There are two ways of viewing definitions of mathematical properties that I am comfortable with. By property, I mean a statement about one or more sets, that forms a proposition when the subjects are fixed. Let $P$ be an $n$-arry property and $x_1,\ldots,x_n$ be sets, then $P(x_1,\ldots,x_n)$ is a proposition which is the case when $x_1,\ldots,x_n$ has the property $P$, and false otherwise. When I take classes and work on the proofs, it becomes relevant to how these properties are treated.
The current sense I have of definitions of properties is that they are abreviations or "macros". For example, $A\subseteq B$ is an abbreviation for $\forall x(x \in A\to x \in B)$, so the property is $\_\subseteq \_$ and $A\subseteq B$ is the proposition. In this way though, the property is not a component of the underlying theory/formal system/logic/ZFC, but just a shorthand of the logical sentence that the notion "subset" is said to mean. I like this view since it makes proofs unconfusing, as proving a statement in the form of $A\subseteq B$, is exactly the same as proving $\forall x(x \in A\to x \in B)$. In addition, it makes the concept of a "mathematical theory" less segregated, since it doesn't necessarily force additional logical machinery to handle all of the definitions and axioms. However, this is in contrast to the way first order logic is explained in many presentations.
The alternative view (which I suspect is incorrect) is where
- there is an underlying logic or formal system: ZFC with some notion of predicates (FOL?),
- each definition of a property corresponds to (1) predicate that has a name, and (2) a logical sentence schema for deciding that properties truth, for which an axiom is appended that relates the predicate with sentences of the form of the schema, via bidirectional implication,
- to prove a property, you must prove the sentence that the property is assumed to be if and only if the sentence is true.
This differs from the first in the sense that properties are now separate from the logical statement they describe, but linked uniquely via $(\iff)$ to its defined statement. Another way to make sense of the distinction is that in this case, $A\subseteq B$ would not mean the same thing as $\forall x(x \in A\to x \in B)$, as they would be separate statements/sentences in the logic, but would be logically equivalent.
Is there any merit to the second view?
