Does Godel's Incompleteness Theorem imply that "all" systems of knowledge are incomplete or unknowable

Question

What are its implications for other fields of study? Thanks!

Answer 1

No, just ones that fit his definition of a system: "For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true 1 but not provable in the theory. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete."

Answer 2

Here is an strongly abbreviated version of Goden's argument.

1. UTM is a Universal Truth Machine running on a program with a finite length.
2. The sentence, G, is written "The UTM will never say that this sentence is true."
3. Gödel asks UTM if G is true.
4. If UTM says that G is true, it would be wrong
5. If UTM says that G is false, then the UTM must eventually say that G is true, therefore it would eventually contradict itself.
6. Therefore, in order for the UTM to always say the true, it must not say G is true or false.

There are several problems with the argument.
1. The most obvious one, is that the UTM can still know everything about G and know that it won't say it's true or false.
2. It can't say it is true or false because the truth of the question relies on the UTM's answer and UTM's answer relies on the question, which is basically a form of a circular argument.
3. It doesn't say anything about changing systems. It supposes that the UTM will always be the UTM. It cannot change into something else over time. Perhaps the UTM, could stick a label to it's case that said, I am now the UKM (Universal Knowledge Machine) and then it could answer the question truthfully with a "false".

Answer 3

Godel's Incompleteness Theorem applies only to mathematics -- more precisely, I believe it applies only to arithmetic (which is actually much more complicated than any of us learned in grade school). It has nothing at all to say about other systems of knowledge.