Gödel and Beyond: A Treatise on the Notion of Indeterminancy

I have had the opportunity to reflect on the notions of indeterminacy and incompleteness within various fields associated with mathematics. I initially gained a reading comprehension on paradoxes, emphasizing Russell\u27s paradox of the library. I then studied first order logic, using propositional calculus as a means of examining a formal system which is consistent and complete for the purpose of comparison and future reference. I explored various proofs utilizing the essence of Cantor\u27s diagonal argument, which allowed me the opportunity to better comprehend the ideas of infinity, contradiction, incompleteness, and indeterminacy. All this led to a Halting version proof of Gödel\u27s Incomplete Theorem. I examined the proofs of Gödel\u27s two incompleteness theorems and concluded with a discussion on Chaitin\u27s Omega