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An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of geometric propositions can be deduced from a set of definitions, postulates, and common notions: all three types constitute first principles.
This method resembles the modern axiomatic method but with a big philosophical difference: axioms and postulates were supposed to be true, being either self-evident or resulting from experiments, while no other truth than the correctness of the proof is involved in the axiomatic method. So, for Aristotle, a proved theorem is true, while in the ...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time. An axiom or postulate is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a definition, which gives the meaning of a word or a phrase in terms of known ...
The resulting structure, a model of elliptic geometry, satisfies the axioms of plane geometry except the parallel postulate. With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis ...
An axiomatic system is said to be consistent if it lacks contradiction.That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explo
Adding axioms to K gives rise to other well-known modal systems. One cannot prove in K that if "p is necessary" then p is true. The axiom T remedies this defect: T, Reflexivity Axiom: p → p (If p is necessary, then p is the case.) T holds in most but not all modal logics. Zeman (1973) describes a few exceptions, such as S1 0.