Löwenheim–Skolem: When a Theory Cannot Control the Size of Its Models

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.

Gödel: Completeness, Incompleteness, and the Limits of Formal Reason

This essay is a product of the Department of Philosophical Theology at Christ School of Theology, Institute of Lutheran Theology, and is part of the Disputationes series on important results in logic and their significance for theology.

The foundational work of Frege, Peirce, Cantor, Russell, and Zermelo made possible a remarkable hope. Perhaps mathematics could be placed upon completely explicit foundations, so that one could specify a formal language, identify axioms, formulate rules of inference, and determine exactly what followed from what. Logic would then no longer depend merely upon intuitive judgments concerning valid argument, since proofs themselves could become mathematically tractable objects whose structure and consequences could be precisely investigated.

Kurt Gödel changed our understanding of this project forever, and he did so by proving two results that initially appear to point in opposite directions. His completeness theorem of 1930 demonstrated the extraordinary power of first-order logic, while his incompleteness theorems of 1931 demonstrated equally extraordinary limitations upon sufficiently strong formal theories. To understand why these results are not in conflict, and why both matter for philosophical theology, one must distinguish with some care the logic within which proofs are constructed from the particular theories formulated within that logic.

Completeness: Proof and Truth Meet

Suppose we have a collection of sentences Γ and another sentence φ. We can then ask two different questions, one syntactical and the other semantical. Does φ follow syntactically from Γ, so that φ can be derived from Γ by the rules of a specified proof system, or does φ follow semantically from Γ, so that φ is true in every structure in which all the sentences belonging to Γ are true?

In compact notation we distinguish

Γ ⊢ φ

from

Γ ⊨ φ.

The first expression says that φ is formally derivable from Γ, whereas the second says that every model satisfying all the sentences in Γ also satisfies φ. The distinction is basic, for the first concerns what can be proved by operating with formulas according to formal rules, while the second concerns what must be true in any structure in which the premises are true.

Soundness tells us that proof cannot outrun semantic consequence:

If Γ ⊢ φ, then Γ ⊨ φ.

In ordinary English, if φ can be correctly proved from Γ, then φ is true in every model in which Γ is true. A sound proof system therefore never certifies as a consequence something that fails to hold in a model satisfying the premises.

Gödel's completeness theorem establishes the converse for first-order logic:

If Γ ⊨ φ, then Γ ⊢ φ.

That is, if φ is true in every model satisfying Γ, then there is a formal proof of φ from Γ. Thus, for first-order logic,

Γ ⊨ φ if and only if Γ ⊢ φ.

Semantic consequence and formal derivability therefore coincide at the level of first-order logical consequence. This is a magnificent result, not because every mathematical truth becomes formally provable, but because first-order logic possesses a proof system powerful enough to capture every consequence that follows purely in virtue of first-order logical form.

Then Comes Incompleteness

Only a year later Gödel proved something that can sound contradictory if the distinction between a logic and a theory formulated within that logic is ignored. Consider a formal mathematical theory sufficiently strong to express elementary arithmetic, and suppose that its axioms can be effectively specified, so that there is a mechanical procedure for determining whether a given expression is an axiom.

Gödel showed, roughly speaking, that if such a theory is consistent, there will be statements expressible in its language that the theory can neither prove nor disprove. The first incompleteness theorem therefore says, in simplified form, that for any consistent, effectively axiomatized formal theory strong enough to express elementary arithmetic, there are sentences that the theory cannot decide.

There will be a sentence G such that, under the relevant assumptions,

T ⊬ G

and

T ⊬ ¬G.

Read this as saying that the theory T proves neither G nor its negation. The theory is therefore incomplete in the technical sense that some sentence expressible in its language is neither provable nor refutable within the theory.

Gödel achieved this by discovering how arithmetic could, in effect, speak about its own formulas and proofs. Expressions and finite sequences of expressions were assigned numbers—what we now call Gödel numbers—so that claims about formulas, derivations, and provability could themselves be represented arithmetically. This made possible the construction of a sentence which, in a carefully defined sense, says of itself that it is not provable within the theory.

If the theory proved that sentence, the theory would thereby become inconsistent; yet if the theory is consistent, it cannot prove the sentence. The system therefore contains a statement that escapes its own power of demonstration, not because the rules of inference are defective, but because a sufficiently expressive formal theory cannot, under the relevant conditions, settle every sentence formulable within its own language.

Gödel's second incompleteness theorem deepens the point. A sufficiently strong consistent theory cannot, using only its own formal resources, prove its own consistency; thus the limits disclosed by incompleteness concern not merely this or that recalcitrant sentence, but also the capacity of a formal theory to certify from within itself the consistency upon which its deductive enterprise depends.

Why Completeness and Incompleteness Do Not Conflict

The apparent paradox disappears once we notice that Gödel's two results concern different objects. The completeness theorem concerns first-order logic itself and says that every semantic consequence at the level of first-order logical validity can be captured by formal proof, whereas the incompleteness theorem concerns particular formal theories formulated in languages sufficiently rich to express arithmetic and says that no suitably effective, consistent theory of the relevant strength can decide every sentence expressible within it.

First-order logic can therefore be complete as a logic even though particular first-order theories are incomplete as theories. Logic may provide entirely adequate rules for capturing logical consequence without thereby guaranteeing that a given axiomatic theory will settle every question formulable in its language, and failure to keep these two claims distinct has been responsible for a great deal of confusion in both popular and theological appropriations of Gödel.

What Might Theology Learn?

