ABSTRACT

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given that there are no algebraic structures that one could associate with those minimal axiom systems.

中文翻译：

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几何不可约化为代数的案例†

摘要

本文基于从一阶逻辑得出的考虑，对初等几何的现状，初等几何是否可以简化为代数的问题提供了明确的答案。我们得到的答案是否定的，并且是基于一系列只能在几何形式理论内部提出的结构问题，以及对逆几何的考虑，这是找到足够强大的最小公理系统以证明的艺术某些几何定理，因为没有代数结构可以与那些最小公理系统相关联。