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Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in elliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry).
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge. His constructive approach appears even in his geometry's postulates, as the first and ...
Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of ( second order ) axioms for Euclidean geometry, called Hilbert's axioms , and between 1926 and 1959 Tarski gave some ...
Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, [7] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. [8]
This is a list of notable theorems.Lists of theorems and similar statements include: List of algebras; List of algorithms; List of axioms; List of conjectures
This is a list of axioms as that term is understood in mathematics. In epistemology , the word axiom is understood differently; see axiom and self-evidence . Individual axioms are almost always part of a larger axiomatic system .
An axiom P is independent if there are no other axioms Q such that Q implies P. . In many cases independence is desired, either to reach the conclusion of a reduced set of axioms, or to be able to replace an independent axiom to create a more concise system (for example, the parallel postulate is independent of other axioms of Euclidean geometry, and provides interesting results when negated ...