Undecidable questions are problems for which no algorithm can always provide a correct answer for every possible input in a finite amount of time. A classic example in computer science is the Halting Problem, which asks whether a given program will halt or run forever. In set theory, Kurt Gödel's incompleteness theorems highlighted the undecidability of certain statements within standard set theory, notably the Continuum Hypothesis (CH).

To prove a problem is undecidable, one can use reducibility, by showing it is "harder" than a known undecidable problem. After Gödel's foundational work, a program emerged to resolve such questions by finding natural extensions of ZFC. Key methods developed to address these problems include forcing, a powerful technique introduced by Paul Cohen and reformulated by Dana Scott and Solovay, which can introduce new solutions to previously undecidable questions like CH [8, 179, conversation history]. Additionally, the use of large cardinal axioms postulates the existence of very large infinite sets to decide statements.

Philosophical approaches also explore re-evaluating the frameworks in which decisions are made, rather than solely focusing on algorithmic solutions. Modern set theory continues to explore this "landscape of foundational axioms," including the consistency strength hierarchy, to answer questions that remain undecided within the standard framework. Relevant academic papers and course materials discussing undecidability can be found in academic repositories like CORE or ResearchGate, university websites, and researchers' personal sites.

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