Gödel's theorem is frequently abused in theological argument. It does not prove that God exists, establish that theological truth transcends reason, show that every worldview must contain mysteries, or entail that because arithmetic is incomplete, systematic theology must be incomplete in Gödel's technical sense. Such conclusions trade upon analogy without first establishing that the formal conditions governing Gödel's results apply to theological systems in the required way.

The genuine theological lesson is subtler, for theology itself regularly identifies primitive concepts, formulates doctrinal commitments, draws distinctions among them, and asks what follows from what. Precisely for that reason, it has something to learn from Gödel about the difference between the rigor with which consequences are drawn and the adequacy or completeness of the theory from which they are drawn.

One may possess completely precise rules of reasoning without thereby possessing a theory capable of deciding every question formulable within the theory's language. The rigor of an inferential procedure and the completeness of a theory are therefore different achievements, and formal precision should never be confused with exhaustive conceptual capture.

This distinction matters particularly for philosophical theology because there is no need to oppose formal reasoning to mystery, as though careful logic were somehow hostile to theological depth, nor is there any warrant for supposing that once doctrines have been sufficiently formalized every further theological question becomes mechanically decidable. Formalization can clarify commitments, expose hidden assumptions, reveal inconsistency, and determine consequences; yet what a theory can express is not identical with what that theory can prove, and what follows rigorously from a set of assumptions does not by itself establish that those assumptions are sufficient for everything theology wishes to say.

The validity of an inference must therefore be distinguished from the adequacy of the theory within which the inference is made. A theological argument may be formally impeccable while the assumptions from which it proceeds remain too weak, too narrow, or otherwise insufficient to determine all that theologians wish to assert; conversely, expanding one's theological vocabulary or axiomatic commitments does not by itself guarantee that the resulting system will be more adequate unless one also asks what structures satisfy it and what consequences genuinely follow.

Gödel consequently belongs in theological education not because he supplies an apologetic shortcut, but because he teaches intellectual discipline. His work forces us to distinguish syntax from semantics, logic from theory, truth from proof, and the expressive resources of a language from the demonstrative resources of an axiomatic system; these are distinctions theology needs whenever it attempts to state with precision what it believes and what follows from those beliefs.

Those distinctions become still more important when we turn from Gödel to model theory, for the next question is unavoidable: even when a first-order theory has models, how tightly can its language determine what those models are like? The Löwenheim–Skolem results show that the answer is stranger than one might initially suppose, and with them the problem shifts from the limits of proof to the relation between a theory and the structures capable of satisfying it.

Bibliographical Note

Gödel's completeness theorem appeared in his 1930 dissertation and in the related publication “Die Vollständigkeit der Axiome des logischen Funktionenkalküls.” His incompleteness results appeared the following year in “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I” (1931).

For philosophical orientation, the crucial distinction to preserve is between the completeness of first-order logical consequence and the incompleteness of sufficiently strong, effectively axiomatized formal theories. Standard introductions to mathematical logic and the philosophy of logic provide fuller treatments of both results, while Gödel's original papers remain models of mathematical economy and conceptual force.

When Logic Became Dangerous: Russell, Zermelo, and the Discipline of Totality

This is the second 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.

The great achievements of Frege, Peirce, and Cantor vastly enlarged the expressive resources available to logic and mathematics. Quantification made it possible to speak formally about all or some objects satisfying a condition; the logic of relations permitted systematic treatment of structures involving two or more objects; Cantor showed that infinity itself could be mathematically articulated and that infinite collections could differ in cardinality. Yet precisely this increase in expressive power produced a new problem, for once logic could speak about collections defined by conditions, what prevented us from forming a collection corresponding to any condition whatsoever?

The answer, discovered with particular force by Bertrand Russell, was contradiction.

Consider the apparently innocent idea of the collection of all collections that are not members of themselves. Let us call this collection R. We can describe it this way:

R = {x : x ∉ x}.

Read: R is the collection of all objects x such that x is not a member of itself.

Now ask whether R itself belongs to R. By the very condition defining R, we obtain:

R ∈ R if and only if R ∉ R.

Read: R is a member of itself if and only if R is not a member of itself.

Either answer produces its opposite. If R belongs to itself, then by definition it must not belong to itself; but if R does not belong to itself, then it satisfies the condition for membership in R and therefore does belong to itself.

This was not merely an amusing puzzle. Russell communicated the paradox to Frege in 1902 while the second volume of Frege's Grundgesetze der Arithmetik was in press, and Frege immediately recognized the seriousness of the difficulty. His project had aimed to show that arithmetic could be derived from logical principles, but the paradox exposed a defect in the assumptions governing the formation of extensions or classes. The new logical machinery had become powerful enough to undermine one of its own foundational constructions.

The philosophical lesson is easy to state but profound in its consequences: not every condition that can be expressed determines an object corresponding to that condition.

Suppose we can specify a condition F and say intelligibly what it would be for something to satisfy F. It does not follow merely from this that there exists an object consisting of all and only the things satisfying F, for the movement from a condition to a corresponding totality requires justification. This distinction between specification and object formation became one of the central lessons of twentieth-century logic.

Zermelo and Restricted Set Formation

One response came through Ernst Zermelo and the subsequent development of axiomatic set theory. Instead of assuming a general principle according to which every condition determines a set, Zermelo restricted the circumstances under which sets may be formed. One does not simply move from a condition F to “the set of all Fs”; rather, set formation proceeds according to specified axioms.

The dangerous principle can be expressed simply as:

For every condition F, form the set {x : Fx}.

Read: for any condition F whatsoever, there exists a set containing exactly those objects that satisfy F.

