Equalfinality

Equalfinality is a concept in logic and set theory that refers to the property of two structures being equally final with respect to a given class of structures. In other words, it means that both structures are the "most complete" or "most saturated" members of that class. This concept is particularly relevant in the study of models of set theory and the foundations of mathematics.

The term "equalfinality" was introduced by the logician Saharon Shelah in the context of his classification theory of models of set theory. It is closely related to the notion of "finality," which refers to the property of a structure being the "most complete" or "most saturated" member of a given class. Two structures are said to have equalfinality if they are both final with respect to the same class of structures.

Equalfinality is a useful concept in the study of large cardinals and the consistency strength of various axioms in set theory. It allows mathematicians to compare the complexity of different models of set theory and to understand the relationships between different large cardinal axioms. For example, two models of set theory may have equalfinality if they satisfy the same large cardinal axioms and have the same cardinal characteristics.

In summary, equalfinality is a concept in logic and set theory that refers to the property of two structures being equally final with respect to a given class of structures. It is a useful tool in the study of large cardinals and the consistency strength of various axioms in set theory.