TL;DR
This paper introduces the concept of uniformity through axioms, explores its relation to ordered sets, and demonstrates the efficiency of axiom schemes in clarifying mathematical theories.
Contribution
It presents a new formulation of uniformity using axiom schemes, simplifying the understanding of axiomatic systems in mathematics.
Findings
Axiom schemes effectively replace traditional axiom systems
Uniformity concept linked to ordered sets
Clarifies the main ideas of mathematical theories
Abstract
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing axiom systems is shown. With such a replacement, the main idea of the system of axioms of the given mathematical theory immediately becomes clear. The reasons for using number systems in mathematics are also shown.
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Taxonomy
TopicsHistory and Theory of Mathematics