
Shepherdson
above, as recursive definitions of the predicates in their heads, it is the model
which corresponds to these recursive definitions.
In general, however, T
p
is not monotonie and may not have fixpoints. For
example, if Ρ is q -iq, -ir then Τ
ρ
(φ) = {q} but T
p
({q}) = φ; there is
no fixpoint of T
p
and no least Herbrand model—there are two minimal
Herbrand models {q}, {r} (i.e., models with no proper submodel), but their
intersection is not a model.
Although all that is required for I to be a model of Ρ is that T
P
(I) C I i.e.,
that I is a pre-fixpoint of T
p
, fixpoint models have a special interest, par-
ticularly in the context in whic ...