M Theory Lesson 278
From a logos perspective, the axioms for fields are rather messy, and they should not be considered in the context of ordinary sets. Set theory doesn't even know the difference between the continuum cardinality and other choices, so why do we use it to inspire definitions of categories? Actually, category theorists have thought about this for a long time, and there are many kinds of category capable of all the important things that sets are capable of, but which aren't at all like the usual category of sets.
M theorists need to learn more about these alternatives. For example, today David Corfield brings our attention to the concept of pretopos. If one delves a little into this idea, the rationals and finite fields start to look even more remote from the (not uniquely defined) reals than they do in a topos!