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@XenaProject @KamerynJW I actually just came across this passage from Conway's "On Numbers and Games" I'm helping a student work through. It seems he'd suggest the solution is to wait until you have R^+ before constructing R if you're using Dedekind cuts.
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@XenaProject @KamerynJW Or you can define R as a subset of the surreal numbers, where addition and multiplication are already defined, by taking Dedekind cuts of dyadic rationals.