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The Pythagorean theorem has at least 370 known proofs. [1]In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. [a] [2] [3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
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'.
The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures.
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n -ary truth tables ), and in the exact complete choice of axioms over the chosen basis of connectives.
Its defining method can be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF ...
In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line. This axiom became a theorem proved by Emil Artin in his book Geometric Algebra.
However, E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902. [36] Hilbert moved the axiom to Theorem 5 and renumbered the axioms accordingly (old axiom II-5 (Pasch's axiom) now became II-4).