Continuity Axioms

``The'' continuity axiom is an additional Axiom which must be added to those of Euclid's Elements in order to guarantee that two equal Circles of Radius $r$ intersect each other if the separation of their centers is less than $2r$ (Dunham 1990). The continuity axioms are the three of Hilbert's Axioms which concern geometric equivalence.

Archimedes' Lemma is sometimes also known as ``the continuity axiom.''

See also Congruence Axioms, Hilbert's Axioms, Incidence Axioms, Ordering Axioms, Parallel Postulate

References

Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 38, 1990.

Hilbert, D. The Foundations of Geometry. Chicago, IL: Open Court, 1980.

Iyanaga, S. and Kawada, Y. (Eds.). ``Hilbert's System of Axioms.'' §163B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 544-545, 1980.