Sent Draft of Decidability Investigations

In the mid-1930s, Alan Turing engaged in foundational work on the concept of decidability, which is central to mathematical logic and computer science. This period was marked by Turing's collaboration with prominent logician Alonzo Church, as both sought to explore the implications of Gödel's incompleteness theorems. These theorems demonstrated inherent limitations in formal systems, suggesting that not all mathematical truths can be proven within a given system. Turing's investigations aimed to address the decidability of certain problems, which refers to the ability to determine, through a finite procedure, whether a given statement is provable within a formal system. This work laid the groundwork for what would later become known as the Turing machine, a theoretical construct that formalizes the concept of computation and algorithmic processes. In April of the same year, Turing sent a draft typescript of his findings to Max Newman, a fellow mathematician and collaborator. This draft represented a significant step in articulating the principles of computability and decidability. Turing's work would eventually lead to the formulation of the Church-Turing thesis, which posits that any function that can be computed algorithmically can be computed by a Turing machine. The implications of these investigations extend beyond mathematics into the realms of computer science, artificial intelligence, and philosophy, influencing how we understand computation and the limits of what can be computed. Turing's contributions during this period are recognized as pivotal in establishing the theoretical foundations of modern computing.