Gödel’s incompleteness theorem

Peano axioms or
ZFC apply only to sufficiently complex and meaningful
Axiomatic system, and both can be avoided if there is only a single element or only two truth values.

If you have a finite axioms in

Axiomatic system, no matter how much you build and establish theories within that system, it can be like the continuum hypothesis where it is not inconsistent whether this is true or false.

In a consistent mathematical system, there must exist at least one proposition that cannot be proven. No matter how meticulously an

Axiomatic system is designed to avoid this, it cannot be avoided. This is proven by the incompleteness theorem. The set of propositions provable in first-order logic precisely has a model. In other words, there necessarily exists a case where the truth defined by proof theory and the truth defined by model theory coincide.

2nd Gödel’s incompleteness theorem

A sufficiently complex axiomatic mathematical system cannot prove its own consistency. In other words, when a mathematical system tries to prove that it is not contradictory, it becomes incomplete.

Gödel’s Incompleteness Theorems

Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system \(F\) within which a certain amount of arithmetic can be carried out, there are statements of the language of \(F\) which can neither be proved nor disproved in \(F\). According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial. The present entry surveys the two incompleteness theorems and various issues surrounding them. (See also the entry on Kurt Gödel for a discussion of the incompleteness theorems that contextualizes them within a broader discussion of his mathematical and philosophical work.)