Russell's paradox showed that this principle cannot be accepted without restriction. Zermelo's alternative was more cautious: begin with a set already given, and then select from it those members satisfying a specified condition. Thus, given a set A, one may form:

{x ∈ A : Fx}.

Read: the set of those members x of A that satisfy the condition F.

The difference is crucial because one is no longer permitted to range freely over absolutely everything and then collect into a set whatever satisfies an arbitrary condition. Set formation takes place relative to sets already available within an axiomatic framework, so that the transition from a predicate to a set is controlled rather than automatic.

Russell and the Theory of Types

Russell pursued a different strategy through the theory of types. The underlying intuition was that certain forms of self-reference arise because expressions belonging to different logical levels have been allowed to interact indiscriminately. Individuals occupy one level; predicates of individuals another; predicates of predicates another still. On such an approach, a predicate should not simply be allowed to take itself as an argument.

The slogan is crude but useful: things of one logical type cannot simply be treated as things of every logical type. Russell's solution therefore imposes hierarchy where unrestricted logical construction had permitted self-application, and the point is not merely technical. Logical grammar itself must be disciplined if expressions are not to generate combinations that the theory cannot consistently sustain.

These developments suggest a distinction that remains philosophically important: expressibility is not the same thing as admissible predication, and admissible predication is not the same thing as objecthood. The fact that we can describe something does not yet show that the description determines a genuine object, nor does the grammatical availability of an expression settle the ontological commitments of a theory.

Why This Matters for Theology

The theological relevance is greater than it may first appear because theology regularly speaks in terms that invite totalization: all truths, all possibilities, everything God knows, everything God can do, the totality of creation, or even everything that is not God. Russell's paradox does not show that such expressions are illegitimate, but it does force a distinction between quantifying over things and reifying the domain of quantification into another thing.

Suppose, for example, that we say:

For every x, if x is a creature, then God knows x.

Symbolically:

∀x (Cx → Kgx).

The symbols are simply an abbreviated way of saying: for every object x, if x is a creature, then God knows x.

Nothing in that assertion requires there to be an additional object called the set of all creatures, for the quantifier may range over creatures without thereby packaging the domain over which it ranges into a further entity. The same point applies when theologians speak of divine omniscience. One may say:

For every truth p, God knows p.

That claim does not by itself commit us to the existence of a further object called the set of all truths. Similarly, one may say:

For every creature x, x depends upon God.

Again, nothing in the quantificational structure of the sentence requires that there be one further object called the totality of everything other than God. Quantification alone does not force reification.

The methodological lesson can therefore be stated compactly: quantification should not be confused with reification. To say something of every member of a domain is not yet to say that the domain itself exists as one additional member of one's ontology, and theology is especially susceptible to overlooking this distinction because its characteristic subject matter repeatedly calls forth universal expressions.

There is a deeper theological resonance as well. Christian theology has long had to distinguish between what can legitimately be said of God and what ontological assumptions may be smuggled in by the forms of language used to say it. Russell, Zermelo, and type theory remind us that grammatical or logical form may tempt us into constructing objects that our theory neither requires nor can consistently sustain.

The suspect inference has this form: we can specify what it is to be an F; therefore, there exists one object consisting of all Fs. Yet the first statement does not entail the second. To put the point more carefully, being able to determine of each object whether it is F does not entail the existence of a set containing all and only the Fs. That is precisely the gap Russell's paradox forces us to notice.

The foundational crisis thus yielded a constructive philosophical lesson. Modern logic had acquired enormous power through Fregean quantification and Cantorian set theory, but Russell's paradox demonstrated that expressive power requires discipline. Zermelo supplied axiomatic restrictions upon set formation, while Russell supplied logical hierarchy through types; both responses forced philosophers to distinguish more carefully among language, predication, collection, and existence.

For theology, these distinctions are invaluable precisely because theology regularly attempts to speak about the ultimate, the universal, and the all-encompassing. Whenever theology speaks about all, it should therefore ask a further question: have we merely quantified over everything in some domain, or have we quietly turned that domain into one more thing?The question matters whenever theology attempts to speak of God, creation, possibility, truth, or totality.

Bibliographical Note

The classic primary source for the paradox is Bertrand Russell's 1902 letter to Gottlob Frege, together with Frege's discussion in the appendix to the second volume of Grundgesetze der Arithmetik (1903). Russell developed the theory of types most fully with Alfred North Whitehead in Principia Mathematica (1910–1913). Ernst Zermelo's “Untersuchungen über die Grundlagen der Mengenlehre I” (1908) provided the first major axiomatization of set theory designed in part to avoid the paradoxes generated by unrestricted set formation. For historical orientation, Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 remains invaluable.

Two Revolutions in Modern Logic: Quantification and the Infinite

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.

Most theologians who have studied logic have encountered quantifiers and perhaps heard that Georg Cantor proved that there are different sizes of infinity. What is easier to miss is just how revolutionary these developments were. In the final decades of the nineteenth century, logic ceased to be primarily a theory of propositions of the Aristotelian sort—“All men are mortal,” “Some Greeks are philosophers”—and acquired the resources needed to describe indefinitely complicated structures of objects and their relations. At roughly the same time, mathematics learned that the infinite was not a single undifferentiated beyond, but possessed an articulated internal structure.

Both developments matter for theology, although neither proves anything theological. What they do is enlarge the conceptual space within which theological claims can be formulated, distinguished, and assessed, making it possible to ask with much greater precision what follows from what, what sorts of relations are being asserted, and what kind of infinity is actually at issue.

