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In the "Type Theory for the Working Rustacean" video, Dan Pittman explains how type theory works in Rust, and how dependent types can help to improve the precision and formality of semantics. He demonstrates how to use a dependent type to represent the natural numbers, and how this can be used to avoid type mismatch errors.

**00:00:00**Type theory is a formal system that helps mathematicians reason about mathematical objects. In the early days of type theory, Russell found that the theory was susceptible to self-referential problems. To address this problem, he developed the idea of stratification, which divides the theory from the objects it describes. However, this idea was later found to be problematic, and so subsequent type theorists developed a hierarchy of types that can be self-referenced.**00:05:00**In this video, Dan Pittman explains how type theory works in Rust, and how dependent types allow for more precise, formal semantics. Pittman also discusses an example of a dependent type in Rust, and how it enables proof-carrying.**00:10:00**The "Type Theory for the Working Rustacean" video by Dan Pittman discusses the natural numbers, which are a set of numbers that are defined using the 0 and successor constructors. Embedding the natural numbers into a data sight that carries a proof using it allows Rust code to use the numbers without having to worry about the proofs. The video also demonstrates how to add elements to a vector using the push function.**00:15:00**This video discusses the problem of type mismatch errors in Rust programs, and how to avoid them by writing parts of the program in the type system. It also introduces the idea of implementing dependent types in Rust, which can help to catch errors early.

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