This essay is part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results in modern logic and their significance for philosophical and systematic theology.
Gödel’s completeness theorem established a remarkable correspondence between syntax and semantics: if a sentence follows semantically from a set of first-order premises, then it can also be formally proved from those premises. The incompleteness theorems then showed that sufficiently strong formal theories cannot decide every sentence expressible within them.
The Löwenheim–Skolem theorems reveal a different limitation, one not primarily concerning proof but models. Even when a first-order theory says enough to describe an infinite structure in considerable detail, the theory may prove unable to determine how large its models must be. A theory possessing one infinite model will, under the usual conditions, possess models of very different infinite sizes.
The result is one of the deepest lessons of modern logic: a theory may say a great deal about a structure without uniquely determining the structure that satisfies it.
From Sentences to Structures
A first-order theory consists of sentences in a formal language. A model of that theory is a structure in which all those sentences are true.
Suppose, for example, that a language contains a two-place relation symbol R and that a theory says various things about how objects are related by R. One model might contain ten objects, another a thousand, and another infinitely many. Whether all these structures are possible models depends upon what the theory actually says.
If a theory explicitly says that there are exactly three objects, then a model containing four objects will not satisfy it. But infinite structures behave differently. Once a first-order theory has an infinite model, the Löwenheim–Skolem results severely restrict the theory’s ability to determine the cardinality of its models.
This is sometimes described as the elasticity of first-order theories.
The Downward Löwenheim–Skolem Theorem
The result begins historically with Leopold Löwenheim and was subsequently sharpened and clarified by Thoralf Skolem.
In one familiar form, the downward Löwenheim–Skolem theorem says:
If a first-order theory in a countable language has an infinite model, then it has a countable model.
Here “countable” means that the members of the model can, in principle, be placed into one-to-one correspondence with the natural numbers:
1, 2, 3, 4, …
This is surprising because the original model might be enormously larger than countable. It might contain uncountably many objects. Nevertheless, if the language is countable and the theory has an infinite model at all, then there is also a countable structure satisfying exactly the same theory.
A more structural formulation says that an infinite structure in a suitably small language has a smaller elementary substructure. We sometimes write:
M ≺ N.
Read: M is an elementary substructure of N.
This means much more than merely saying that M is contained within N. The smaller structure preserves the first-order truths of the larger structure, at least with respect to elements belonging to M. If a first-order formula with parameters from M is true in N, it is also true in M, and conversely.
The smaller structure can therefore be genuinely smaller while remaining indistinguishable from the larger one by the relevant first-order formulas evaluated on its members.
That is already philosophically striking.
The Upward Löwenheim–Skolem Theorem
The result also runs in the other direction.
In simplified form:
If a first-order theory has an infinite model, then it has models of arbitrarily large infinite cardinalities.
Thus a theory that has one infinite model ordinarily does not merely admit a countable alternative. It has models larger and larger without end.
Suppose a theory T has an infinite model. Then, under the appropriate conditions, T will have a model of cardinality ℵ₀, another of cardinality ℵ₁, another of still greater cardinality, and so forth through arbitrarily large infinite sizes.
We should be careful about what this does and does not mean. It does not follow that every structure can be enlarged or reduced arbitrarily while preserving all of its properties. Nor does it follow that cardinality is irrelevant. The theorem concerns what can be controlled by first-order theories.
The point is instead that first-order description has a remarkable inability to pin down the size of an infinite model.
This has an important consequence. If a first-order theory has an infinite model, it cannot be categorical across all infinite cardinalities. That is, it cannot have exactly one model up to isomorphism when models of every infinite size are considered, because models of different cardinalities cannot be isomorphic.
The theory may characterize much, but it cannot characterize everything.
The Skolem Paradox
The most famous philosophical puzzle associated with these results appears when they are applied to set theory.
Standard set theory proves that there are uncountable sets. The real numbers, for example, are uncountable: there can be no one-to-one correspondence between the natural numbers and the real numbers.
Yet set theory can be formulated in a countable first-order language. If that theory has a model, the downward Löwenheim–Skolem theorem tells us that, under the relevant assumptions, it has a countable model.
We now seem to have a contradiction.
The countable model satisfies the sentence:
The real numbers are uncountable.
Yet from outside the model we can count all the objects in its domain, including the objects that the model takes to constitute the real numbers.
How can a countable model contain something it correctly describes as uncountable?
The answer lies in understanding what “uncountable” means inside the model.
To say that a set R is uncountable is to say that there is no bijection between the natural numbers and R. But when the model says that no such bijection exists, its quantifiers range only over functions and objects available within the model.
From outside the model, we may be able to define or identify a correspondence that enumerates the members that the model calls “the reals.” But that correspondence need not itself be an object belonging to the model.
Consequently the model can correctly satisfy:
There is no bijection between the natural numbers and the real numbers
even though someone standing outside the model can enumerate all the members of the model.
There is therefore no formal contradiction. What appears paradoxical arises because “there exists a function” is interpreted relative to the structure in which the sentence is being evaluated.
The Skolem paradox is thus not really a contradiction but a lesson in semantics.
What the Paradox Teaches
The lesson is easy to underestimate. Truth in a model depends not only upon the sentence being considered but also upon the domain over which its quantifiers range and the interpretations assigned to its nonlogical vocabulary.
