The term 'indecidable' describes a statement or proposition within a formal system (like mathematics or logic) whose truth or falsity cannot be proven or disproven based on the system's axioms and rules of inference. This doesn't necessarily mean the statement is inherently meaningless; rather, it signifies a limitation in the system itself. The undecidability arises from the incompleteness of the system, its inability to resolve certain questions based on its defined structure. Discovering an undecidable statement often reveals fundamental limits to what a particular system can accomplish and can lead to deeper investigations into its underlying assumptions and capabilities. Often undecidable means that given the current set of rules, any judgement would result in the system being incomplete. Indecidability is a core concept within the context of theoretical computer science.
Indecidable meaning with examples
- Gödel's incompleteness theorems demonstrated the existence of undecidable statements within sufficiently complex formal systems. This meant that certain true mathematical statements within these systems could not be proven within the system's axioms, leading to a re-evaluation of the foundations of mathematics. These findings challenged the very concept of complete axiomatic systems that could fully describe all truths.
- In theoretical computer science, the Halting Problem is a classic example of an undecidable problem. The question of whether a given computer program will eventually halt or run forever, for a specific input, cannot be solved by any general algorithm. This fundamental limitation influences what can be computed or determined by machines.
- The decision problem for first-order logic is undecidable, which implies that there is no general algorithm to determine whether a formula is valid or satisfiable. This has substantial implications on the expressibility of statements and demonstrates the inherent challenges of automated reasoning within more complex systems.
- The quest for a complete and consistent axiomatic system for mathematics encountered hurdles when dealing with undecidable problems in set theory. Certain statements about sets cannot be proven or disproven from standard set theory axioms, like the axiom of choice, revealing incompleteness.
- When we define a new system, the undecidable statements are those statements that are true but cannot be derived from the system. The definition is internal to the system, for example, the statement “this sentence is false” within the same system is undecidable and leads to paradoxes. This demonstrates limitations in the system itself.