Questions: First-Order Types and Partial Descriptions

Question 1

Theory T is complete. A set of formulas p(x) = {φ₁(x), φ₂(x), φ₃(x), ...} with infinitely many formulas is consistent with T, but no single element in any countable model of T satisfies all formulas in p simultaneously. What can we conclude?

• A. p(x) is inconsistent with T, since no model realizes it
• B. p(x) is a complete type, since T is complete and decides every formula
• C. p(x) is a consistent partial type that is omitted in all countable models of T — whether this is possible is characterized by the Omitting Types Theorem
• D. p(x) cannot exist for a complete theory, since complete theories realize all consistent types

Answer: (C)

Consistency with T means every finite subset of p(x) is satisfiable in some model of T — but this doesn't imply simultaneous satisfaction by a single element. A type that is consistent with T but not realized in any countable model is called *omitted*. The Omitting Types Theorem characterizes exactly when this is possible: a consistent type p(x) can be omitted in a countable model if and only if p is not 'isolated' (not implied by any single formula consistent with T). This shows that non-isomorphic countable models of the same complete theory can exist — they differ in which types they realize.

Question 2

In which sense does a complete 1-type over T give a 'finer' description of an element than any single formula φ(x) consistent with T?

• A. A complete type is a single formula that uniquely identifies an element, while φ(x) may be satisfied by many elements
• B. A complete type is a maximal consistent set of formulas — for every formula ψ(x), either ψ or ¬ψ is in the type — so it decides every first-order question about the element
• C. A complete type includes second-order formulas that describe the element's role in all possible models, not just in T
• D. A complete type assigns a probability to each formula being satisfied, unlike a single formula which is all-or-nothing

Answer: (B)

A complete 1-type is a maximal consistent set of first-order formulas: for every formula φ(x) in the language of T, either φ or ¬φ is in the type. This means the type settles every first-order question about the element — no formula in the language is left undecided. A single formula φ(x) may be satisfied by many structurally different elements; the complete type pins down the element as precisely as the first-order language allows. Two elements have the same complete type if and only if the theory T cannot distinguish them using any first-order formula.

Question 3

If two elements a and b in models of the same complete theory T have the same complete 1-type, then T cannot distinguish them using any first-order formula.

• T. True
• F. False

Answer: True

This is the definition of having the same complete type. A complete 1-type is the collection of all formulas (in the language of T) satisfied by an element. If a and b have identical complete types, then for every formula φ(x), φ(a) is true iff φ(b) is true — T's language has no formula that holds for a but not b, or vice versa. Note this does not mean a and b are the same element, or even in the same model — it means that within the descriptive reach of T's first-order language, they are indistinguishable.

Question 4

A complete type over a theory T is just a single formula φ(x) such that T ⊢ ∀x(φ(x) → ψ(x)) for most other formula ψ(x) consistent with T.

• T. True
• F. False

Answer: False

A complete type is not a single formula but a maximal consistent *set* of formulas — typically infinite. The description 'T ⊢ ∀x(φ(x) → ψ(x)) for all consistent ψ' would describe an *isolated* type (one implied by a single formula), which is a special case. Most complete types are not isolated — they cannot be generated by any single formula. The Omitting Types Theorem turns on exactly this distinction: non-isolated types can be omitted in countable models, while isolated types are realized in every model of T. Confusing a type with a single formula misses the fundamentally infinitary character of type theory.

Question 5

What is the difference between a realized and an omitted type, and why does this distinction show that two models of the same complete theory need not be isomorphic?

Model answer: A type p(x) is realized in model M if some element a ∈ M satisfies every formula in p simultaneously. It is omitted if no such element exists in M. Since a complete theory T can have consistent types that are not realized in every model — the Omitting Types Theorem characterizes when this is possible — two models of T can differ in which types they realize. If model M₁ realizes type p but model M₂ omits it, then M₁ has an element with a specific 'description' that M₂ lacks entirely, so M₁ and M₂ cannot be isomorphic: an isomorphism would have to map the element realizing p in M₁ to something in M₂, but no element of M₂ satisfies p.

This is one of the core reasons that first-order theories can have non-isomorphic models (beyond the trivial reason of different cardinalities). Even among countable models of the same complete theory, type realization and omission creates structural diversity. Classifying which types are realized in which models — and constructing models that realize all types (saturated models) or omit specific ones — is a central technique in model theory.