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Then John H. Conway's Surreal Numbers might be of interest to you. Donald Knuth wrote a book about the subject and it's actually in the form of a story with dialogue, not your typical math book. I'll attempt a summary. It is possible to build all number systems at once including all integers, rationals, algebraic, irrational, infinitesimal, hyper-reals and transfinite all at once using a single procedure which resembles Dedekind cuts. From that foundation one can recreate many of the results of non-standard analysis. Conway discovered his Surreal Numbers while investigating Combinatorial Game Theory. He wasn't even trying to create a new foundation for mathematics, he was just playing at his games which many of his colleagues looked down upon as useless. Apparently playing games and having fun can occasionally result in extraordinary discoveries.

Conway's left and right "cuts" look like sets but then again I'm not really sure what they are. Is this alien mathematics accidentally discovered on Earth by perhaps the most intelligent Earthling who ever lived. I don't know. All I can say is I took a detour from my usual math studies to look into Surreal numbers and it was an interesting trip. I'm quite conservative when it comes to math. I don't go around preaching Surreal numbers are the way to do math but it is a curiosity none-the-less worth at least the small price of reading Knuth's short book about it from cover to cover.




If you listen to Conway talk about the Surreals you'll note that he considered it a game (in the game theory sense). He was interested in games of a certain kind and what properties they had, a long the way he stumbled upon this game that we now call the surreal numbers.




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