Explore the logical weaknesses in using Cauchy sequences as a foundation for real number theory in this 52-minute video lecture. Delve into topics such as infinite sequences of rationals, choice and algorithmic approaches to infinite sequences, ambiguities in Cauchy requirements and equivalence classes, and the implications for rational numbers. Examine the problems with "Cauchy sequence real numbers" through detailed explanations and examples. Gain critical insights into the foundations of mathematics and challenge conventional understanding of real number theory.