Kurt Gödel's paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" discusses the formalization of mathematics towards greater precision, particularly focusing on the Principia Mathematica (PM) and Zermelo-Fraenkel's axiomatic system of set theory. Gödel argues that while these systems are comprehensive and can formalize all known mathematical proof methods, they are not sufficient to decide all mathematical questions that can be expressed within them. Specifically, he demonstrates that there exist relatively simple problems in the theory of ordinary whole numbers that cannot be decided by the axioms of PM or ZFC. This conclusion holds for a wide class of formal systems, including those derived from PM or ZFC by adding finitely many axioms, provided these additional axioms do not introduce false statements of a specific kind. The paper outlines the main idea of the proof, which involves defining metamathematical concepts in terms of natural numbers and their sequences, and showing that certain sentences within PM are undecidable.Kurt Gödel's paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" discusses the formalization of mathematics towards greater precision, particularly focusing on the Principia Mathematica (PM) and Zermelo-Fraenkel's axiomatic system of set theory. Gödel argues that while these systems are comprehensive and can formalize all known mathematical proof methods, they are not sufficient to decide all mathematical questions that can be expressed within them. Specifically, he demonstrates that there exist relatively simple problems in the theory of ordinary whole numbers that cannot be decided by the axioms of PM or ZFC. This conclusion holds for a wide class of formal systems, including those derived from PM or ZFC by adding finitely many axioms, provided these additional axioms do not introduce false statements of a specific kind. The paper outlines the main idea of the proof, which involves defining metamathematical concepts in terms of natural numbers and their sequences, and showing that certain sentences within PM are undecidable.