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A closed term contains no variables, so its value in a first-order structure does not depend on an assignment. In a language containing only relations and the constant zero, zero is the only closed term. A term-generated model has every element denoted by such a term.

A term-generated model is a structure in which every element is the value of a closed term of the original language. Equivalently its substructure generated by the empty set, including interpretations of named constants, is the entire structure. This is the relevant meaning of “term model” for a model of an arbitrary complete theory, rather than a claim about one particular syntactic construction.

Each closed term $t$ gives the parameter-free defining formula $x=t$. ThusThe next parts show that pointwise definability implies atomicity, and countable atomicity is equivalent to primeness. Therefore property (i) implies all three other properties.