Godel's incompleteness theorem is one of the biggest mind-shaking truths about mathematical reality. Detailed overview

• No set of finite axioms whose theorem is presented in an effective procedure or algorithmic way can be used to prove all arithmetic truths.

• It also states that there will always be true statements that can not be proved to be true with a finite number of axioms. IF we start doing that we would, in turn, have to use another unprovable truth eg the Tiling problem of Fermat's Last Theorem. This really puts [[Platonism]] in a different stage as they already believe in the pureness of Mathematics.

There is a loop - if a system is complete, it’s not consistent, and if it’s consistent then it’s not complete. It’s kinda like Heisenberg's uncertainty principle in this way where you have to trade off one thing for another.

It was based as an answer to Hilbert's program to build a definitive framework of axioms and procedures to prove a given statement or a Mathematical truth. Hilbert's hope was that there must be a way to use a framework for a whole topic that applies to every statement or proof of that topic. The hope was to define every existing Mathematical structure and theory in a finite set of axioms and the axioms could, in turn, become Arithmetic, the main goal was to reduce most complex Mathematical theories into arithmetic. It would have helped Mathematics gives the foundational framework it needed to become absolutely provable and true in a sense.

• It was deemed to solve the crisis of paradoxes and inconsistencies in the 20th century.

• These are the goals of the program in bullet points

  • A formulation of all mathematics: in other words, all mathematical statements should be written in a precise formal language, and manipulated according to well-defined rules.

  • Completeness: a proof that all true mathematical statements can be proved in formalism.

  • Consistency: proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.

  • Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.

  • Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

People who thought that every mathematical statement was provable were left in a hard place as they could not come up with a solution to Godel's statements.

• This is a great video that fully describes the theorem with examples.

But when Gödel published the paper containing his Incompleteness theorem, it turned the mathematical world upside down. There were Debates on the practicality of mathematics and its absoluteness.

• Now you might think that showing that there is a big hole in the Maths itself might have decreased our exploration capabilities but it did the opposite.

A young chap named Alan Turing set upon solving Hilbert's problem and Gödel's Theorem which resulted in the development of a Turing Machine and subsequently the device you are reading this on. It’s intertwined with Turing's Halting Problem

• It helped spark the innovation in notion systems of mathematics and truly showed that some things are beyond human comprehension and which leads to one of my favorite conclusions of Humanity needing [[AGI]].

PS: As always thank you dear reader who has stuck to the end of this short blogpost. I am thinking of writing some book posts in the future as I recently started taking notes on the books I read and if you liked the blogpost you can buy me a book to show some gratitude as books are the places where I find most of my ideas. or maybe send me some eth for fun at madhavg.eth .

Godel's incompleteness theorem is one of the biggest mind-shaking truths about mathematical reality. Detailed overview

• No set of finite axioms whose theorem is presented in an effective procedure or algorithmic way can be used to prove all arithmetic truths.

• It also states that there will always be true statements that can not be proved to be true with a finite number of axioms. IF we start doing that we would, in turn, have to use another unprovable truth eg the Tiling problem of Fermat's Last Theorem. This really puts [[Platonism]] in a different stage as they already believe in the pureness of Mathematics.

There is a loop - if a system is complete, it’s not consistent, and if it’s consistent then it’s not complete. It’s kinda like Heisenberg's uncertainty principle in this way where you have to trade off one thing for another.

It was based as an answer to Hilbert's program to build a definitive framework of axioms and procedures to prove a given statement or a Mathematical truth. Hilbert's hope was that there must be a way to use a framework for a whole topic that applies to every statement or proof of that topic. The hope was to define every existing Mathematical structure and theory in a finite set of axioms and the axioms could, in turn, become Arithmetic, the main goal was to reduce most complex Mathematical theories into arithmetic. It would have helped Mathematics gives the foundational framework it needed to become absolutely provable and true in a sense.

• It was deemed to solve the crisis of paradoxes and inconsistencies in the 20th century.

• These are the goals of the program in bullet points

  • A formulation of all mathematics: in other words, all mathematical statements should be written in a precise formal language, and manipulated according to well-defined rules.

  • Completeness: a proof that all true mathematical statements can be proved in formalism.

  • Consistency: proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.

  • Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.

  • Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

People who thought that every mathematical statement was provable were left in a hard place as they could not come up with a solution to Godel's statements.

• This is a great video that fully describes the theorem with examples.

But when Gödel published the paper containing his Incompleteness theorem, it turned the mathematical world upside down. There were Debates on the practicality of mathematics and its absoluteness.

• Now you might think that showing that there is a big hole in the Maths itself might have decreased our exploration capabilities but it did the opposite.

A young chap named Alan Turing set upon solving Hilbert's problem and Gödel's Theorem which resulted in the development of a Turing Machine and subsequently the device you are reading this on. It’s intertwined with Turing's Halting Problem

• It helped spark the innovation in notion systems of mathematics and truly showed that some things are beyond human comprehension and which leads to one of my favorite conclusions of Humanity needing [[AGI]].

PS: As always thank you dear reader who has stuck to the end of this short blogpost. I am thinking of writing some book posts in the future as I recently started taking notes on the books I read and if you liked the blogpost you can buy me a book to show some gratitude as books are the places where I find most of my ideas. or maybe send me some eth for fun at madhavg.eth .