Complexity of well-ordered sets in an ordered Abelian group

Abstract: We consider the following three basic problems, plus some variants.

1. How hard is it to say of a countable well-ordering that it has type at least $\alpha$?

2. How hard is it to say of well-ordered sets $A,B$ in an ordered Abelian group $G$ that the set $A+B = \{a+b:a\in A\ \&\ b\in B\}$ has type at least $\alpha$?

3. How hard is it to say of a well-ordered set $A$ of non-negative elements in an ordered Abelian group $G$ that the set $[A]$ consisting of finite sums of elements of $A$ has type at least $\alpha$?

Each problem asks the complexity of membership a smaller class $K$, assuming membership in a larger class $K^*$. We want to measure complexity in the Borel and effective Borel hierarchies. However, the classes $K^*$ and $K$ are not Borel. Calvert's notions of complexity and completeness within allow us to measure complexity in the way we want, setting upper bounds, and showing that the bounds are sharp.

Authors: Chris Hall, Julia Knight, and Karen Lange

logic

Audience: researchers in the topic

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Online logic seminar

Series comments: Description: Seminar on all areas of logic

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