The Formal Limits of Speaking About God: From Cantor’s Absolute Infinity to Inexhaustible Representation

*This exploratory essay grew from personal theological reflection and sustained dialogue with AI. It investigates mathematical possibilities and limitations without claiming to present a new proof or completed formal theory.

I began with a question that seemed much simpler than the one I am now trying to formulate. Mathematics has studied infinity rigorously for more than a century and has shown that infinite collections can have different cardinalities. Theology, meanwhile, has spoken about the infinity and incomprehensibility of God for much longer. I wondered whether these two traditions had been brought together adequately. If mathematics had discovered increasingly large infinities, could those discoveries contribute to a more precise theological understanding of the infinite God?

Behind that question was an intuition I had not yet examined. I was treating mathematical infinity as a possible route of approach: perhaps the greater the infinity, the closer the mathematical representation would come to divine infinity. I did not think a mathematical object could simply be God. I imagined it instead as a projection or creaturely reflection of God—something that could disclose the divine without being identical with the divine.

At this stage, my question was mainly historical. Surely theologians, for whom the nature of God is fundamental, must have discussed these matters extensively. Mathematicians might have developed more rigorous languages, but divine infinity is only one problem among many in mathematics, whereas for theology it belongs to the heart of the discipline. I therefore expected to find a long and active exchange between the two fields.

What I found was more uneven. Theological traditions had indeed developed profound accounts of infinity, transcendence, analogy and incomprehensibility. Modern mathematics had developed powerful theories of transfinite numbers, reflection, incompleteness, independence and uncomputability. Yet after a remarkable moment in the work of Georg Cantor, mathematics became increasingly capable of studying infinity without continuing to ask explicitly what infinity ultimately is—or what its relation to God might be.

That observation produced the first important change in my inquiry. I was no longer asking only what mathematics had said about infinity. I was asking what mathematics had learned by suspending the theological question, and whether the results of that suspension could now be returned to theology.

The surprise of Cantor’s Absolute Infinity

The turning point began when I encountered Cantor’s distinction between transfinite numbers and Absolute Infinity. I had known that Cantor established different cardinalities of infinity. I did not know how explicitly he connected Absolute Infinity with God.

Cantor did not place God at the summit of the transfinite hierarchy as its greatest mathematically available member. Transfinite numbers remained mathematically determinable and could be ordered, compared and subjected to arithmetic. Absolute Infinity occupied a different position. Cantor associated it with God, the ens simplicissimum and actus purissimus, while assigning it to speculative theology rather than ordinary transfinite mathematics (Gutschmidt and Carl, 2024).

My first reaction was surprise, followed almost immediately by another question: what happened after Cantor? Mathematicians continued to investigate infinity intensely. Set theory developed enormous hierarchies of cardinals and ordinals. Logic discovered incompleteness and undefinability. Model theory studied relations between languages and structures. Computability theory identified problems no general algorithm can decide. Yet relatively few major mathematicians continued Cantor’s explicit theological interpretation of Absolute Infinity.

At first this absence seemed strange. If mathematical infinity had once been placed so close to the doctrine of God, why did later mathematics not continually return to that connection?

A provisional explanation emerged: after Cantor, mathematics learned to use infinity while suspending the question of what, if anything, the infinite ultimately is. This did not mean that mathematicians ceased thinking philosophically, nor that theology became irrelevant to every mathematician. It meant that the technical study of infinity no longer depended upon resolving its final metaphysical interpretation. Mathematicians could investigate sets, cardinals, ordinals, models and proofs while bracketing the identity of Absolute Infinity.

I found this intellectually exciting because the suspension did not make the theological question disappear. It created a century of mathematical results that had usually been developed without theological conclusions but might nevertheless constrain which theological conclusions can responsibly be drawn.

The new question became:

What can the mathematical study of infinity after Cantor contribute to Cantor’s theological idea of Absolute Infinity, even when later mathematicians were not themselves trying to develop a doctrine of God?

What Cantor’s theorem actually establishes

Before trying to interpret Cantor theologically, I needed a more exact account of the mathematical construction that had made the question possible.

For any set A, its power set P(A) is the collection of all subsets of A. Cantor’s theorem states:

For every set A:

|A| &lt; |P(A)|

Equivalently:

There is no surjective function f : A → P(A).

The proof constructs a subset that escapes every proposed enumeration. Suppose that f : A → P(A) is claimed to be surjective. Define the diagonal set:

D_f = { a ∈ A | a ∉ f(a) }.

Because D_f is a subset of A, it belongs to P(A). If f were surjective, there would be some d ∈ A such that:

f(d) = D_f.

But then:

d ∈ D_f
↔ d ∉ f(d)
↔ d ∉ D_f.

The contradiction shows that no such surjection exists. For every attempted enumeration of all subsets of A, diagonalization constructs a subset absent from the enumeration.

In cardinal notation:

2^|A| = |P(A)| &gt; |A|.

Consequently, there is no greatest cardinal number:

For every cardinal κ, there exists a cardinal λ such that κ &lt; λ.

One possible choice is:

λ = 2^κ.

This is the exact mathematical reason that “the largest set-sized infinity” cannot be obtained in ordinary set theory. Given any cardinal candidate κ, the power-set operation yields the strictly greater cardinal 2^κ.

The result does not establish that God is beyond every mathematical representation. It establishes a precise non-maximality result within set theory. The theological significance arises only when this formal pattern is interpreted through an additional account of Absolute Infinity.

The first attractive mistake

My initial attempt to answer the theological question returned to magnitude. Perhaps the later hierarchy of increasingly powerful infinities could supply progressively better projections of divine infinity. Even if no mathematical object were God, one of them might be the nearest available approximation.

The thought seemed plausible for several reasons. Theology already accepts that finite realities can mediate knowledge of God without containing God. A created sign may communicate divine reality without becoming divine reality. A theological proposition may be true without exhausting its subject. Why could a mathematical infinity not function similarly?

I began considering universal sets, self-referential structures and indefinitely iterated systems. Could a set contain every set, including itself? Could a self-referential mathematical object exceed the usual distinction between container and contained? Could repeated iteration eventually produce something capable of representing all possible levels?

An AI system with which I was developing the question responded that even a consistent theory containing a universal set would not thereby have represented God. The universal set would remain an object defined by a theory, satisfying the rules of that theory and interpreted through a surrounding metalanguage.

I agreed that such a set would not be God, but I resisted ending the inquiry there. My proposal was representational rather than identificatory. Could it be understood as a projection of God—a structure through which some aspect of divine infinity became accessible to us?

The AI then answered that self-reference does not automatically produce absolute transcendence. A recursively defined object can refer to itself while remaining an object within a formal system. Mathematical theories often permit carefully controlled circularity, fixed points, recursive processes and self-description. None of these operations by itself converts an object into the ground of the system that defines it.

Formally, there is a difference between a system containing a code for itself and a system becoming identical with its own metalogical ground. If ⌜T⌝ is a code or arithmetized description of a theory T, then:

⌜T⌝ ∈ Domain(T)

does not imply:

T contains a complete semantic account of T,

or:

T = the metatheory in which T is interpreted.

