Logical Difficulties With the Modern Theory of Limits - Real Numbers + Limits Math Foundations
Explore the logical challenges in the modern theory of limits in this 37-minute video lecture. Delve into the formal definition of sequence limits, initiated by Bolzano, Cauchy, and Weierstrass, and uncover the serious logical problems it presents. Examine extended harmonic sequences, including those formed by the Collatz game, to reveal the generality and inaccessibility of limit discussions. Investigate the fundamental issue of infinite computations required by the definition and question their mathematical validity. Learn about definitions in modern analysis courses, inductive definitions of extended harmonic sequences, and the concept of sequences and multiplicities. Discover the great dichotomy with 'sequences' and explore the choice approach to sequences in this thought-provoking mathematical exploration.