When a model says:
There is no function f with property P,
the quantifier “there is no function f” ranges over what the model recognizes as functions. It does not automatically range over every object that some external observer might regard as a possible function.
The distinction between the internal and external standpoint therefore becomes crucial.
From within the model:
R is uncountable.
From outside the model:
The collection of objects that the model takes to constitute R is countable.
Both statements can be true because they are made relative to different domains of quantification.
This is one reason model theory proved philosophically explosive. Formal semantics forces us to ask not merely whether a sentence is true, but true in what structure, under what interpretation, and with quantifiers ranging over what domain?
What Might Theology Learn?
The Löwenheim–Skolem theorems do not show that theological language is hopelessly indeterminate, nor do they prove that religious doctrines can have any interpretation one wishes. Still less do they establish theological relativism. Such conclusions would greatly outrun the mathematics.
Their theological importance lies elsewhere.
Whenever theology is formalized, one must distinguish between a theory and the structures satisfying that theory. A set of theological sentences may impose substantial constraints upon its models without uniquely determining one model. The fact that several structures satisfy the same sentences therefore need not indicate ambiguity or inconsistency; it may instead disclose something about the expressive resources of the language in which the theory has been formulated.
Suppose, for example, that a theological theory T contains propositions concerning creatures, divine action, dependence, justification, or participation. We can ask whether a proposed structure M satisfies T:
M ⊨ T.
Read: the model M satisfies the theory T.
But suppose another structure N also satisfies T:
N ⊨ T.
It does not follow merely from these two facts that M and N are the same structure, or even that they are isomorphic. The same formal theory may admit genuinely different models.
This matters because theology often moves too quickly from the claim that a doctrinal formulation is true to the assumption that the formulation uniquely determines the metaphysical structure making it true. Model theory forces those claims apart. A theory may constrain reality without exhausting every structural feature of the reality that satisfies it.
The point becomes particularly important when theology employs language about totality, infinity, divine knowledge, created orders, or relations among persons. The Löwenheim–Skolem theorems remind us that what a formal language can distinguish depends upon its expressive resources. Two structures may differ substantially while remaining indistinguishable with respect to the sentences available in a particular first-order theory.
This does not imply that reality itself is indeterminate. It implies that description and determination are different things.
A map can fail to distinguish two terrains without the terrains themselves becoming identical. In much the same way, a formal theological language may fail to distinguish structures that differ in respects the language cannot express.
There is consequently a methodological warning here. The theologian should not infer:
Our theory has a model; therefore we have uniquely described the reality under discussion.
Nor should one infer:
Two models satisfy the same theological theory; therefore there is no fact of the matter about which structure is correct.
Neither conclusion follows.
The first overestimates the expressive power of the theory; the second confuses limitations upon description with limitations upon reality.
Intended Models and Theological Reference
The Löwenheim–Skolem results therefore raise a question that becomes increasingly important in the philosophy of logic: if many structures satisfy the same theory, what makes one of them the intended interpretation?
Mathematics encounters this question when it speaks of the natural numbers or the set-theoretic universe. Theology encounters an analogous problem whenever formal representations are used to speak about God, creation, Christ, justification, or the Trinity. The formal theory does not itself guarantee that every model satisfying its sentences captures everything the theologian intends to say.
Something more may be required: historical usage, semantic intention, causal relations, practices of reference, further axioms, richer logical resources, or substantive metaphysical commitments.
The important point is not that formalization fails. Quite the contrary. Formalization succeeds precisely by revealing where the formal theory ends and further philosophical questions begin.
Löwenheim and Skolem thus teach theology something different from Gödel. Gödel showed that formal proof has limits even within sufficiently strong theories. Löwenheim–Skolem shows that semantic description has limits of another sort: an infinite first-order theory may be satisfied by structures of radically different sizes.
The resulting lesson is both modest and profound. A theory is not its model, and a model satisfying a theory need not be the only model capable of doing so.
For philosophical theology, that distinction is indispensable whenever we ask what our doctrines say, what structures make them true, and how much of theological reality those doctrines formally determine.
The natural next step is compactness, for compactness explains another remarkable feature of first-order theories: if every finite portion of a theory can be satisfied, then the entire theory can be satisfied. Together with Löwenheim–Skolem, this result will show just how surprising the relation between local consistency and global model existence can become.
Bibliographical Note
Leopold Löwenheim’s foundational result appeared in “Über Möglichkeiten im Relativkalkül” (1915). Thoralf Skolem subsequently reformulated and strengthened the result in several papers, including “Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze” (1920) and “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre” (1922), the latter containing the discussion that gave rise to what came to be called the Skolem paradox.
For modern treatments, the Löwenheim–Skolem theorems are standard results in model theory and mathematical logic. Useful sources include C. C. Chang and H. Jerome Keisler, Model Theory; Wilfrid Hodges, A Shorter Model Theory; and standard introductions to mathematical logic treating elementary substructures, cardinality, and first-order theories. Philosophically, the Skolem paradox has remained important because it raises enduring questions concerning reference, intended interpretation, internal and external perspectives, and the relation between formal theory and mathematical structure.