A theory may represent its own syntax without internalizing every semantic fact about that syntax. Gödel’s arithmetization of syntax depends precisely upon a system being able to encode formulas and proofs while still encountering limits on what it can prove about those encodings.

This correction was convincing, but disappointing. If self-reference did not provide the passage beyond every boundary, what mathematical operation could? Was there some more advanced construction? Could category theory, type theory, non-well-founded set theory or a hierarchy of metalanguages eventually produce a final perspective?

The answer remained negative: no familiar iteration obviously produces a final “view from nowhere.” A language may be interpreted in a metalanguage, which may in turn become the object of a stronger metalanguage. A universe can contain codes or descriptions of structures in a lower universe. A theory can reason about fragments of its own syntax. Each advance provides a new standpoint, but the new standpoint remains a standpoint.

A simplified hierarchy can be written:

L_0  &lt;  L_1  &lt;  L_2  &lt;  ...

L_1 contains semantic resources for L_0.
L_2 contains semantic resources for L_1.
...

Moving from L₀ to L₁ may solve a semantic limitation of L₀. It does not establish that L₁ is semantically closed with respect to itself.

At that moment I experienced the absence of a final construction primarily as a failure. I had been trying to approach divine infinity mathematically, and the path seemed continually to retreat. Only later did I realize that this failure might contain the more interesting idea.

When the scale itself became questionable

The decisive reformulation came through one sentence proposed during the dialogue:

God would not be the largest point on the scale. God would instead be outside—or the ground of—the scale itself.

I immediately recognized the theological significance of this formulation. My previous search had silently placed God and mathematical infinities within a common genus. Smaller infinities appeared at lower positions and divine infinity at the highest possible position. Yet classical theology often refuses precisely this arrangement. God is not one being among beings, distinguished from the others by possessing a greater quantity of existence. God is not the final item obtained by extending a creaturely sequence far enough.

Christian Tapp has shown that substantially different concepts operate under the word “infinity” in mathematics and theology. They have significant relations, but they cannot be identified without further argument (Tapp, 2011). In particular, divine infinity need not mean spatial or numerical magnitude continued without limit. Within classical theology, infinity is connected with the absence of creaturely limitation, composition and finite determination.

The mathematical hierarchy of cardinalities can be represented as an open-ended progression:

ℵ₀ &lt; ℵ₁ &lt; ℵ₂ &lt; ...

and, for every cardinal κ:

κ &lt; 2^κ.

A theological mistake occurs if one adds a final symbol G and assumes, without argument:

ℵ₀ &lt; ℵ₁ &lt; ℵ₂ &lt; ... &lt; G,

therefore G = God.

The ordering relation < is a mathematical relation between cardinalities. Divine transcendence is not already contained in its definition. A proposed theological interpretation would have to supply a bridge such as:

Greater cardinal magnitude
corresponds to
greater adequacy as a representation of divine infinity.

But that bridge is precisely what had not been established.

The theological problem therefore changed the mathematical one. If divine infinity is not fundamentally quantitative, then finding a larger cardinal does not automatically provide a better representation of God. “Larger” already belongs to a defined mathematical ordering. Before using it theologically, I would have to prove that this ordering corresponds to some relevant dimension of divine reality.

A larger photograph is not necessarily a more faithful photograph. A longer theological book is not necessarily closer to God. An ontology containing more objects may still represent the relevant relation less adequately. I had treated mathematical enlargement as theological improvement without defining the criterion of improvement.

What survived from my original intuition was the idea of projection. What had to be abandoned was the assumption that projection improves through magnitude alone.

The new problem became relational:

What kind of formal relation permits a finite representation to be true of God without pretending to contain, determine or exhaust God?

The mirror and the mathematical problem it concealed

The image that repeatedly returned to me was simple and theologically familiar:

A mirror can reflect the sky truthfully without containing the sky.

There is nothing historically new about this intuition. Theology has long distinguished between God and creaturely knowledge of God, between divine essence and divine self-communication, and between comprehension and genuine participation. Robert John Russell describes the God of Western monotheism as Absolute Mystery, the incomprehensible ground and source of being, while maintaining that God can nevertheless be known through revelation received and interpreted by finite creatures (Russell, 2011).

What became new for my inquiry was the attempt to translate this familiar theological distinction into a question about formal representation. Mathematics normally asks whether a sentence is true in a model, derivable from axioms, definable in a language or computable by a procedure. Theology adds another relation: a representation may be true and warranted while remaining non-exhaustive with respect to its referent.

Ordinary model-theoretic satisfaction is written:

M ⊨ φ.

This means that the sentence φ is true in the model M. It does not mean:

M is identical with the reality to which φ ultimately refers,

or:

M exhausts every truth about that reality.

To make the theological distinction explicit, one could introduce two meta-level relations:

Adeq_A(R, G)
Exh(R, G)

Here:

R is a formal representation;
A is a specified theological aspect;
G denotes the theological referent at the metatheoretical level;
Adeq_A(R,G) means that R is adequate with respect to aspect A;
Exh(R,G) means that R exhaustively represents its referent.

A proposed apophatic non-exhaustion principle could then be written schematically:

ANEP:

For every representation R and aspect A:

Adeq_A(R, G) → ¬Exh(R, G).

This is not an established mathematical theorem. It is a proposed theological bridge axiom expressed formally. Its purpose is to prevent the inference:

R adequately represents some aspect of G
therefore
R exhaustively contains G.

The distinction can also be represented epistemically. Let K_A(R) be the set of warranted claims preserved by representation R with respect to aspect A. Then:

K_A(R) may be nonempty and truth-bearing

without requiring:

K_A(R) = Th(G),

where Th(G) would denote a complete theory of the referent. The latter notation is itself philosophically dangerous because it presupposes that the total truth about God forms an available formal totality. It is therefore better understood as a schematic limit concept than as an already constructed set.

This is more complicated than dividing reality into two binary regions—what we know and what we do not know. I initially wondered whether the distinction between the divine essence and God’s self-communication ad extra was simply such a binary model. But the boundary is not static. What is received can be deepened, reinterpreted and related to other aspects. New knowledge may become possible without eliminating the distinction between knowledge and comprehension.

The mirror does not divide the universe into an entirely known image and an entirely unknowable sky. It preserves selected relations under particular conditions. Change the mirror, its position, its curvature or its field of view, and the image changes. Some features remain stable; others appear or disappear. This led to a more exact question:

What theological relations remain invariant across multiple non-equivalent representations?

At this point, stability and transcendence appeared as two different research directions. One direction asks what persists when perspectives change. The other asks whether every perspective can be surpassed. I initially treated them separately: invariants seemed to concern what does not change, while transcendence concerned the possibility of going beyond.

Gradually I saw that they belong to one structure. If representations can be refined indefinitely, invariants are the relations that survive those refinements. “Going beyond” without preservation would amount only to replacing one discourse with another. Stability without extensibility would risk turning one representation into an idol. The formal problem requires both.

Reflection as a positive route

When mathematical reflection first entered the conversation, I did not understand why it was described as a positive route. Negative theology seemed to emphasize that God cannot be comprehended. How could reflection transform uncharacterizability into positive knowledge?

