Undecidable

Undecidable refers to a problem or statement for which it's impossible to construct an algorithm or proof that definitively determines its truth or falsity, within a given formal system. This doesn't imply the problem is inherently meaningless, just that the rules of the system are insufficient to provide a solution. It's a fundamental concept in computability theory and logic, highlighting limitations of formal systems. The problem's nature means no finite sequence of operations will universally lead to an answer. The question remains beyond the scope of the system, meaning we cannot resolve it definitively.

Undecidable meaning with examples

• The halting problem in computer science, which asks whether a program will eventually stop or run forever, is a classic example of an undecidable problem. There's no general algorithm that can analyze any program and predict its behavior. No matter how clever the logic, a solution cannot be found.
• In set theory, the continuum hypothesis, which concerns the size of infinite sets, is an undecidable statement within the standard axioms of Zermelo-Fraenkel set theory (ZFC). It can be neither proven nor disproven within ZFC; additional axioms would be needed to reach a conclusion. This highlights how different mathematical systems have different strengths.
• Certain logical systems, such as those incorporating Gödel's incompleteness theorems, can contain undecidable statements. These statements are true, but they cannot be derived from the system's axioms. This demonstrates that the system lacks sufficient axioms to define every fact inside it.
• Philosophical questions about consciousness or free will might be considered undecidable in a scientific framework, as definitive, verifiable proof, using current methods, may not exist or be unattainable. Some would propose science is not the system to be used in these cases.