Gödel's Incompleteness Theorems revolutionized mathematical logic by demonstrating inherent limitations in formal systems, showing they cannot be both consistent and complete. Alan Turing's concept of a universal machine furthered this understanding by establishing computability theory, proving that certain problems, like the halting problem, are undecidable. Church's Thesis reinforced these limits, asserting that any effectively calculable function is Turing-computable. These insights highlight the boundaries of algorithmic problem-solving and the foundations of mathematics, challenging Hilbert's program and influencing fields from philosophy of mind to artificial intelligence.
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