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hyperds
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I saw this explanation of Gödel's incompleteness theorems in another thread:
Is this correct? And does this mean that all unprovable statements in math, will take the form of a self-referential paradox? If this is true, I don't get why the incompleteness theorem is considered deep, if it is basically irrelevant to the rest of math, and only applies to carefully constructed paradoxical statements.
Godel found a way of encoding a statement to the effect of "This statement is unprovable" into the symbolic logic system defined in Principia Mathematica (PM). The notable aspect of the statement is that it is self-referential, which Godel managed to accomplish by encoding statements in PM into "Godel Numbers." Thus the actual statement in PM refers to its own Godel Number.
To boil it down into a nutshell, I'd say it means that any system which is expressive enough to be consistent and complete is also expressive enough to contain self-referential statements which doom it to incompleteness.
Is this correct? And does this mean that all unprovable statements in math, will take the form of a self-referential paradox? If this is true, I don't get why the incompleteness theorem is considered deep, if it is basically irrelevant to the rest of math, and only applies to carefully constructed paradoxical statements.