If you love mathematics then you would definitely love this article but if you hate mathematics then you would love this even more. This is a story about the quest to find a complete system of mathematics, something similar to the ‘theory of everything’ in modern Physics.
Before moving on this story we must learn a bit of mathematical jargon.
So mathematics can be seen as a set of axioms from which all mathematical ‘principles’ or ‘theorems‘ which simply means ‘rules‘ are directly or indirectly derived. Now lets understand the meaning of axiom, an axiom is a basic or fundamental truth that acts as the starting point for deducing other theorems and rules within the given system of mathematics. Let’s take an easy example to understand this idea – “All bachelors are unmarried.” Consider this statement to be true by definition, and it now serves as a basic assumption upon which other principles can be drawn. It’s not something we prove; instead, we accept it as a starting point for an argument.
Let’s try to build an argument using the above stated axiom “All bachelors are unmarried” and see how the an axiom is used:
Premise 1 (Axiom): “All bachelors are unmarried.”
Premise 2: “John is a bachelor.”
From these premises, we can deduce a conclusion using logical reasoning:
Conclusion: “Therefore, John is unmarried.”
And that is all we need to enjoy the story.
“Mathematics is a game played according to certain simple rules with meaningless marks on paper.”
— David Hilbert
The Quest
Now back to our story. Somewhere near the end of 19th and the beginning of the 20th century some of the prominent mathematicians embarked on a difficult journey, led by one of the most influential mathematicians of that era David Hilbert – to find a complete and consistent set of axioms for all of mathematics. He started the project for completing mathematics, which later came to be known as Hilbert’s Program. It called for a formalization of all of mathematics in axioms along with proof that these axioms are not inconsistent, i.e., one does not counter another. This theory would, in a sense, complete the domain of mathematics. Those mathematicians working to formalize mathematics were known as ‘formalists.’
But then came a guy named Kurt Gödel who pulled one of the best intellectual masterstrokes of the century and gave us ‘Gödel’s Incompleteness Theorems‘. It is considered as the most significant achievement in the field of logic and philosophy of mathematics. And it doomed the formalists forever as it showed that Hilbert’s Program can never achieve its objective and to find a complete and consistent set of axioms for all mathematics is impossible.
“Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.”
— Kurt Gödel
Gödel’s Incompleteness Theorems
First Incompleteness Theorem: IT says that in any consistent formal system which can express basic arithmetic, there will be some statement that cannot be proven either true or false. So, this means that the system remains incomplete as there will be true mathematical statements that cannot be derived using the already known given the rules of the system.
To understand it better, consider this: suppose you visit a fictional city called ‘Logicville’. Upon arrival, you get a book from the city’s mayor, which contains rules for everything in the town, from building houses to baking cakes. So everything that you could do or need to do is already in the book, at least according to the mayor. Now, going by Gödel’s first theorem, regardless of how long the book is, there will always be some new things that could be done in Logicville that the book couldn’t account for. Thus there will be truths about Logicville that are true but not in the book because they go beyond its scope.
Second Incompleteness Theorem: It builds on the first one and says that no consistent system can prove its own consistency. This means that a system cannot contain proof that it does not contain any contradictions or discrepancies i.e. — no system can prove it is complete using its own rules.
Let’s continue with ‘Logicville’. According to the second theorem, let’s say the book says that: “Everything written in this guidebook is true.” The theorem suggests that the book cannot prove this by itself, i.e., using things written within it. If it tried, it would need another book to confirm the truth of the first one, and so on, creating an infinite regress. So, the book can’t prove its own infallibility or consistency that it will work for everything that one could do or needs to do in the city. There’s always room for things that could be done which are not there in the book.
In simpler terms, the first theorem tells us that no set of rules can capture all truths about a system, and the second one tells us that a system can’t prove its own consistency without relying on something outside itself. Thus proving the system of mathematics will always be incomplete within itself.
Self-Referential Loop
Gödel’s proof of his theorem is based on the concept of self-reference. Let’s consider the statement: “This sentence is false.” If we look from a logical perspective, it is both true and formally unprovable. As If the statement is true, it’s false and if it’s false, it’s true. And yes, logically, this does not make any sense, and that’s why they are also called paradoxical.
This dialogue from the famous show ‘The Office’ beautifully captures the essence of the ‘self-reference’ paradox:
“Jim is my enemy. But it turns out that Jim is also his own worst enemy. And the enemy of my enemy is my friend. So Jim is actually my friend. But, because he is his own worst enemy, the enemy of my friend is my enemy, so actually Jim is my enemy. But…”
— Dwight Schrute
There are quite a few cases where the ideas of self referential paradox are relevant. Consider the case for the modern legal system: it can’t guarantee it will resolve all future disputes using the same laws, as unforeseen cases may arise that will require future additions or amendments. Thus any legal system was, is, and always will remain incomplete. Also, consider the case of anti-malware programs that are common on a PC or laptop. It suggests there can never be an anti-malware program that could guarantee to protect your system from all future malware attacks.
Apart from mathematics and computing there are various other domains where Gödel’s theorems have left their mark, from philosophy to cognitive sciences. And not only has it made a profound impact on mathematics, but it also changed how modern physics and we perceive and understand the nature and the universe itself.
Curious Corner 
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