In my view, Godel's theorem means a lot to philosophy of mathematics and can do no more than that.
I have a simpler description of the theorem that includes the details:
You can choose 3 out of the 4 things within a axiom system (You can think of it as a language):
1. Completeness: No expressible statement can be neither proved nor disproved.
2. Consistency: There is no statement such that both the statement and its negation are provable from the axioms.
3. The system must be able to describe natural number
4. The system must use formal methods. (The proofs are finitistic)
For mathematicians nowadays, some of them have a natural Platonist mathematics view and take away 4.
Some ultrafinilist like Edward Nelson, wants to take away 3.
In practical cases we can take away 1, e.g. total functional language.
Some philosophers just wants to take away 2 without justify all the other assumptions.
In terms of moral philosophy, a much simpler alternative to derive the undecidability of nature, Chaos theory, can be used instead of assuming 1, 3, and 4.
The case here, in my view, is much less interesting philosophical than the interpretations of quantum mechanics, where most physicists refuse to discuss.
I have a simpler description of the theorem that includes the details:
You can choose 3 out of the 4 things within a axiom system (You can think of it as a language):
1. Completeness: No expressible statement can be neither proved nor disproved.
2. Consistency: There is no statement such that both the statement and its negation are provable from the axioms.
3. The system must be able to describe natural number
4. The system must use formal methods. (The proofs are finitistic)
For mathematicians nowadays, some of them have a natural Platonist mathematics view and take away 4. Some ultrafinilist like Edward Nelson, wants to take away 3. In practical cases we can take away 1, e.g. total functional language.
Some philosophers just wants to take away 2 without justify all the other assumptions.
In terms of moral philosophy, a much simpler alternative to derive the undecidability of nature, Chaos theory, can be used instead of assuming 1, 3, and 4.
The case here, in my view, is much less interesting philosophical than the interpretations of quantum mechanics, where most physicists refuse to discuss.