Explore the complex relationship between arithmetical expressions and natural numbers in this thought-provoking 27-minute video lecture. Delve into the challenges of defining and working with numbers expressed through elaborate arithmetical expressions, including towers of exponents and hyperoperations. Examine the limitations of primitive natural numbers and Hindu-Arabic numerals, and investigate the potential difficulties that arise when attempting to establish a consistent arithmetic system for complex expressions. Analyze various types of mathematical expressions as objects, discuss logical issues surrounding numerical expressions, and consider the concept of mid-level expressions. Conclude by exploring the possibility of developing a coherent system for arithmetical expressions, gaining valuable insights into the foundations of mathematical thinking and the nature of numbers themselves.