm{I}0\) function as \(\textrm{TR}(\textrm{ZFC}+\textrm{I}0,n)\). Here, \(\textrm{I}0\) denotes the axiom of the existence of a rank-into-rank cardinal, which is a very strong large cardinal axiom. As Friedman does not coin a specific transcendental integer, Fish does not coin a value of \(\textrm{I}0\) function.

Analysis
By the definition, \(\textrm{TR}(T,n)\) grows faster than any computable function which is provably total in \(T\). It implies that if a given computable total function is "known to be total", then it is bounded by \(\textrm{TR}(T,n)\) for a specific choice of \(T\). For example, almost all known total computable function is bounded by \(\textrm{I}0\) function.

Although it is arguable whether it is a naive extension of the notion of a transcendental integer, it is significant because it explicitly gives an explanation that a stronger theory directly yields a larger number in a further stronger theory, as Fish pointed out. Therefore it is reasonable to fix and clarify the base theory if we work in a theory stronger than \(\textrm{ZFC}\) set theory. Otherwise, any total computable function is weaker than \(\textrm{TR}\) function in the sense above.

A function like \(\textrm{TR}(T,n)\) with a specific \(T\) is sometimes "approximated" to \(\textrm{PTO}(T)\), i.e. the proof-theoretic ordinal of \(T\), in the fast-growing hierarchy, but the "approximation" does not make sense because the fast-growing hierarchy is well-defined not for an ordinal but for a tuple of an ordinal and a system of fundamental sequences. Unlike smaller ordinals, \(\textrm{PTO}(T)\) does not possess a fixed system of fundamental sequence, and hence the comparison is meaningless. Since the fast-growing hierarchy heavily depends on the choice of a system of fundamental sequences, the comparison would be quite doubtful even if we could fix a system of fundamental sequences.