What is Gödel’s incompleteness theorem, and why is it so infamous in Mathematics?
In: Mathematics
Perhaps it would be useful to add that, at the core and in simple terms, Gödel showed that there will always be a theorem stating “I can’t be proved” (self-referential):
* if true, it can’t be proved;
* if false, it can’t be proved either as it is false.
Which means that as soon as you have the conditions (axioms) to do a modicum of mathematics (find theorems), there will always be *true but “unreachable” results*. (Others have discussed the “conditions”.)
And also that you can’t just put a computer on the task of churning out all true results, it will always at some point hit a snag and never return a true/false result. (That’s Turing’s “halting problem”.)
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