Godel's Incompleteness Theorem

Godel's incompleteness theorem- a theorem that proves the "Un-Provable". A mathematical proof that it is possible or impossible to prove some given mathematical problems. Gödels theorem is the basis of modern mathematics to state that axioms exist without the formal proof. Kurt Gödel proposed two incompleteness theorems and these theorems are also one of the most important theorems in modern logic.

First incompleteness theorem states that for a consistence and formal system, there are statements which can be neither proved and nor disproved.

Formal system is a system that contains axioms, based on which new theorems can be defined. A number of mathematical approaches today are based on axiomatic definition. e.g. Probability theory. The number of axioms should be finite or there are ways which determine if the given statement is axiom or not. An axiomatic (formal) system is consistent if it doesn't contain the contradiction or if it is possible to prove that the statement and its negation are both true.

The Second incompleteness theorem is the extension of the first theorem which states that, any consistence axiomatic system, where certain elementary arithmetic operations are carried out, does not demonstrate its own consistency. Or in other words, if a formal system forms its own consistency then the system itself is inconsistent.

One of the interesting questions that always arose in my mind is that- "Is the universe analog or digital?" Is it only the outcome of 0 or 1 ? In other words, are there any possibilities that anything exist and do not exist at the same time? Godel's theorem could  answer them to some extent.