1. Frege, Peirce, and the Revolution in Quantification

The decisive breakthrough associated with Gottlob Frege’s Begriffsschrift of 1879 was not simply the invention of some new logical symbols. Frege supplied a new analysis of the logical structure of propositions, replacing the limitations of traditional subject-predicate analysis with a framework of functions, arguments, variables, and quantification.

Consider a simple statement:

Every human being is mortal.

In contemporary notation we write:

∀x(Hx → Mx).

Read aloud, this says: for every object x, if x is human, then x is mortal.

The statement does not name some peculiar object called “every human being.” Rather, it says that anything whatsoever in the relevant domain that is human is also mortal, so that its logical form becomes visible only when we distinguish the predicates—being human and being mortal—from the variable over which the quantifier ranges.

The real power of quantification appears when quantifiers are nested. Compare:

∀x∃y Rxy

with

∃y∀x Rxy.

The first reads: for every x, there is some y such that x bears relation R to y.

The second reads: there is some single y such that every x bears relation R to that y.

The difference lies only in the order and scope of the quantifiers, yet the difference in meaning can be enormous. “Everyone loves someone” does not entail “There is someone whom everyone loves,” and once this machinery becomes available logic can represent patterns of dependence that traditional syllogistic logic could scarcely express. Mathematics, science, metaphysics, and theology thereby acquire a far more exact language for describing structures.

Frege was not alone in bringing about this transformation. Charles Sanders Peirce and his collaborators, especially Oscar Howard Mitchell, independently developed powerful systems of quantification during roughly the same period, while Peirce’s work on the logic of relations deserves particular notice. Aristotelian logic had been especially comfortable with one-place predicates such as “is human,” “is mortal,” or “is wise,” whereas Peirce emphasized relations among two, three, or more objects.

Thus:

Rxy

can be read simply as: x bears relation R to y.

A three-place relation,

Rxyz,

says that x, y, and z stand in some specified three-place relation. One might represent “x gives y to z,” for example, by such a structure.

This sounds elementary to us precisely because the revolution succeeded.

For theology the significance is immediate, since Christian theological vocabulary is saturated with relations: the Father begets the Son; the Son is begotten of the Father; the Spirit proceeds; God creates the world; God justifies the sinner; Christ assumes a human nature; believers participate in Christ; promise is addressed to hearer. Such claims cannot adequately be represented merely by assigning properties to isolated objects, for their logical articulation requires relations, often asymmetric relations, and sometimes relations whose formal properties themselves become doctrinally significant.

Consider merely the difference between saying that each divine person is God and saying that the divine persons stand in particular relations to one another. The first set of claims concerns predication, whereas the second concerns relational structure. No piece of predicate logic solves the doctrine of the Trinity, but modern quantificational and relational logic allows theologians to see with far greater precision what kinds of claims they are making and where apparently similar formulations differ logically.

There is another consequence that may be even more important. Quantification makes explicit the distinction between what is true of some object and what is true of every object, and theology constantly moves among existential, universal, and uniqueness claims: there is a God; everything other than God depends upon God; there is exactly one God; every human being is a creature; some human beings believe; every justified sinner stands in a particular relation to Christ. The logical differences among these claims are not stylistic variations, for they determine what follows from them and what further commitments they carry.

The theological payoff, then, is not that Frege or Peirce secretly supplied Christian doctrine with its proper metaphysics. It is rather that modern logic makes possible a level of structural clarity that theological argument badly needs, since quantifier scope, relational order, dependence, uniqueness, and identity can all be made explicit. Much theological disagreement that appears initially to concern “concepts” turns out, upon examination, also to concern logical form.

2. Cantor and the Discovery of Different Infinities

The second revolution came from Georg Cantor. Before Cantor, philosophers and mathematicians had certainly discussed infinity, but infinity was commonly treated as though it were a single notion: the indefinite, the unbounded, or that which simply exceeds every finite magnitude. Cantor showed that infinite collections can themselves differ in size.

The key idea is deceptively simple. Two sets have the same cardinality when their members can be paired one-to-one: every member of the first set is paired with exactly one member of the second, and none is left over. For finite sets this seems trivial. A set containing five books has the same cardinality as a set containing five chairs because each book can be paired with exactly one chair.

Cantor applied the same criterion to infinite sets. Consider the natural numbers:

1, 2, 3, 4, …

and the even numbers:

2, 4, 6, 8, …

At first the even numbers seem to form a smaller collection, since they constitute only part of the natural numbers. Yet every natural number n can be paired with exactly one even number, namely 2n: 1 with 2, 2 with 4, 3 with 6, and so on without end. Thus the natural numbers and the even numbers have the same cardinality, and infinity therefore behaves differently from finite magnitude because a proper part of an infinite set can have the same number of members as the whole.

Cantor’s deeper result was more startling. Not all infinite sets have the same cardinality. His diagonal argument establishes that the real numbers cannot be paired one-to-one with the natural numbers. In ordinary language: there are strictly more real numbers than natural numbers, even though both collections are infinite.

Cantor proved something stronger still. Given any set S, form its power set, written:

P(S),

which simply means the set of all subsets of S. Cantor’s theorem tells us that:

|S| < |P(S)|.

Read in words: the set of all subsets of S is always strictly larger than S itself.

This means that there can be no greatest cardinal number. Begin with any infinity whatsoever, and one can specify a still greater infinity. Cantor’s discovery therefore forced philosophy to reconsider what it meant by “the infinite,” since there is no single mathematical infinity but rather an ordered hierarchy of infinite cardinalities.

