What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.
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Ever seen the statement, “This statement is false.”?
Now codify that using the language of pure math.
If you can do so, this statement means your language can have contradictions.
If you remove the statement, then the language is incomplete, because it no longer contains that statement.
So now decide which math language you want: one that is complete with contradictions, or one that is incomplete with no contradictions, because you can’t have both.
Cool, huh?
Up to his theorem, it was believed that a pure math language could be written that described everything, and was consistent with no contradictions. Bertrand Russel and Alfred Whitehead even wrote a giant book called The Principia Mathematica to formalize all know math. Bummer for them.
Ultimately it means there will be some contradictions a complete language. The implications are that math isn’t perfect, or the definition of perfect must contain contradictions. An epistemology crisis, with no real impact to the average person.
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