Explore the complexities and controversies surrounding the concept of real numbers in this thought-provoking lecture from the Famous Math Problems series. Delve into the various approaches used to establish real numbers, including infinite decimals, nested intervals, Cauchy sequences, and Dedekind cuts. Examine the logical and technical difficulties inherent in these methods, challenging conventional wisdom about the foundations of calculus and analysis. Gain a critical perspective on the computational limitations and philosophical implications of attempting to model the continuum mathematically. Consider the importance of concrete examples and explicit computations in developing a more logically coherent understanding of mathematics.