The theological temptation at this point is obvious and should generally be resisted. Cantor’s transfinite mathematics does not provide a mathematical model of divine infinity, and God’s infinity should not simply be identified with some enormously large cardinal number. Indeed, because Cantor proved that there is no greatest cardinality, saying that God possesses “the largest mathematical infinity” would not even make mathematical sense.

Precisely here, however, Cantor becomes theologically useful, because his mathematics teaches theologians to distinguish different senses of infinity rather than allowing the word infinite to do undisciplined work. Mathematical infinity concerns the cardinality or ordering of mathematical structures, whereas divine infinity has traditionally concerned the absence of creaturely limitation in God. To say that God is infinite is not ordinarily to say that God has infinitely many parts, occupies infinitely many locations, or instantiates some transfinite cardinality.

Cantor therefore offers theology a conceptual warning: do not infer from the single word “infinite” that two uses of the term belong to the same logical or ontological category. At the same time, his work expands our imagination, for human reason can rigorously distinguish structures beyond every finite magnitude without thereby collapsing into incoherence. The infinite need not simply mean “something too large for us to think,” since certain infinities can be precisely defined, compared, and ordered.

This has interesting consequences for classical theological discussions of divine knowledge. Suppose we say that God knows every natural number, every real number, every subset of the natural numbers, and every mathematical truth. The point is not that divine knowledge thereby acquires some particular cardinality, but rather that Cantorian mathematics demonstrates how quickly apparently simple talk of “all things” becomes structurally complicated. A theology of omniscience must therefore take seriously the logical character of the totalities over which its claims range.

Cantor also helps us resist an old rhetorical maneuver in theology: invoking infinity whenever conceptual analysis becomes difficult. “God is infinite” cannot function as a license for contradiction, for Cantor’s work is a striking historical demonstration that infinity and rigor are not opposites. Modern mathematics became more rigorous precisely by refusing to leave infinity unanalyzed.

Why These Two Revolutions Belong Together

Frege, Peirce, and Cantor transformed different areas of mathematics and logic, yet the two revolutions belong together. Fregean and Peircean logic gave us vastly more powerful ways of saying what is true of objects and structures, while Cantor revealed that the domains over which our variables range may themselves possess unsuspected structure.

Together they prepared the ground for twentieth-century model theory. Once we possess quantifiers ranging over a domain, predicates and relations interpreted over that domain, and a mathematically disciplined conception of infinite structures, questions that would previously have been nearly impossible to formulate become natural: Which sentences are true in a structure? Can two different structures satisfy exactly the same sentences? Must an infinite theory have models of different cardinalities? Can a formal language uniquely characterize the structure it is intended to describe?

Those questions lead eventually to Löwenheim, Skolem, Gödel, Tarski, compactness, and some of the most philosophically unsettling results of modern logic. What begins with the enrichment of logical language therefore becomes a problem about the relationship among language, theory, model, and world.

For theology, however, the first lesson is already substantial. Modern logic teaches us that form matters, because who quantifies over what, which relations hold between which objects, what depends upon what, and what sort of infinity is being invoked are not technical details added after theology has finished its real work. They belong instead to the work of determining what theological claims actually say and what follows from them.

Bibliographical Note

For Frege, the obvious starting point is Gottlob Frege, Begriffsschrift: A Formula Language, Modeled upon That of Arithmetic, for Pure Thought (1879), conveniently available in translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Harvard University Press, 1967). Van Heijenoort’s volume remains especially valuable because it allows readers to encounter many of the foundational texts of modern logic themselves.

The parallel American development should not be overlooked. See Oscar Howard Mitchell, “On a New Algebra of Logic” (1883), and Charles S. Peirce, “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885). For the importance of Peirce and his circle in the development of quantification and the logic of relations, see Geraldine Brady, From Peirce to Skolem: A Neglected Chapter in the History of Logic (North-Holland, 2000).

For Cantor, the classic sources include his 1874 paper establishing the non-denumerability of the real numbers and his 1891 diagonal argument, “On an Elementary Question of the Theory of Manifolds.” Both are widely reprinted and translated. A particularly accessible philosophical introduction remains Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford University Press, 1984). For the broader historical setting, see Joseph W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton University Press, 1979).

Where Have the Theologians Gone? Christ School of Theology and the Changing Geography of Doctoral Theological Education

I recently asked myself what I thought was a fairly straightforward question: How large is the PhD program at Christ School of Theology (CST) when compared with doctoral programs at the institutions that have traditionally trained systematic, philosophical, and historical theologians?

The question arose because CST has 42 PhD students enrolled in 2026–27. They are concentrated in three areas: Philosophical Theology, Systematic Theology, and Historical Theology. Forty-two seemed to me like a substantial number, but substantial compared with what?

When I entered academic theology, the geography was familiar. Yale, Chicago, Harvard, Princeton, Duke, Notre Dame, and the Hyde Park consortium immediately came to mind. If one wanted to become a systematic theologian, historian of theology, or philosopher of religion, these were among the places one considered. They possessed distinguished faculties and large communities of doctoral students. Their graduates populated university and seminary faculties throughout North America.

I assumed that CST's 42 students would still constitute a relatively small program measured against these older centers. That assumption appears to be wrong.

Comparing the Same Thing

One has to be careful with the numbers. A PhD in Religion is not necessarily a PhD in theology.

Consider Duke. Duke currently reports 45 students in its entire PhD program in Religion. Those students are distributed among Asian Religions, Christian Theological Studies, Early Christianity, Hebrew Bible/Old Testament, History of Judaism, Islamic Studies, New Testament, Religion, Aesthetics & Society, and World Christianity. CST has forty-two students in three areas of theology alone.

