Transcendental integer

The transcendental integers are a class of huge numbers, defined by Harvey Friedman.

If \(n\) is an integer then we call it transcendental iff the following holds: let M be a Turing machine, such that there is proof in ZFC of length at most \(2^{1,000}\) showing that M halts. Then M halts in at most \(n\) steps. In other words, \(n\) is greater than halting time of every Turing machine with ZFC proof of halting of length at most \(2^{1,000}\).

Properties
Notion of transcendental number is meaningful iff ZFC is consistent. We will assume that in this article.

It is clear that if \(m>n\) and \(n\) is transcendental, then so is \(m\). Equivalently, transcendental integers are upwards closed, and consequently there is a smallest transcendental integer.

Friedman suspects that this kind of numbers should naturally arise in Boolean relation theory. It is very likely that transcendental integers quickly appear in Friedman's finite promise games, because corresponding functions outgrow all provably recursive functions of SMAH, which is strong extension of ZFC. However, no specific examples of nontrivial transcendental integers are known.