In set theory, reflection principles say, roughly and with important differences among their formulations, that properties attributed to the vast universe of sets already appear in some smaller portion of that universe. The universe is not captured as an ordinary object inside itself, yet structures within it can reflect features of the wider whole. Welch and Horsten connect such principles with Cantor’s conception of Absolute Infinity and argue that reflection can support strong axioms of infinity (Welch and Horsten, 2016). Barton develops an explicitly theological reflection principle in dialogue with apophatic mathematics (Barton, 2024).

The ordinary cumulative hierarchy of sets is defined recursively:

V₀ = ∅

V_(α+1) = P(V_α)

V_λ = ⋃_(β&lt;λ) V_β      when λ is a limit ordinal

V = ⋃_(α∈Ord) V_α.

Here V denotes the universe of sets, while each V_α is a set-sized rank-initial segment. The Lévy–Montague reflection scheme can be expressed approximately as follows. For every finite collection of formulas Φ, there are ordinals α such that:

(V_α, ∈) ≺_Φ (V, ∈).

The notation ≺_Φ means elementarity only with respect to the formulas in Φ. More explicitly, for the relevant parameters a⃗ ∈ V_α and every φ ∈ Φ:

V ⊨ φ(a⃗)
↔
V_α ⊨ φ(a⃗).

The qualification by Φ is essential. The theorem does not provide one set-sized V_α that captures every truth about V. Given a finite family of formulas, an appropriate stage reflects those formulas. When the family changes, a different or larger stage may be required.

This gave mathematical content to the mirror analogy:

Truth in V
is reflected in
a bounded V_α

for a specified family Φ,

without:

V_α = V.

The theological bridge became clearer when I returned to the mirror. Reflection does not mean that the smaller structure contains the whole. It means that something true of the wider reality is genuinely present or reproduced in a restricted domain. Negative theology therefore need not leave us only with negation. It can protect the difference between knowing and possessing while allowing positive, limited knowledge.

This produced a tension I had not adequately expressed before:

Does Absolute Infinity give us positive mathematical reflection, or does it principally disclose the impossibility of total comprehension?

I no longer think these alternatives exclude each other. Reflection may yield positive knowledge precisely because it does not require exhaustive identity. The theological pattern would be that God is truly known in divine self-communication while remaining incomprehensible in the divine essence. The mathematical pattern would be that structures reflect selected truths of a wider domain without becoming that domain.

Yet the analogy remains an analogy. A set-theoretic reflection principle is a mathematically specified statement. Divine revelation is a theological concept involving agency, relation and history. Moving between them requires an explicit account of relevant similarity. The word “reflection” cannot perform that work by itself.

Every answer generated another standpoint

The idea of reflection immediately produced another question. If a smaller structure reflects a greater one, from where do we judge that the reflection is adequate? That judgment appears to require a new standpoint. But what exactly is this standpoint?

I considered several possibilities. Perhaps it was simply a new concept created in natural language. Perhaps it was a stronger mathematical theory. Perhaps it was the existence of reflective thinking itself: the mind’s ability to turn a previous act of representation into an object of further thought. Perhaps it was a metalanguage capable of describing the relation between a language and its models.

Each possibility captured part of the movement. A standpoint can be a richer vocabulary, a stronger theory, an enlarged semantic domain, a new model, or an interpretive act from which an earlier limitation becomes visible. But none is automatically absolute. The metalanguage can become a new object language. The stronger theory can be investigated from a further theory. The reflective act can itself be reflected upon.

At the simplest schematic level:

T₀ ⊆ T₁ ⊆ T₂ ⊆ ...

Meta(T₀) may be represented in T₁.
Meta(T₁) may be represented in T₂.
...

But the union of an ascending sequence does not automatically produce an absolutely final system. Even if one defines:

T_ω = ⋃_(n&lt;ω) T_n,

one can ask whether T_ω is consistent, complete, effectively axiomatizable, semantically closed or capable of proving its own relevant metatheory. Depending on how the sequence was constructed, one or more of these properties may fail. Passing to a limit stage changes the system; it does not abolish the need for metatheoretical analysis.

This recursive pattern initially seemed to repeat the original frustration. If every representation can be exceeded, perhaps mathematical approach to God is forever impossible. Does mathematics contain theorems proving that God can never be reached? Has mathematical theology already been falsified before it begins?

The correction was essential. Mathematics proves the limitations of specified formal systems under specified assumptions. It does not prove that God lies beyond every logically possible representation. A theorem about arithmetic, truth or formal derivability cannot silently change its subject and become a theorem about divine ontology.

The recursive ascent nevertheless matters. It shows that many natural candidates for a final formal standpoint fail to achieve finality on their own terms. Theology may interpret this as an analogy of transcendence, but the mathematical result and the theological interpretation must remain distinguishable.

From negative theology to non-idolatry

Another formulation proposed during the dialogue described Absolute Infinity as a principle of non-idolatry:

No mathematical world may identify itself with the whole.

I found this formulation compelling because it translated an apophatic discipline into a rule governing representations. A mathematical universe may be extraordinarily rich. It may contain structures modelling almost all mathematics used in ordinary practice. It may even describe other universes. None of this alone licenses the claim that it is identical with absolutely everything.

The principle can be written schematically at the metatheoretical level:

For every formal representation R:

¬Exh(R, Totality).

Or, in theological form:

For every formal theological representation R:

¬Exh(R, G).

Again, this is not a theorem of ordinary mathematics. It is a proposed apophatic constraint. If adopted as an axiom, its theological justification must be defended independently. The interesting mathematical question is whether restricted versions of non-exhaustion can instead be derived from independently motivated properties of the representation.

Gutschmidt and Carl interpret diagonalization as a modern via negativa. Whenever a putative totality is represented in the relevant way, diagonal construction can expose something omitted by the representation. On their account, this does not deliver another positive description of Absolute Infinity. It performatively undermines the attempt at closure and may cultivate methodological humility concerning the boundedness of mathematical practice (Gutschmidt and Carl, 2024).

I initially understood humility here as an ethical attitude added after the mathematics. Their argument is more interesting. The repeated failure of closure can influence what mathematicians regard as an appropriate foundational aspiration. Negative theology may therefore contribute to the philosophy of mathematical practice by proposing that non-totalization is not always a temporary embarrassment. It may be a disciplined response to a structural feature.

Still, this does not imply that mathematics is “only relative” or that proof has no authority. A theorem can be entirely rigorous within its domain while the domain itself remains incapable of being represented as one more ordinary object inside itself. Formal certainty and ontological exhaustion are different ambitions.

The point at which precision became necessary

As the connection with negative theology became more attractive, the danger of overstatement increased. Statements such as “Gödel proves mystery,” “mathematics proves that God is incomprehensible,” or “every formal theology must be incomplete” sounded plausible within the developing analogy. They were also mathematically unsafe.

