Every sentence derivable from a theory is true in every model of that theory. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.
Set-up
A first-order language \(\mathcal L\) supplies symbols, formulas, and structures. Semantic consequence \(T\models\varphi\) is truth in every model, while provability \(T\vdash\varphi\) is a finite formal derivation.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
The calculation
An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.
What survives abstraction
The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.
The boundary
First-order compactness does not apply to arbitrary second-order properties. Finiteness and categorical characterization of the natural numbers lie beyond its direct reach.
This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.