Harvard supplies another revealing comparison. Harvard Divinity School currently reports 63 students in the PhD program. Yet this is the Harvard Graduate School doctorate administered through the Committee on the Study of Religion, encompassing the broader academic study of religion rather than sixty-three students doing Christian theology.

Yale has undergone a similar development. Its Religious Studies doctorate now contains ten fields of study. Theology remains one of them, and Philosophy of Religion remains another. Indeed, Yale's Philosophy of Religion program explicitly includes philosophical theology and requires engagement with both analytic and continental philosophical traditions.

These remain extraordinary doctoral programs. The point concerns what they are programs in. Much of what was once concentrated in theology has become part of the broader academic study of religion. That changes the comparison.

Notre Dame Is Particularly Interesting

Notre Dame provides a cleaner case because its doctorate remains explicitly a PhD in Theology. Its current public roster lists 77 PhD students. I counted the students according to Notre Dame's own classifications. Twenty-two are in Systematic Theology and twelve are in History of Christianity. That gives 34 students in the two fields most directly comparable with CST's Systematic and Historical Theology programs. CST currently has 42 across Systematic Theology, Historical Theology, and Philosophical Theology.

This does not tell us that CST has somehow overtaken Notre Dame. It tells us something much more precise and, to my mind, more interesting. The concentration of doctoral students at CST in these traditional theological disciplines is already on the same numerical order as the corresponding concentration at one of North America's premier departments of theology. I did not expect that.

Princeton and Chicago

Princeton Theological Seminary remains another important comparison because its doctorate still bears a recognizably theological shape. Its current students work in Theology, Ethics, and Politics; History and Ecumenics; Biblical Studies; Practical Theology; and Religion and Society. Current profiles confirm active doctoral work in systematic and constructive theology as well as the several historical fields.

Our preliminary count of Princeton's current public profiles produced roughly eighteen students across Theology, Ethics, and Politics and the broadest possible construction of History and Ecumenics. Even that comparison is generous, since Religion in the Americas and World Christianity do not always constitute historical theology in the sense in which CST uses the term.

Chicago may display the historical shift still more dramatically.

The University of Chicago Divinity School remains one of the world's distinguished places for the study of religion. Its current PhD describes itself as interdisciplinary research into the “human phenomenon of religion,” encompassing constructive, historical, social-scientific, literary, and other approaches. Its current directory shows students across a wide range of areas. Theology and Philosophy of Religions continue, but they now occupy places within a much broader conception of the discipline.

Anyone who remembers Hyde Park several decades ago will recognize how significant the change is. The Chicago theological ecosystem once concentrated an extraordinary number of theologians within a few blocks. The institutions remain, but the distribution of doctoral work has changed.

So What Has Happened?

I think at least two developments are visible.

First, many of the strongest university programs have moved toward the academic study of religion conceived very broadly. That move has intellectual justification. Religion is a global human phenomenon, and serious scholarship cannot simply identify the study of religion with the study of Christianity.

Yet there is an institutional consequence. Forty or sixty doctoral students distributed among Christianity, Judaism, Islam, Asian religions, biblical studies, anthropology, ethics, history, philosophy, and other fields do not constitute forty or sixty systematic, philosophical, and historical theologians.

Second, seminary contraction has reduced the number of institutions maintaining large research doctorates in the classical theological disciplines. Programs have closed, contracted, consolidated, or broadened their purposes.

The result is a curious situation. There may now be fewer places in North America where a substantial number of doctoral students gather specifically to do theology. However, CST has moved in precisely the opposite direction.

What CST Has Accidentally—or Deliberately—Become

Our forty PhD students are concentrated in three neighboring disciplines. A student working in philosophical theology regularly encounters systematic theologians. A systematician encounters historians of doctrine. Historical work raises constructive questions; constructive work encounters philosophical ones. That density matters.

There is a considerable difference between enrolling forty people pursuing doctorates and possessing a community of forty doctoral theologians. The latter can become an intellectual environment. But what follows from the numbers?

Certainly not that CST is now “better than” Yale, Harvard, Princeton, Chicago, Duke, or Notre Dame. Such a claim would be silly. These institutions possess resources accumulated over generations: great libraries, endowed faculties, enormous universities, selective admissions, funding structures, placement networks, and reputations generated through thousands of graduates. Reputation is real, and so is institutional capital. But another fact is real as well:

Christ School of Theology has achieved doctoral scale before it has achieved commensurate reputation. That is what surprised me in looking at the numbers.

I think we can now say, with reasonable caution, that CST has developed one of the larger concentrated communities of PhD students in North America working specifically in systematic, philosophical, and historical theology.

Perhaps further research will reveal several programs we have overlooked. I would welcome the correction. This is an empirical question. One must count the students, identify their fields, and compare like with like.

But Duke has forty-five doctoral students in its entire Religion program. Harvard has sixty-three. Notre Dame has thirty-four in Systematic Theology and History of Christianity combined. Yale organizes its doctorate across ten fields. Once those facts are placed beside forty students concentrated in CST's three theological fields, something worth noticing has happened.

Scale Is Only the Beginning

Numbers do not produce scholarship, because 42 disconnected students are simply 42 students. The next question for CST is therefore more important than the enrollment question: What must happen for these students to become a genuine community of theological scholarship?

They must read one another's work. They must present dissertation chapters and defend arguments before colleagues. Philosophical theologians must learn enough historical theology to know when a supposedly new problem is an old one. Historical theologians must be pressed about what follows from the texts they study. Systematic theologians must learn to state clearly what they claim, what makes the claim true, and what follows if it is.

