Explore the challenges and limitations of Dedekind cuts in defining real numbers in this 40-minute mathematics lecture. Delve into Richard Dedekind's 1870 approach to conceptualizing real numbers through partitions of rational numbers. Examine the concept of Dedekind cuts, their purported role in establishing a logical foundation for real number theory, and critically analyze whether this method truly succeeds in creating a workable arithmetic of real numbers. Investigate key topics including cuts of rationals, principles of mathematical analysis, subsets of rational numbers, theories of infinite sets, and the complexities of arithmetic with Dedekind cuts. Gain insights into the philosophical and practical difficulties surrounding this foundational concept in mathematical analysis.