Non-derivable

Non-derivable, in a mathematical or logical context, refers to something that cannot be obtained or deduced from a set of initial axioms, rules, or other established premises through a series of logical steps. It signifies an independent or fundamental concept that cannot be explained or proven using pre-existing knowledge or existing methods. It also can refer to something that cannot be formed from some set of prior things. Its nature resists reduction or decomposition, and relies on an independent and perhaps unique construction. This is to say that it cannot be derived in a logical, mathematical, or programmatic sense.

Non-derivable meaning with examples

• In axiomatic set theory, the Axiom of Choice is often considered non-derivable from other standard axioms like ZFC. This means it cannot be proven using the other axioms, and its acceptance is a matter of postulation. Therefore, the truth of the axiom is separate from the base assumptions, unlike other theorems derived from those same axioms.
• Consider the definition of a primitive concept in geometry, like 'point' or 'line'. The definition is non-derivable; a point is simply accepted as an undefined element. Attempts to define it using more fundamental terms inevitably lead to circular reasoning. Its meaning stems from intuition and acceptance within the context of other geometric elements and axioms.
• In programming, some operations or data structures might be designed to be non-derivable or non-computable from basic programming language constructs. For example, determining whether a program will halt (the halting problem) is a non-derivable problem, as there's no general algorithm to solve it.
• When creating an encryption key, a truly random key should be non-derivable from any public or previously-known information. The key must be generated independently, perhaps by quantum randomness, ensuring no relationship or dependency with any of the other parameters that it uses to generate its own security profile.