They should meet major scholars. They should publish. They should participate in conferences. Their dissertations should matter. Their graduates should eventually teach the next generation. At this point enrollment begins to generate intellectual capital.

This also suggests why CST should be cautious about proliferating doctoral concentrations. There is nothing inherently wrong with PhDs in practical theology, leadership studies, biblical studies, missiology, or other fields. But institutional identity comes partly from decisions about what one will not do. CST presently possesses something unusual: a large doctoral program centered quite deliberately upon theology. We ought to know what we have.

A Changed Map

The most interesting conclusion therefore concerns neither CST nor enrollment statistics by themselves. The map has clearly changed.

The great institutions remain great. Yale is still Yale. Harvard is still Harvard. Notre Dame, Princeton, Duke, and Chicago remain institutions with extraordinary scholars and enormous intellectual resources. Yet the concentration of doctoral students within the classical theological disciplines is no longer distributed as it once was.

New centers can consequently emerge. Whether CST becomes one of them will not be settled by enrollment numbers. It will be settled by the quality of the scholarship produced by its faculty and students, by the rigor of its dissertations, by publications, by intellectual exchange, and ultimately by the scholars its PhD program sends into the church and academy.

But one necessary condition is already present. There are enough theologians in the room, and that is something I had not fully realized until I counted them.

Model Theory in Theology: From Descartes to Luther

This short talk given on August 12, 2026, at the 15th International Luther Congress in Aarhus, Denmark, introduces model theory to historians. Like all my papers, it arises from work done at the Department of Philosophical Theology, ILT Christ School of Theology. 

Two Problems We Have Not Escaped

I want to begin some distance from model theory, with two problems associated above all with Descartes. The first is the problem of the external world. How do I know what the world is like apart from my representations of it? The second is the problem of other minds. How do I know what another person thinks, means, intends, believes, or experiences when I never occupy that person’s consciousness?

Descartes formulated these problems with unusual clarity. Four centuries later, we have not made them disappear. We have mostly learned how to live intelligently with them. We do not solve either problem by somehow getting outside ourselves. Rather, we encounter something over against us, something that resists us, and we try to render it intelligible. The world does not always behave as our theories predict. Other persons say and do things our accounts of them do not anticipate. Our interpretations can fail because something confronts us that is not simply at our disposal.

We cannot solve either problem by looking down at reality from nowhere, as though we could step outside ourselves and check our account against the thing itself. But we are remarkably good at a more modest maneuver: looking left and right, comparing one account with another and both with what resists them. That is the maneuver this essay is about.

Historians Live with the Problem of Other Minds

Historians confront the second Cartesian problem every day. At a Luther Congress we ask what Luther meant, what Melanchthon believed, what a late-medieval logician was doing with suppositio, or what theological possibilities were available to an author in 1517. But we cannot climb into Luther’s mind. We have texts, vocabulary, grammatical practices, controversies, other authors, institutional settings, manuscripts, and subsequent writings.

What historians actually do is reconstruct. We construct an account under which this evidence becomes intelligible together. Different historians may agree about the words on the page and nevertheless disagree about the best account of what those words are doing. The text and its historical setting constrain us: they possess what, borrowing loosely from Charles Peirce’s language of Secondness, I shall call an over-and-againstness or “bump-up-againstness.” They do not allow us responsibly to make them say whatever we wish.

This means that the mens auctoris need not function as an independently accessible semantic court of appeal. Authorial intention remains a legitimate historical hypothesis, but what we can reconstruct is the intentionality of a textual act from publicly available historical evidence. Our reconstruction may be excellent without becoming incorrigible.

The Special Sciences Face the External-World Version

The special sciences confront an analogous problem from the other Cartesian direction. The physicist does not have uninterpreted physical reality sitting on one side of the laboratory and a theory on the other. The biologist does not inspect life in itself. Scientists encounter phenomena, measurements, regularities, anomalies, and recalcitrant observations, and they construct theoretical structures that make these intelligible.

The historian’s problem and the scientist’s problem are not identical. A manuscript, another mind, and a physical process are not the same kind of object. But structurally the inquiries share something important: neither historian nor scientist possesses a God’s-eye standpoint. Both construct accounts of something that confronts them and is capable of proving those accounts inadequate. Theories are answerable to what they seek to understand without our having to suppose that finite knowers can compare their theories with an entirely uninterpreted world.

Theology Has Both Problems at Once

Theology inherits both problems. Theologians are historians. We ask what Paul meant, what Augustine meant, what Luther meant, what Nicaea meant, and what inherited doctrinal language was doing in its original settings. But theologians also make assertions about reality: God raised Jesus from the dead; God justifies the ungodly; Christ is present in the Supper.

Theology therefore interprets witnesses to other minds while also speaking about a world in which God is soteriologically ingredient. It asks both what inherited theological assertions meant and what account of God, world, Christ, sinner, promise, sacrament, and salvation renders the subject matter intelligible. The two Cartesian problems meet inside theological inquiry.

This is why theology cannot simply borrow its method from the historian or the scientist alone. It needs a single account of inquiry general enough to cover both tasks at once—one that tells us what we are doing when we build a model of a mind we cannot enter and a model of a God we cannot view from above. That is the account model theory can supply, and it is why I turn to it now.

A Small Luther and Trutvetter Example

Let me give one small example from another paper I presented here. Luther entered the University of Erfurt in 1501, the year Jodocus Trutvetter’s Summule totius logice appeared. We possess Trutvetter’s texts and Luther’s texts. The historical question is not merely whether Luther remembered particular sentences from Trutvetter. It is what semantic and inferential possibilities belonged to the intellectual world within which Luther’s later disputation became intelligible.

