One way to get infinitely many terms is to allow our vocabulary to have infinitely many object constants. While there is nothing wrong with this in principle, it makes the job of axiomatizing things effectively impossible, as we would have to write out infinitely many sentences in many cases.
In this lesson, we explore an alternative to Relational Logic, called Herbrand Logic, in which we can name infinitely many objects with a finite vocabulary. The trick is to expand our language to include not just object constants but also complex terms that can be built from object constants in infinitely many ways. By constructing terms in this way, we can get infinitely many names for objects; and, because our vocabulary is still finite, we can finitely axiomatize some things in a way that would not be possible with infinitely many object constants.