I had to ask what “formal theology” meant. A finite list of doctrinal propositions can form a consistent, complete and decidable theory. Gödelian incompleteness does not apply to every system that happens to contain theological vocabulary. It applies when the system is effectively axiomatized, consistent and sufficiently expressive to represent elementary arithmetic. Even then, “incompleteness” means that some sentences in the system’s language can be neither proved nor refuted within that system. It does not mean that every theological truth is inaccessible.

The claim therefore had to become narrower before it could become stronger.

Let a formal theological system be represented as:

F = (L, T, ⊢),

where:

L is a formal language;
T ⊆ Sent(L) is a theory or set of axioms;
⊢ is the derivability relation generated by the proof rules.

Several properties must then be distinguished.

Consistency:

Cons(T) iff there is no sentence φ such that:

T ⊢ φ
and
T ⊢ ¬φ.

Syntactic completeness:

Complete(T) iff for every sentence φ ∈ Sent(L):

T ⊢ φ
or
T ⊢ ¬φ.

Effective axiomatizability:

Eff(T) iff the axioms, or equivalently the proofs,
can be generated by an effective procedure
under the relevant standard conditions.

The mathematically responsible theological formulation is then:

Let F=(L,T,⊢) be a formal theological system whose axioms are computably enumerable, whose deductive apparatus is consistent, and whose expressive resources suffice to interpret elementary arithmetic. By the Gödel–Rosser incompleteness theorem, there exists a sentence σ∈Sent(L) such that T⊬σ and T⊬¬σ. Under the standard conditions required by Gödel’s second incompleteness theorem, T⊬Con(T). Moreover, a Tarskian truth predicate satisfying every biconditional Tr(⌜φ⌝)↔φ cannot be defined within the same sufficiently expressive language. These results do not prove divine incomprehensibility. They establish only that certain effectively governed, arithmetically expressive formal systems cannot simultaneously possess consistency, completeness, internal self-verification and unrestricted semantic closure. Any theological interpretation of this formal non-closure therefore requires an additional and explicitly defended bridge principle.

In compact form:

Let F = (L, T, ⊢).

Assume:
1. T is computably axiomatizable.
2. T is consistent.
3. T interprets a sufficient fragment of elementary arithmetic.

Then:
∃σ ∈ Sent(L) such that (T ⊬ σ) ∧ (T ⊬ ¬σ).

Under the standard conditions for the second incompleteness theorem:
T ⊬ Con(T).

The first conclusion can be summarized as the following incompatibility:

Eff(T) ∧ Cons(T) ∧ Arith(T)
→
¬Complete(T),

where Arith(T) means that T has enough expressive and deductive strength to interpret the required elementary arithmetic.

For the second incompleteness theorem, define a formal provability predicate:

Prov_T(x)

meaning that x codes a proof in T. If ⊥ is a contradiction, such as 0=1, the standard formal consistency statement is:

Con(T) := ¬Prov_T(⌜⊥⌝).

Under the required derivability conditions, Gödel’s second incompleteness theorem yields:

If T is consistent, effectively axiomatized
and sufficiently arithmetically strong, then:

T ⊬ Con(T).

This does not mean that no stronger theory can prove Con(T). A stronger metatheory S may satisfy:

S ⊢ Con(T),

while still facing the corresponding question about its own consistency:

Does S ⊢ Con(S)?

Gödel’s theorems concern derivability relative to particular formal systems; they do not establish an absolute realm of propositions unprovable in every possible system. A sentence undecidable in one theory may become an axiom or theorem in a stronger one. The second theorem also concerns a formally constructed consistency sentence and specified derivability conditions (Raatikainen, 2025).

Tarski and the movement to a metalanguage

Tarski’s result introduces a related but distinct hierarchy. Suppose that a language L contains a predicate Tr intended to express truth for every sentence of L. Material adequacy would require the biconditionals:

Tr(⌜φ⌝) ↔ φ

for every sentence φ of the relevant language.

For a consistent and sufficiently expressive formal language, there is no internally definable predicate satisfying all such biconditionals for the language itself. Schematically:

There is no formula Tr_L(x) in L such that,
for every sentence φ ∈ Sent(L):

T ⊢ Tr_L(⌜φ⌝) ↔ φ.

A truth definition for the object language L can instead be formulated in a sufficiently strong metalanguage ML:

Truth(L) is definable in ML,

where:

L &lt; ML.

The solution therefore has a hierarchical form:

L₀ receives a truth definition in L₁.
L₁ receives a truth definition in L₂.
L₂ receives a truth definition in L₃.
...

The move to a metalanguage is genuinely successful. It gives a rigorous truth definition for the lower language. But it does not produce one language that automatically contains its own unrestricted truth predicate. Tarski’s account therefore supplies a precise example of positive semantic knowledge through ascent without final semantic self-containment (Hodges, 2022).

Löwenheim–Skolem and the failure of unique determination

The discussion then introduced another limitation that I initially grouped too quickly with incompleteness. The Löwenheim–Skolem theorems do not concern unprovable sentences. They concern the sizes and plurality of structures satisfying first-order theories.

In a simplified form, if a first-order theory T in a language L has an infinite model, then it has models in multiple infinite cardinalities. Under the usual hypotheses:

If T has an infinite model, then for every infinite cardinal κ
with κ ≥ |L|, there is a model M_κ such that:

M_κ ⊨ T
and
|M_κ| = κ.

For a countable first-order language, this includes a countable model:

If T has an infinite model
and L is countable,

then there exists M such that:

M ⊨ T
and
|M| = ℵ₀.

Consequently, one first-order theory may be satisfied by non-isomorphic models:

M ⊨ T,
N ⊨ T,

but:

M ≇ N.

The same formal propositions can therefore fail to determine one uniquely intended infinite structure. This does not imply that every theory is hopelessly ambiguous. Categoricity can be studied at specified cardinalities, and stronger logical resources can change the situation. But first-order satisfaction alone does not guarantee unique ontological determination (Hodges and Scanlon, 2024).

The theological analogy is striking but conditional:

Same expressible theory
≠
necessarily one uniquely determined model.

Therefore:

A formal model may satisfy every theological proposition
expressible in a chosen language

without thereby:

being identical with God,
uniquely determining God,
or exhausting divine reality.

The last three conclusions do not follow directly from Löwenheim–Skolem. They require theological interpretation. The theorem supplies a precise warning against assuming that formal satisfaction automatically secures unique reference.

Several limits rather than one omnibus theorem

The mathematical results can now be displayed as a family of conditional limitations:

Gödel–Rosser:
Effective axiomatizability
+ consistency
+ sufficient arithmetic
→ syntactic incompleteness.

Gödel II:
Consistency
+ effective axiomatizability
+ sufficient arithmetic
+ standard derivability conditions
→ no internal proof of the standard consistency sentence.

Tarski:
Sufficient expressivity
+ unrestricted internal truth biconditionals
→ undefinability or inconsistency.

Löwenheim–Skolem:
First-order description
+ an infinite model
→ models in multiple infinite cardinalities.

Cantor:
Any set-sized cardinal candidate κ
→ a strictly greater cardinal 2^κ.

These are not instances of one theorem called “the impossibility of exhaustive representation.” They block different ambitions for different reasons. Their conjunction can motivate a research programme only after the shared term “exhaustive” has been divided into more precise properties.

