Proven in 1931 by Kurt Gödel, his incompleteness theorems say that with any given set of logical rules, except for the most simple, there will always be statements that are undecidable, meaning that they cannot be proven or disproven due to the inevitable self-referential nature of any logical systems that is even remotely complicated. This is thought to indicate that there is no grand mathematical system capable of proving or disproving all statements. An undecidable statement can be thought of as a mathematical form of a statement like “I always lie.” Because the statement makes reference to the language being used to describe it, it cannot be known whether the statement is true or not. However, an undecidable statement does not need to be explicitly self-referential to be undecidable. The main conclusion of Gödel’s incompleteness theorems is that all logical systems will have statements that cannot be proven or disproven; therefore, all logical systems must be “incomplete.”

The philosophical implications of these theorems are widespread. The set suggests that in physics, a “theory of everything” may be impossible, as no set of rules can explain every possible event or outcome. It also indicates that logically, “proof” is a weaker concept than “true”; such a concept is unsettling for scientists because it means there will always be things that, despite being true, cannot be proven to be true. Since this set of theorems also applies to computers, it also means that our own minds are incomplete and that there are some ideas we can never know, including whether our own minds are consistent (i.e. our reasoning contains no incorrect contradictions). This is because the second of Gödel’s incompleteness theorems states that no consistent system can prove its own consistency, meaning that no sane mind can prove its own sanity. Also, since that same law states that any system able to prove its consistency to itself must be inconsistent, any mind that believes it can prove its own sanity is, therefore, insane.

(Source: mohandasgandhi)