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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If you make a formal logical system powerful enough to derive the rules of arithmetic, then it will always be possible to construct a sentence of the system that is true but which cannot be proved in the system.
The arithmetic part is important. If someone ever tells you that Godel’s result applies to all logical systems, then they do not know what they are talking about. The sentential calculus, aka propositional logic, for example, is complete: all true statements within it can be proved by it. But you cannot derive arithmetic from it.
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