Let:

Comp(T)       = syntactic completeness,
SelfCon(T)    = internal proof of the standard consistency statement,
TruthCl(T)    = internally definable unrestricted truth,
Cat(T)        = unique determination of the intended model,
Max(T)        = possession of a greatest set-sized infinity,
OntExh(T,G)   = ontological exhaustion of the divine referent.

For an appropriately restricted class of arithmetic-capable effective theories, existing mathematical results support a schema such as:

Eff(T) ∧ Cons(T) ∧ Arith(T)
→
¬Comp(T).

Under further standard conditions:
Eff(T) ∧ Cons(T) ∧ Arith(T)
→
¬SelfCon(T).

Under Tarskian conditions:
Cons(T) ∧ Arith(T)
→
¬TruthCl(T).

But mathematics does not supply:

¬OntExh(T, G)

unless OntExh, G and the relevant bridge assumptions have first been given a formal interpretation. That final movement belongs to the proposed mathematical theology, not to Gödel’s or Tarski’s theorem by itself.

The bridge principle I could no longer leave hidden

At an earlier stage I had moved too quickly from formal non-closure to divine incomprehensibility. The correction introduced a three-part structure:

Formal theorem
    +
Explicit theological bridge principle
    =
Conditional theological interpretation

For example:

Formal result:
A consistent, effectively axiomatized and arithmetically expressive
theory T is syntactically incomplete.

Bridge principle:
Any exhaustive formal representation of God would need to decide
every truth expressible in its own relevant language.

Conditional conclusion:
T does not provide an exhaustive formal representation of God.

In logical form:

1. FormalLimit(T).

2. Exh(T,G) → ¬FormalLimit(T).

Therefore:

3. ¬Exh(T,G).

The inference from statements 1 and 2 to statement 3 is valid. The controversy lies in statement 2. Mathematics may prove FormalLimit(T); theology and philosophy must defend why an exhaustive representation of God would imply the absence of that formal limit.

Another bridge might concern model plurality:

1. T has non-isomorphic models M and N.

2. If T exhaustively and uniquely determined G,
   every admissible model of T would determine the same referent
   in the relevant strong sense.

Therefore:

3. T does not exhaustively and uniquely determine G.

Again, the mathematical result establishes model plurality. The second premise supplies the philosophical account of what unique determination would require.

Theology must explain why exhaustive representation would require the relevant form of completeness, whether the undecidable sentences are theologically significant, and why the formal system should be treated as a candidate representation of God rather than a limited calculus for reasoning about selected doctrines.

This bridge may ultimately fail. A theologian could argue that no doctrine of divine comprehension ever required a formal theory to decide every arithmetical sentence. A mathematician could point out that changing the logic, language or semantic framework changes the available limit results. These are serious objections. They do not destroy the programme; they determine what the programme must prove.

The central methodological question therefore became:

Which limitations belong merely to a chosen mathematical language, which belong to broad classes of formal representation, and which—if any—can responsibly be interpreted as traces or analogies of divine transcendence?

Could theology contribute back to mathematics?

Having reached this point, I became concerned that the exchange remained one-directional. Perhaps theology could borrow precise mathematical limit results, while mathematics received only metaphors in return. If so, the project might be useful theology but would contribute nothing to mathematical research.

This question mattered personally because I have formal training in theology and computer science rather than advanced mathematical training. The problems I had reached—reflection principles, model theory, self-reference, undefinability and abstract relations between logical systems—belonged to specialized areas of mathematics. I wondered why my theological question had moved so quickly into technically difficult territory, and whether I had any legitimate role there.

The answer was both encouraging and limiting. My questions became mathematically advanced because they concerned the boundary of formal representation itself. Questions about “everything,” final languages, self-description, unique models and complete truth move almost immediately into the foundations of mathematics. Computer science contributed an intuition for recursion, interfaces, formal languages and verification. Theology contributed the distinction between true knowledge and exhaustive comprehension. Their intersection naturally reached mathematical logic even though I had not begun from a mathematical research problem.

But arriving at a research-level question is not the same as producing a new mathematical result. At present, my thinking is a possible research design. It would become a mathematical contribution only if the theological distinctions required new formal definitions, generated a non-trivial conjecture, produced a theorem or countermodel, or motivated a formal system with properties not already studied elsewhere.

The most promising theological contribution may be the replacement of quantitative maximality with representational inexhaustibility. Theology can propose that divine infinity is better approached through relations such as:

truth without exhaustive possession;
participation without identity;
refinement without final closure;
stability across transformations;
and manifestation without reduction of the source to the manifestation.

These are theological distinctions, but they can function as design requirements for a new formal semantics. Mathematics would then be asked to determine whether such requirements are coherent, which combinations are possible, and what limitations follow from them.

Gutschmidt and Carl explicitly suggest that mathematics can learn from a performative interpretation of negative theology and that the resulting humility might influence mathematical practice (Gutschmidt and Carl, 2024). Computational metaphysics provides another example of reciprocal influence: formal metaphysical questions have led to computer-assisted philosophical discoveries while also motivating techniques relevant to logic and computer science (Kirchner, Benzmüller and Zalta, 2019).

The possibility of feedback is therefore real, although it remains programmatic in my own proposal. Theology cannot contribute to mathematics simply by attaching the word “God” to an existing theorem. It can contribute by giving mathematics a problem it did not previously formulate in quite the same way.

A formal theory of non-exhaustive representation

The possible programme that gradually emerged can be described as a limit theory of mathematical and computational theology: a formal study of what theological models preserve, what they decide, how they can be refined, and which ambitions of exhaustive representation are blocked under explicit assumptions.

Suppose a formal theological representation is written as R_A, where A denotes the aspect or group of aspects being represented. One model might concern divine knowledge, another divine simplicity, another freedom, and another the relation between God and creation.

At first I called these “dimensions.” I then wondered whether each dimension could itself contain multiple dimensions, generating another indefinite hierarchy. That question revealed an ambiguity. “Dimension” can suggest a fixed coordinate system already containing every possible theological aspect. “Aspect” or “formal profile” leaves open whether new kinds of theological relevance can emerge that were not coordinates in the previous system.

The adequacy of a representation should therefore be indexed to specified aspects. A model may represent one relation well and another poorly. There is no initial license to combine every criterion into a single number called “closeness to God.”

A refinement relation might be written:

R_A ≼ R_B.

This means that R_B is an admissible extension or refinement of R_A. I originally treated refinement as simple enlargement. That proved insufficient. Adding propositions can introduce contradiction. Increasing expressive power can destroy decidability. A more complicated model can obscure a theological distinction that a simpler representation preserved.

“Refinement” must therefore be defined through adequacy criteria rather than size alone. It might require preservation of selected truths, correction of identified distortion, increased expressive capacity, compatibility with specified commitments, or improved explanatory power.

If ≼ is treated as a preorder, it must satisfy:

Reflexivity:
R ≼ R.

Transitivity:
If R ≼ R' and R' ≼ R'',
then R ≼ R''.

