In this post, we attempt to better understand the questions posed in this post and its follow-up by formalizing them generally in the language of universal algebra.
Tag: Series: Algebraic Structures Applicable to All Sets
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General Field Applicability to Infinite Sets
In this post, we undertake a more focused investigation into the conjecture that every infinite set can be turned into a field, as stated in this post. (We assume the Axiom of Choice throughout.)
Categories
Algebraic Structures Applicable to All Sets
This exploration is inspired by the fact that, given the Axiom of Choice, any set can be given a group structure. (Fine print: throughout the rest of this post, we assume all sets are nonempty.) In fact, these two statements are equivalent. Questions about similar statements have popped up in other contexts; for example, in Topology as an Algebraic Structure, the question arises as to whether any infinite set can be given a field structure. (Clearly, not every finite set can be given a field structure, since we have the classification of finite fields.)
