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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It means that any fixed Turing-recognizable model of arithmetic will either fail to prove some factually true things about arithmetic, or else it will be inconsistent, meaning that it will prove everything, including false things. In that sense, every Turing-recognizable and consistent model of arithmetic is “incomplete,” since it will have “blind spots” in terms of true things for which it actually provides a proof. It means that no fixed formal computational model of arithmetic can account for all of the things that are actually true about arithmetic.
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