A strict refinement relation can then be defined:

R ≺ R'
iff
R ≼ R' and not(R' ≼ R).

The hypothesis of indefinite representational extensibility becomes:

IE:

For every R ∈ Rep,
there exists R' ∈ Rep such that:

R ≺ R'.

If IE is assumed directly, the absence of a maximal representation follows immediately. That is mathematically trivial. A substantial theorem would need to derive IE from other properties rather than include it as a premise disguised as a conclusion.

A refinement semantics for stable theological claims

A translation could map sentences from one representational language to another:

τ_AB : L_A → L_B.

If R_A ≼ R_B, a sentence φ ∈ L_A is preserved under that refinement when:

R_A ⊨ φ
→
R_B ⊨ τ_AB(φ).

A stronger notion of stability would require preservation through every admissible future refinement. Introduce a refinement modality □_≼:

R ⊨ □_≼ φ

iff

for every R' such that R ≼ R':

R' ⊨ τ_RR'(φ).

The dual possibility operator ◇_≼ can express availability in some refinement:

R ⊨ ◇_≼ φ

iff

there exists R' such that R ≼ R'
and
R' ⊨ τ_RR'(φ).

Indefinite extensibility could then be expressed modally as:

For every representation R:

R ⊨ ◇_≼ ProperExtension.

An invariant theological claim would satisfy:

If R ⊨ φ,
then R ⊨ □_≼ φ.

This proposed logic distinguishes two movements that I initially treated separately:

Stability:
truth survives legitimate refinement.

Extensibility:
a proper refinement remains possible.

A theological representation would avoid stagnation if it remained extensible, and avoid mere replacement if it preserved warranted invariants. A mature theory would need to determine which claims deserve modal stability and which remain revisable.

For example, a tradition might propose that divine non-dependence should remain invariant:

R ⊨ NonDependent(G)
→
R ⊨ □_≼ NonDependent(G).

Another claim might be treated as revisable rather than invariant:

R ⊨ φ
and
R ⊨ ◇_≼ ¬τ(φ).

This formal possibility matters because theological development may correct earlier representations rather than simply accumulate them. Requiring every later model to preserve every earlier proposition would preserve errors as well as truths.

Institution theory as a possible mathematical home

The question of truth-preserving translation connects naturally with institution theory, developed in theoretical computer science to compare logical systems independently of one fixed logic (Goguen and Burstall, 1992).

An institution is a structure:

I = (Sign, Sen, Mod, ⊨),

consisting of:

a category Sign of signatures;
a functor Sen : Sign → Set assigning sentences to each signature;
a contravariant functor Mod : Sign^op → Cat assigning models to each signature;
a satisfaction relation ⊨_Σ between Σ-models and Σ-sentences.

Given a signature morphism:

σ : Σ → Σ',

the sentence functor translates a Σ-sentence into a Σ'-sentence:

Sen(σ) : Sen(Σ) → Sen(Σ').

The model functor moves in the opposite direction by taking a Σ'-model to its Σ-reduct:

Mod(σ) : Mod(Σ') → Mod(Σ).

The institution satisfaction condition requires:

M' ⊨_(Σ') Sen(σ)(φ)

iff

Mod(σ)(M') ⊨_Σ φ.

In plain language, translating the sentence forward and reducing the model backward preserve truth. Satisfaction remains invariant under a legitimate change of notation.

A proposed theological institution might be written:

I_Theo = (
    Sign_Theo,
    Sen_Theo,
    Mod_Theo,
    ⊨_Theo
).

Its components could be interpreted as:

Sign_Theo:
formal vocabularies for theological aspects.

Sen_Theo(Σ):
theological sentences expressible in signature Σ.

Mod_Theo(Σ):
formal structures interpreting Σ.

⊨_Theo:
the satisfaction relation between those structures and sentences.

This would not yet solve the theological problem. Institution theory preserves satisfaction under translation; it does not determine whether the chosen sentences are revealed truths, whether the models adequately represent God, or whether the translations preserve the intended theological meaning. Those questions would enter through additional constraints on admissible signatures, models and morphisms.

Nevertheless, institution theory offers a rigorous language for the question that had emerged through the mirror analogy:

What remains true when theological discourse is translated between different formal representations?

No final representation and the terminal-object question

The earlier search for a final “view from nowhere” can also be expressed categorically.

Let Rep be a category whose objects are theological representations and whose morphisms are admissible translations or refinements:

Objects:
R, R', R'', ...

Morphisms:
f : R → R'.

A terminal object R_⊤ would satisfy:

For every object R in Rep,
there exists a unique morphism:

f_R : R → R_⊤.

Symbolically:

Terminal(R_⊤)
iff
for every R ∈ Ob(Rep),
there exists exactly one f : R → R_⊤.

If arrows represent movement toward a final refinement, R_⊤ could be interpreted as a representation into which every other representation maps canonically. A proposed non-finality claim would be:

There does not exist R_⊤ ∈ Ob(Rep)
such that Terminal(R_⊤).

In a preorder rather than a general category, uniqueness of arrows is automatic. The corresponding maximality condition is:

Greatest(R_⊤)
iff
for every R:

R ≼ R_⊤.

The no-final-representation claim becomes:

There does not exist R_⊤
such that for every R:

R ≼ R_⊤.

But neither version is currently a theorem. Everything depends on how representations and morphisms are defined. If the category is constructed to contain a terminal object, it has one. If indefinite extensibility is inserted as an axiom, it does not. The research task is to find independently defensible theological and logical conditions from which a non-terminal result follows.

A non-trivial target might have the form:

No-Terminal-Representation Theorem Schema

Let Rep be a category of formal theological representations.

Assume:
C₁. Representations are effectively specifiable.
C₂. They are sufficiently expressive for semantic self-reference.
C₃. Morphisms preserve a defined class of theological invariants.
C₄. Every representation admits a diagonal extension
    satisfying an independently justified adequacy condition.

Then:
Rep has no terminal adequate representation.

At present, condition C₄ is only a research placeholder. The central difficulty is defining a diagonal extension that is theologically meaningful and not simply stipulated to be better. This is exactly where theological conceptual work might generate a new mathematical problem.

Why “better” may require several directions

My earlier search for the largest infinity assumed a single ordering. Once magnitude and adequacy were separated, this became untenable.

A representation may be more expressive while less computationally tractable. It may be more faithful to one theological tradition while less capable of translation into another. It may gain doctrinal precision while losing the narrative or transformative function of theological language. It may preserve consistency only by excluding difficult theological claims.

Adequacy may therefore need to be represented as a profile. Let A be a set of currently identified theological aspects. For each a ∈ A, let Q_a be a partially ordered space of adequacy values. Then:

q_a : Rep → Q_a

assigns an aspect-relative adequacy value to each representation. The full adequacy profile is:

a_A(R) = (q_a(R))_(a∈A)

with:

a_A(R) ∈ ∏_(a∈A) Q_a.

In a simplified finite example:

a(R) = (
    expressivity,
    consistency,
    explanatory power,
    traditional continuity,
    computational tractability,
    transformational adequacy,
    apophatic restraint
).

