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A mathematical proof is a deductive argument for a mathematical statement, ... not a form of inductive reasoning. In proof by mathematical induction, ...
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. [1]
Mathematical logic is the study of formal logic within mathematics.Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory).
Deductive reasoning plays a central role in formal logic and mathematics. [1] In mathematics, it is used to prove mathematical theorems based on a set of premises, usually called axioms. For example, Peano arithmetic is based on a small set of axioms from which all essential properties of natural numbers can be inferred using deductive reasoning.
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally ...
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. [5] The field was founded by Harvey Friedman . Its defining method can be described as "going backwards from the theorems to the axioms ", in contrast to the ordinary mathematical practice of deriving ...
A method of proof used in mathematics to prove statements about natural numbers, based on proving a base case and an inductive step. mathematical induction schema Synonym of mathematical induction. [182] [183] mathematical logic The study of logic within the framework of mathematical reasoning, focusing on the mathematical properties of logical ...
In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems. [7] The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. [8]