Turing Machines & the Halting Problem
Alan Turing · 1936
"Turing proved that no general algorithm can decide in advance whether an arbitrary program will ever halt — some questions are not just hard, they are uncomputable, which sets a hard limit on what automation can ever guarantee."
In 1936 a 24-year-old at Cambridge asked a disarmingly simple question: can we build a machine that looks at any program and tells us, beforehand, whether it will ever finish running? His answer — no — is the most important negative result in computer science.
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Russell's Paradox & the Foundations Crisis
"Russell showed that naive set theory lets you define 'the set of all sets that don't contain themselves,' which both contains and doesn't contain itself — forcing mathematics to rebuild its foundations with stricter rules."