Yes, you read that right. One of the most complicated subjects in the world will never be complete thanks to Kurt Godel, the mastermind behind the very aptly named Gödel’s incompleteness theorems. Being an avid enthusiast of mathematics, I found this proof to be extremely mind-boggling as it not only removes all barriers in the way of mathematics but also enables passionate mathematics students such as myself to explore the field without any restraint. Without further ado, let us delve deeper into how Kurt broke mathematics.
What are Gödel’s incompleteness theorems?
These are two logical statements made by Gödel that essentially state the limits of mathematics in terms of axiomatic proofs. This hence shows how all axiomatic proofs are unlimited and infinite, therefore making mathematics forever incomplete.
The first incompleteness theorem in its completely mathematical form states that “any consistent formal system 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.” Wait, what? The flurry of mathematical jargon initially confused me as well but what Kurt was trying to convey was that in any given set of axioms, there will always be a statement using those axioms that can never be proven or disproven using the given axioms. He used formal systems as they are defined by a set of axioms alongside some symbols for operations to derive new axioms. Kurt used the following infamous example:
‘This statement is false”
As confusing as it sounds, the statement above can never be true, because if it is then it should be false. It can never be false because then it should be true. Despite having made fun of the logical fallacies in the English language, Kurt extended this logic to mathematics, converting this very statement into lines of mathematics where each character translates to a number of sorts.
Moreover, Kurt enhanced his study by involving a second theorem that helped solidify his movement to make mathematics forever incomplete. He stated the following:” a formal system following the first theorem cannot prove that the system itself is consistent”. Kurt’s consistent formal systems hence indicated that there was always a theorem that could prove both a statement and its negation. His logic simplified indicates that if a system is infinite, there could always be a singular statement without a negation, hence disproving its consistency. In order for the system to prove that it is consistent, however, it needs to prove that all axioms within the set have negations that disprove them.
In real life, Kurt’s results proved to be extremely beneficial to a lot of students, researchers, and teachers alike by inspiring yes and no problems in computer science such as the halting problem, which cannot halt no matter the input or urgency. Moreover, his results can be linked to the Liar Paradox, enabling his research to extend further.
Overall, Kurt’s incompleteness theorem stands to be a singular theorem that impacts the lives of millions of scientists. The incompleteness theorem is hence currently indicative of certain facets of our lives. The application depends from person to person but it seemed as real as an actual gun; the amount of power we possessed skyrocketed.