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The term "class" is not a technical term with a universally definite meaning, but there are various established meanings in various contexts. In ZFC the established usage as Wojowu mentions in the com …

Here is one way to view the so-called ordering principle as a selection principle. Theorem. The following are equivalent over ZF set theory: Every set admits a linear order. For every set $X$, there …

Therefore, unlesss there is already an established terminology for your ordinals coming from fine-structure theory (see below), I would suggest the terminology: $\alpha$ is sententially categorical with … I am not sure if they isolate your notion exactly with terminology, but that is where I would look. …

The elements of this partition are precisely the atoms of the complete Boolean algebra generated by the family.

In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders) … I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees. …

Let me mention as a counterpoint that there is less need for new terminology than one might expect. … Mathematical exposition is often more successful and clearer without new terminology, and one should consider whether one needs any new terminology at all. …

To my way of thinking, the other answers are missing an important element, a necessary feature for a mathematical tool or method to be called "trick." Namely, in order to be called a "trick," a metho …

In weak set theories, using classical logic and interpreting "small subclass" as "set", this principle amounts to an alternative formulation of the collection axiom. For example, in Zermelo set theory …

Your question amounts to treating construction problems in geometry as decision problems, and so it makes sense to me to adopt the terminology of computability theory. … This same kind of distinction arises in computability theory, where we have the following terminology: A set $A$ is decidable if we can computably verify yes-or-no whether a given input $a$ is in $A$ …

The answer is no for uncountable cardinals $\kappa$. Let $\mathfrak{A}=\langle A,U\rangle$ be a set $A$ of size $\kappa$ with a unary predicate $U\subset A$, where $U$ and $A-U$ both have size $\kappa …

recasting of entire schemes of terminology to focus on what was all along the actual focus, namely, the concept of computability, rather than specifically recursion. … This text was called, "Recursively enumerable sets and degrees," using the old terminology, but Soare preferred to recast it as "Computably enumerable sets and degrees." …

This terminology aligns with the similar terminology for superhuge cardinals and super $n$-huge cardinals, which seems to be fairly established. … For this reason I would recommend against introducing a new terminology without very good reason. …

The subscript $0$ is meant to indicate the amount of induction that the theory has. The wikipedia entry on Reverse mathematics says of the big five theories of reverse mathematics that The …

Define an order on pairs of ordinals $(\alpha,\beta)$ by ordering first by maximum, then by first coordinate, then by second coordinate. That is, one pair preceeds another if the maximum is smaller, o …

This is called an $L$-valued relation, when $L$ is the target of the function, which can be viewed as the collection of possible truth values. Thus, a $2$-valued relation is just an ordinary relatio …