# The Limits of Knowledge  ## Metadata - Author: [[freedium.cfd]] - Full Title: The Limits of Knowledge - Category: #articles - Summary: In the seventeenth century, mathematicians like Leibnitz and Hilbert believed that all mathematical questions could be answered, but Gödel later proved that some true statements cannot be proven. Turing expanded on this idea by showing that certain problems, like the halting problem, cannot be solved by any computer program. Today, the P vs NP question remains unanswered, highlighting limits in both mathematics and computer science. - URL: https://www.freedium.cfd/https://towardsdatascience.com/the-limits-of-knowledge-b59be67fd50a ## Highlights - Here's a simplified way to understand Gödel's theorem. Consider the following statement. *Statement S: This statement is not provable.* Now, suppose that within the context of Mathematics we could prove S to be true. But then, the statement S itself would be false, leading to an inconsistency. Okay, then let's assume the opposite, that we cannot prove S within the context of Mathematics. But that would mean that S itself is true, and that Mathematics contains at least one statement that is true but cannot be shown to be true. Hence, Mathematics must be either inconsistent or incomplete. If we assume it to be consistent (statements cannot be true and false at the same time), this only leaves the conclusion that Mathematics is incomplete, i.e., there are true statements that simply cannot be shown to be true. ([View Highlight](https://read.readwise.io/read/01j6qhm227208wkwmvm9kvkmmd))