Abstract
The aim of this paper is modern establishment of geometric theory of continuity, which is based on two, fourth group axioms - Archimedes ' and Cantor's axiom. Various consequences of Archimedes' and Cantor's axioms are proven such as Cantor's and Dedekind's theorem. The paper gives a special view on Hilbert 's axiomatic establishment of geometry which uses axiom of linear completeness instead of Cantor's axiom. Finally, the paper illustrates the use of the axioms of continuity and their consequences in proving some theorems.
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