# 超越整数

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## 性質 編集

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 有限約束ゲーム, 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.

## 出典 編集

1. https://u.osu.edu/friedman.8/files/2014/01/EnormousInt.12pt.6_1_00-23kmig3.pdf