A complete theory T has a prime model P. A different model M of T is constructed by adding an element that realizes a type not realized in P. What is the relationship between P and M?
AM elementarily embeds into P, because P is the smallest model and must contain M
BP elementarily embeds into M, because a prime model embeds elementarily into every model of T
CP and M are isomorphic, because prime models are unique up to isomorphism
DNeither model embeds into the other, because M realizes a type that P does not
By definition, a prime model P embeds elementarily into every model of T — that is what 'prime' means. The embedding j: P → M is injective and preserves all first-order formulas: φ holds of a tuple in P if and only if φ holds of the corresponding tuple in M. The fact that M realizes an additional type (one not realized in P) means M is strictly larger — it is not isomorphic to P. But P still embeds into M, just not surjectively. Option C is wrong because M has strictly more structure than P (extra elements realizing new types); option D confuses 'not isomorphic' with 'no embedding exists.'
Question 2 Multiple Choice
Which characterization best describes what makes a model of a complete theory T a 'prime model'?
AIt is the largest model of T that satisfies all the axioms with no redundant elements
BIt realizes every type consistent with T, ensuring maximum expressibility
CIt realizes only principal (isolated) types — types forced by a single formula in T
DIt is the unique countable model of T, as guaranteed by the Löwenheim-Skolem theorem
A prime model is an atomic model: every element (and every tuple) in P realizes a principal (isolated) type — one isolated by a single complete formula φ(x) such that T ⊢ ∀x(φ(x) → ψ(x)) for every ψ in the type. Isolated types are 'forced' by the theory — any element satisfying φ must satisfy the entire type, so there is no freedom. A prime model contains only such forced elements, making it as small as possible. Option B describes saturated models (the opposite extreme). Options A and D are incorrect characterizations.
Question 3 True / False
A prime model of a complete theory T is characterized by the fact that it elementarily embeds into every other model of T.
TTrue
FFalse
Answer: True
This is the defining property of prime models: P is prime for T if and only if for every model M of T, there exists an elementary embedding j: P → M. An elementary embedding preserves all first-order formulas (not just quantifier-free ones), so P is not just a substructure of every model — it is an elementary substructure. This universal embedding property is precisely the sense in which P is 'minimal': you cannot remove any element without losing the ability to embed, because every other model must contain an isomorphic copy of P.
Question 4 True / False
A saturated model and a prime model of the same complete theory T are typically isomorphic, because both represent canonical, uniquely determined structures.
TTrue
FFalse
Answer: False
Prime and saturated models are opposite extremes. A prime model is as small as possible — it realizes only principal (isolated) types, containing only what the theory forces. A saturated model is as large as possible — it realizes every consistent type over every finite parameter set, containing far more than any theory forces. They coincide only in degenerate cases, such as a theory with exactly one countable model up to isomorphism (an ω-categorical theory), where the prime, saturated, and universal models all collapse to the same structure. In general, these are very different structures with very different cardinalities.
Question 5 Short Answer
What does it mean for an element in a prime model to realize a 'principal type,' and why does this characterize the prime model as containing nothing beyond what the theory forces?
Think about your answer, then reveal below.
Model answer: A principal type p(x) is a complete type that is isolated by a single formula φ(x): the theory T proves that any element satisfying φ must satisfy every formula in p. In other words, φ alone determines all first-order properties of the element — you get the entire type 'for free' once you have one formula. An element realizing a non-principal type has properties that cannot all be determined by any single formula; its full description is a 'bonus' not entailed by the theory axioms. A prime model contains only elements whose types are principal — every element is entirely determined by some formula in T. This means the prime model includes no extra structure: everything in it is forced by the theory itself, and nothing is added as an unexplained contingent fact.
The contrast with non-principal types clarifies the intuition. A non-principal type might specify, say, 'this element is not definable over the empty set in any specific way' — its existence in a model is optional, not forced. Prime models include no such elements. This is also why prime models are called 'atomic': in the Boolean algebra of definable sets, every element of P is an atom — fully pinned down by a single formula.