The inclusion of “apophatic restraint” is especially significant. A model may fail theologically through deficiency, but it may also fail by claiming too much. A representation that accurately states its own scope may be more adequate than one that covers more propositions while confusing formal success with ontological possession.

An aspect-relative Pareto ordering can be defined:

R ≼_A R'

iff

for every a ∈ A:

q_a(R) ≤_a q_a(R').

A strict improvement would additionally require improvement in at least one aspect:

R ≺_A R'

iff

R ≼_A R'

and there exists a ∈ A such that:

q_a(R) &lt;_a q_a(R').

Two representations may be incomparable:

not(R ≼_A R')
and
not(R' ≼_A R).

One may be stronger in expressivity and another in computational tractability. Neither is therefore unconditionally “closer to God.”

The space of aspects may itself be extended. If A is the present aspect set, a later inquiry may introduce:

A ⊊ B.

An indefinite-extensibility hypothesis for aspect spaces would be:

For every admissible aspect space A,
there exists an admissible B such that:

A ⊊ B.

This is another proposed principle rather than an established theorem. It expresses the possibility that no fixed coordinate system anticipates every theologically relevant aspect.

When A ⊆ B, there should be a restriction or projection map:

π_BA : ∏_(b∈B) Q_b → ∏_(a∈A) Q_a.

A coherent refinement should relate the richer evaluation to the earlier one:

π_BA(a_B(R')) ≥_A a_A(R),

if R' genuinely improves upon R with respect to every preserved aspect in A. This condition formalizes the idea that introducing new aspects should not silently erase the standards under which the earlier representation was judged—unless the later theory explicitly argues that an earlier standard itself requires revision.

Representations may consequently be only partially ordered. One can be better along some dimensions and worse along others. There may be no unique optimum. This produced one of the most precise questions in the entire inquiry:

Under which explicitly defined theological aspects and adequacy criteria does mathematical enlargement correspond to a better representation of divine infinity?

This question does not assume that enlargement is improvement. It asks for the conditions under which the correspondence holds. A possible mathematical contribution would be an impossibility result showing that no single scalar ranking can preserve all the desired adequacy relations. For now, that remains a proposal rather than a theorem.

The AI dialogue became part of the object

The development of this inquiry depended materially on dialogue with an AI system. Making that role invisible would create a false impression that the final structure had been present from the beginning.

I supplied the initial question, the theological context, my surprise at Cantor and my repeated dissatisfaction with answers that stopped too early. The AI articulated several formulations that changed the direction of thought: mathematics had learned to use infinity while suspending what infinity ultimately is; God might be outside or ground of the scale; reflection could permit positive knowledge without comprehension; invariants might replace a single totalizing perspective; and apophaticism might become a formal research programme.

Each useful formulation also generated a new objection. When the AI said that a universal set would not be God, I clarified that I was asking about projection rather than identity. When it said self-reference did not create transcendence, I asked what mathematical operation could go beyond every boundary. When it suggested that no final standpoint exists, I initially experienced the answer as disappointing and asked whether mathematics had proved the impossibility of approaching God. When it proposed invariance across perspectives, I asked how invariance relates to the equally important movement of going beyond. When it distinguished dimensions of divine infinity, I asked whether the space of dimensions was itself indefinitely extensible.

The AI did not solve these questions authoritatively. Its role was closer to a rapidly revisable interlocutor. My questions exposed hidden assumptions in its answers, while its reformulations exposed hidden assumptions in mine. The output of one stage became the input of the next:

Initial intuition
→ AI formulation
→ theological objection
→ mathematical qualification
→ revised question
→ external scholarship
→ new formal proposal
→ further objection

This recursion eventually turned toward AI itself. Can an AI learn to represent the limits of representation, or will it continually convert transcendence into another object within its computational universe? A language model can produce sentences about ineffability, incompleteness and mystery. It can also speak about them with the same fluent confidence it uses for ordinary objects. The capacity to name a boundary is not identical with respecting that boundary.

A computational theology of non-exhaustive representation would therefore need to distinguish at least:

falsehood;
uncertainty caused by insufficient evidence;
undecidability relative to a formal theory;
inexpressibility in a current language;
model-relative truth;
and principled non-exhaustion of a referent.

These categories can be represented as different statuses rather than one generic “unknown” value:

Status(φ, T) ∈ {
    Proven,
    Refuted,
    UndecidedInT,
    UndecidableInT,
    InexpressibleInL,
    EmpiricallyUnresolved,
    TheologicallyNonExhaustive
}.

The categories do not all belong to the same logical level. Proven and Refuted concern derivability. UndecidableInT is metatheoretical. InexpressibleInL concerns the language. EmpiricallyUnresolved concerns evidence. TheologicallyNonExhaustive depends on a theological adequacy relation. A responsible AI system should not collapse them into a single confidence score.

These distinctions may be valuable beyond theology, especially for AI systems that must report the scope and limits of their own representations.

Whether this is mathematics, theology or fake science

As the programme became more formal, another question emerged: would pure mathematicians dismiss it as fake science?

Some versions would deserve dismissal. “Gödel proves God,” “a large cardinal is close to God,” or “mathematical incompleteness is evidence of divine transcendence” all move from formal results to ontology without defending the transition. Mathematical symbols can create an appearance of precision even when the underlying analogy remains undefined.

The appropriate protection is to mark the epistemic status of each claim:

a mathematical theorem follows from explicit formal assumptions;
a proposed definition introduces terminology to be evaluated for usefulness and coherence;
a conjecture states something that still requires proof or counterexample;
a formal model stipulates a representation of selected theological claims;
a computational verification establishes derivability from encoded premises;
a philosophical bridge argues for a relevant structural analogy;
and a theological interpretation evaluates that analogy within a tradition.

The distinction can be displayed formally:

Established:
T ⊢ φ.

Model-theoretic:
M ⊨ φ.

Computational:
A verifies that Proof_T(φ) exists.

Proposed bridge:
FormalRelation(X,Y) is relevantly analogous to
TheologicalRelation(G, creatures).

Theological conclusion:
Accepted only if the bridge is independently warranted.

A theorem prover can verify that a conclusion follows from premises. It cannot establish by that operation alone that the premises are theologically true or that their referent exists. Work in computational metaphysics has shown both the power of mechanized formalization and the importance of distinguishing verification of entailment from verification of ontology (Kirchner, Benzmüller and Zalta, 2019).

A pure mathematics journal would reasonably reject an article that contained no new theorem. That would mean the article was not a contribution to pure mathematics. It would not by itself make the project intellectually illegitimate. Its first academic homes would probably be philosophical logic, philosophy of mathematics, formal ontology, analytic or systematic theology, science-and-religion and computational metaphysics.

If the project later produced a new semantics, a proof calculus, a preservation result, a countermodel or an impossibility theorem, it might contribute directly to mathematical logic or theoretical computer science. Until then, its novelty should be described as a proposed synthesis and research programme rather than an accomplished mathematical theory.

Where the inquiry now stands

I began by asking whether mathematics had found infinities large enough to illuminate the infinity of God. I then discovered that Cantor himself had sharply distinguished transfinite mathematics from Absolute Infinity and placed the latter in relation to God. This made the historical question more urgent: why had later mathematics studied infinity so intensively while largely suspending Cantor’s theological conclusion?