Consider a Trinitarian inference discussed by Trutvetter:

This divine essence is the Father.
This divine essence is the Son.
∴ The Son is the Father.

If I assign the copula est strict numerical identity throughout, the inference is valid. If this divine essence is numerically identical with the Father and numerically identical with the Son, then Father and Son are numerically identical with one another. There is no problem with the logic, but the theology is disastrous.

So what must est, essentia, Father, and Son be doing semantically for Trutvetter’s discussion to become intelligible? And when Luther later reasons with inherited scholastic logical vocabulary, what larger structure of interpretation makes both Trutvetter and Luther intelligible?

Here I need to point out that I am using the word interpretation in a way different from its hermeneutical use familiar to many of us. In formal semantics an interpretation assigns semantic values to the non-logical vocabulary of a language—in the simplest cases, the things to which its terms apply. Two readers may agree completely on the visible words and disagree about the meanings being assigned to them.

In historical work we are not pretending that Trutvetter’s or Luther’s prose is itself a formal language. We construct structured interpretations of the texts, and where useful we can regiment selected claims formally. One historian may model the relevant predication one way, another differently. We can then ask which model makes more of Trutvetter’s text, Luther’s text, and the historical relation between them intelligible. We can even ask whether a larger space of intelligibility can accommodate both without erasing their differences.

Theological Language Presents the Same Problem

The same issue appears when we turn from the historical reconstruction of theologians to the theological tradition and the Word of God that confront us. People of good will can agree on the language of theology while interpreting its terms very differently.

Lutherans may say, “We are justified by grace through faith.” But what is the semantic work of justification? What is grace? What sort of relation between God and the sinner is being asserted? Or Christians may agree that “Christ is present in the Eucharist” while operating with very different understandings of Christ, presence, body, sign, promise, and sacrament.

Theological disagreement therefore frequently occurs because the same sentences are satisfied under different models. A model may preserve familiar vocabulary while changing the larger structure in which the vocabulary functions. The question then becomes whether one model preserves more of the theological subject matter, whether another changes the subject, and whether several admissible models remain possible.

Model Theory Makes the Structure Explicit

Only now do I want to introduce model theory in its technical sense—and I will keep the symbols to the minimum that actually does work for us.

A theory, syntactically understood, is simply a set of sentences. Call it T:

T={φ1​,φ2​,φ3​,…}

Read this as: T is just the set of claims the theory makes.

An interpretation assigns semantic values to the non-logical terms of a language. A model is a structure under such an interpretation in which the relevant sentence or theory is true. When a structure M makes a sentence φ true under the interpretation supplied by M, we say that M satisfies φ, and we write:

$$M\models \phi$$

Read ⊨ simply as “satisfies” or, less technically, “makes true.”

A model of a theory satisfies every sentence in it. The collection of all such models is what logicians call Mod(T):

$$Mod(T)=\{M\mid M\models T\}$$

Read this as: Mod(T) is the class of models that make all of T’s sentences true.

And now we can state formally something historians and theologians encounter constantly. One theory can admit more than one model:

M1​⊨T

and

M2​⊨T,

while

~(M1​ = M2​).

The final expression says only that the two models are not numerically identical. More importantly for our purposes, they may differ substantially in the structures under which the same sentences are true. Both can satisfy the theory without representing its subject matter in the same way.

Satisfaction alone therefore does not tell us which model should be preferred. Some candidate models can be excluded because they violate constraints constitutive of the subject matter. An interpretation of a text that cannot accommodate its actual wording may cease to be an interpretation of that text. A theological model that preserves the word resurrectiononly by abandoning the identity of the one said to have been raised may have changed the subject rather than merely offered a less attractive account.

Among the models that remain admissible, we compare. We ask about consistency, coherence, explanatory scope, fecundity, parsimony, applicability, historical adequacy, capacity to accommodate recalcitrant evidence, and sometimes elegance or beauty. These virtues are not mechanically commensurable. There is no algorithm that adds them together and announces the correct model.

This is where I find Kant’s reflektierende Urteilskraft—reflecting judgment—useful. We compare, discriminate, and judge without pretending that an antecedently possessed universal rule mechanically determines the result. I call the structured field within which such comparative judgments occur teleo-space.

The position is fallibilist. Further texts, evidence, experiences, arguments, or distinctions may force us to reorder the models. But fallibility does not imply that every model has equal epistemic standing. We can give reasons why one is better than another.

Nor do we need a God’s-eye standpoint from which model and uninterpreted reality can be placed side by side. We cannot look down from nowhere. But we are remarkably good at looking left and right. We can compare models. We can see that one explains something another leaves obscure, preserves a distinction another destroys, or accommodates an obstinate text or experience another must suppress.

Model theory therefore does not solve Descartes’ problems by giving finite creatures access to reality without mediation. It does something more useful. It clarifies the structure of responsible inquiry when such access is unavailable.

The historian constructs models of other minds from texts that resist interpretation. The scientist constructs models of a world that resists theory. The theologian does both: we interpret witnesses, and we make claims about a reality in which God is soteriologically ingredient.

Our models are fallible. More than one may satisfy the assertions with which we begin. Some can nevertheless be excluded; others can be comparatively ordered. We can offer what I would call a reasoned account: a publicly criticizable, corrigible judgment about how the subject matter may be independently of our present act of modeling it.

We cannot look down from nowhere. But we can look left and right. And perhaps model theory can help theology become clearer about what it is doing when it does.