Conway’s surreal numbers didn’t come up, which I think is his most interesting idea that I can sort-of understand and which gave new ways of thinking about infinities.

You might like to check out the post on my linked blog: “A Curious Way to Represent Numbers: Ternary Factor Tree Representation”, which connects to some of your interests. TFTR is a simple, n-dimensional, recursive, entirely multiplicative/exponential representation of real numbers (and others), including exact representation of irrational numbers, which resembles the cellular structure that automata live in and which has some interesting potential applications to infinities. The blog post is not rigorous, has a lot of conjectures, but is easily understandable and has brief, readable working computer code for converting from the representation to rational or floating-point. I think it gives an infinite, totally orderable sequence of countable infinite numbers spaced infinitely far apart, but I’m doubtful about my argument as to why Cantor diagonalization does not apply and whether repeating pseudo-random infinite sequences are orderable. Prof. Ono at Emory drew connections between TFTR and the “theory of valuations, heights, and most prominent in the theory of abeles and ideles.” [“ideals”, I think] As far as I can tell these are only vaguely similar. I thought about emailing Conway about TFTR since it seems similar in motivation to the surreals, but chickened out.

The posts: “Thermodynamics, Information and the Afterlife” and “A First Approximation to Mindspace” are also ones you likely would like.

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