An Axiom is a premise or starting point of reasoning. As classically conceived, an Axiom is a premise so evident as to be accepted as true without controversy.
The word comes from the Greek word which means “that which is thought worthy of fit” , or “that which commends itself as evident”. As used in modern -logic- , an Axiom is simply a premise or starting point for reasoning.
Axioms define and delimit the reason of -analysis- . In other words, an Axioms is a logical -statement- that is assumed to be -true- . Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing an inferring other (theory and domain dependent) truth. An Axiom is defined as a -mathematical- satement that is accepted as being true without a -mathematical proof- .
In -mathematics- , the term Axiom is used in two related
distinguishable senses: -logical axioms- and -non-logical axioms- . LOGICAL AXIOMS are usually statements that are taken to be universally true [e.g (A and B) implies A], while
NON-LOGICAL AXIOMS [a b = b a] are actually defining properties for the domain of a specific mathematical theory (such as -arithmetic- ). When used in latter sense, “Axioms” , “Postulate” . “Asumption” may be used interchangeably. In general, A NON-LOGICAL AXIOMS is not a self-evident truth, but rather a formal logical expression use in deducution to build a mathematical theory. To axiomize a system of knowledge is to show that its claims can be derived from a small, well understood set of sentences (the Axioms). There are typically multiple ways to axiomitize a given mathematical domain.
In both senses, an Axioms is any mathematical statement that serves as a starting point from which other statements are logically derived. Axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrate by -mathematical proofs- , simply because they are starting point; there is nothing else from which they logically followed otherwise they would classified as theorems.


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