I next tried to recover that conclusion through universal sets, self-reference and increasingly large infinities. Those attempts failed to provide the desired transcendence. A universal object remained relative to a theory. Self-reference remained formally controlled. Every higher standpoint remained a standpoint.

The phrase “God would instead be outside—or the ground of—the scale itself” changed the direction of the inquiry. Cardinal magnitude ceased to be the obvious measure of theological approximation. Reflection then showed how positive knowledge might remain possible without containment. The mirror could be true without containing the sky.

But this solution generated another problem: how can truth without exhaustion be formalized? The answer required distinctions among completeness, consistency, definability, categoricity, self-verification and ontological adequacy. Gödel, Tarski, Cantor and Löwenheim–Skolem could discipline the discussion, but none could independently prove divine incomprehensibility. A theological bridge principle had to be stated rather than concealed.

Finally, I asked whether theology could offer anything in return. The possible answer is that theology supplies a sophisticated conceptual problem: how to represent truthfully without collapsing representation into possession. This may motivate a formal theory of aspect-relative adequacy, invariant relations, admissible refinement and non-terminal representation. Whether that programme produces new mathematics remains open.

The present formal architecture can be summarized as follows:

1. Formal theological system:
   F = (L, T, ⊢).

2. Formal representation:
   R_A for theological aspect-space A.

3. Satisfaction:
   R_A ⊨ φ.

4. Aspect-relative adequacy:
   Adeq_A(R, G).

5. Non-exhaustion:
   Adeq_A(R, G) → ¬Exh(R, G).

6. Refinement:
   R_A ≼ R_B.

7. Translation:
   τ_AB : L_A → L_B.

8. Preservation:
   R_A ⊨ φ → R_B ⊨ τ_AB(φ).

9. Stable invariant:
   R ⊨ □_≼ φ.

10. Indefinite extensibility:
    For every R, there exists R' with R ≺ R'.

11. Possible categorical target:
    the representation category has no terminal adequate object.

12. Mathematical limitations:
    Gödel–Rosser, Gödel II, Tarski,
    Löwenheim–Skolem and Cantor.

13. Required theological bridge:
    formal non-closure does not by itself entail
    divine incomprehensibility.

Only items derived from existing theorems currently have established mathematical status. The adequacy relation, the apophatic non-exhaustion principle, the refinement logic and the no-terminal-representation theorem remain proposed components of a possible research programme.

I can therefore state a provisional conclusion:

Mathematics cannot currently prove that God is beyond all formal representation. It can prove that many natural candidates for exhaustive representation cannot simultaneously achieve consistency, effective axiomatizability, completeness, internal self-verification, unique determination and unrestricted semantic closure. Theology supplies the further claim that this formal non-closure may correspond analogically to divine incomprehensibility. The legitimacy of that interpretation depends upon an explicitly defended bridge between the formal result and the theological referent.

The question with which I began has not been answered. It has been replaced by a better one:

Can we construct a rigorous theory of representations that permits real knowledge, preserves truth through refinement, remains open to new theological aspects, and formally prevents representational success from being mistaken for exhaustive possession of the referent?

And this question remains accompanied by another:

Which limitations arise from a particular language, which persist across whole families of formal systems, and which—if any—may responsibly be interpreted as analogies of divine transcendence?

The two questions can be combined into a proposed formal research target. Let Rep be a category or preorder of theological representations. Each representation R has an aspect-space A_R, a language L_R, and a set K_R of warranted claims. For every admissible refinement f : R → R', let τ_f : L_R → L_R' translate claims into the refined language. A theory of non-exhaustive representation would seek structures satisfying the following conditions:

Let R, R' ∈ Rep.

1. Aspect-relative adequacy:

   Adeq_(A_R)(R, G).

2. Truth preservation through refinement:

   If f : R → R',
   then:

   τ_f[K_R] ⊆ K_R'.

   Equivalently, for every φ ∈ K_R:

   R ⊨ φ
   →
   R' ⊨ τ_f(φ).

3. Persistent invariance:

   R ⊨ □_≼ φ

   iff

   for every admissible R' with R ≼ R':

   R' ⊨ τ_RR'(φ).

4. Indefinite representational extensibility:

   For every R ∈ Rep,
   there exists R' ∈ Rep such that:

   R ≺ R'.

5. Indefinite extensibility of theological aspects:

   For every aspect-space A_R,
   there exists an aspect-space A_R' such that:

   A_R ⊊ A_R'.

6. Formal separation of adequacy from exhaustion:

   For every R ∈ Rep:

   Adeq_(A_R)(R, G)
   ↛
   Exh(R, G).

   Under an explicitly adopted apophatic bridge principle:

   Adeq_(A_R)(R, G)
   →
   ¬Exh(R, G).

7. Absence of a final adequate representation:

   There does not exist R_⊤ ∈ Rep such that:

   for every R ∈ Rep,
   R ≼ R_⊤

   and

   Exh(R_⊤, G).

Here G functions at the metatheological level as the referent rather than as an ordinary object inside every model. Conditions 1–5 describe how representations may convey knowledge, preserve warranted truths and remain extensible. Condition 6 prevents the formal inference from aspect-relative adequacy to ontological exhaustion. Condition 7 states the desired non-finality result, although it is presently a proposed research condition rather than an established theorem. A substantive mathematical theory would need to derive condition 7 from independently justified conditions rather than assume non-finality from the beginning.

The second question can then be formalized by classifying a limit property P according to whether it changes under admissible translations between formal systems. Let C be a family of formal systems and let F : T → T' be an admissible translation, interpretation or refinement. Then:

Language-relative limitation:

LangRel(P)

iff

there exist T, T' ∈ C and an admissible F : T → T'
such that:

P(T)
and
¬P(T').

Family-stable limitation:

Stable_C(P)

iff

for every T ∈ C:

P(T).

Translation-invariant limitation:

Invariant_C(P)

iff

for every admissible F : T → T':

P(T) ↔ P(T').

Possible theological interpretation:

Stable_C(P)
+
Bridge_P(P, G)
→
Analogy_P(P, Transcendence(G)).

A limitation is language-relative when it disappears after an admissible change of language, logic or expressive strength. It is family-stable when it recurs throughout a defined class of systems, and translation-invariant when legitimate translations preserve it. Even a family-stable or translation-invariant limitation does not by itself become evidence of divine transcendence. That final interpretation requires an additional bridge principle Bridge_P explaining why the formal structure of limitation P is relevantly analogous to a theological meaning of transcendence. The research programme must therefore investigate both the mathematical invariance of each limitation and the theological legitimacy of the bridge by which it is interpreted.

I still do not know whether these questions will produce new mathematics, a new form of computational theology, or primarily a philosophical discipline for using mathematical results responsibly. What has become clear is that the most promising path does not lead toward the greatest object mathematics can construct. It leads toward a more precise understanding of how finite representations can be truthful, transformable and inexhaustibly open without identifying themselves with the whole.

References

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