What is Theorem?
It is considered a proposition that affirms a demonstrable truth. For mathematics, it is any proposition that, starting from an assumption or hypothesis, signs rationality that is not evident by itself. It is also said that a theorem is a formula that can be proved within a system starting from axioms and other theorems. Proving theorems is the main objective of mathematical logic.
The theorems have a number of premises that must be clarified or listed beforehand. The conclusion of a theorem is a logical statement that is true under the given conditions. A theorem is a relationship between the hypothesis and the thesis or conclusion.
What is Axiom?
An axiom is a proposition that is assumed within a theoretical body and other reasonings and propositions deduced from its premises are based on it. This concept was introduced by the Greek mathematicians of the Hellenistic period. The axioms were considered as self-evident propositions and were accepted without requiring prior proof. Then, in the hypothetico-deductive system, the axioms were all those propositions not deduced from others but rather the general rule of logical thought. In logic and mathematics, an axiom is that premise that is assumed regardless of whether it is evident or not, and is used to prove other propositions. The axioms are considered as true statements in any possible world, under any possible interpretation, and under any interpretation of values. suggested video: Axiom vs Theorem
Difference between Axiom and Theorem
- An axiom is a statement that is accepted as true without requiring to be proved. It does not need proof and is universally accepted. Its non-acceptance contradicts any logic.
- The axioms do not have a contradiction and are evident without deep analysis.
- A theorem is a theoretical proposition that requires a test.
- The theorems are not accepted until they are subjected to tests whose results support the theory.