Still have a paper, a programming project, and a final exam left, but this is a nice load off my back (and an interesting topic).
Shattering Hilbert's Spirit: Natural Independence Phenomena and the Goodstein Sequence (or My Function Grows Too Fast For Your Formal System)While Godel's theorem demonstrated that Hilbert's goals were impossible, it did so in a self-referential way which bothered many mathematicians. The discovery of natural propositions which were independent of arithmetic demonstrated that arithmetic did lack power as a formal system, there were true theorems within it that required a more powerful formal system to prove. Such proofs tend to involve fast-growing functions, and there is a poorly understood but potentially fruitful connection between the speed of growth of functions and the power of the formal systems required to prove their properties. We explore these topics and present our own intuitive explanation of the convergence of the Goodstein sequence, which requires little mathematical background.
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