Disputatio XXX: De Veritate Interna et Externa Theologiae

On Internal and External Truth in Theology

Quaeritur

Utrum veritas theologiae sit tantum interna in suo sermone, an etiam externa in relatione ad ipsum Deum, ita ut sermo theologicus non solum sit fidelis in se, sed etiam verus de eo quod est.

It is asked whether the truth of theology is only internal to its own discourse, or also external in its relation to God himself, so that theological language is not only faithful in itself but also true of what is.

Thesis

Theology possesses a twofold truth: internal and external. Veritas interna is the felicity of discourse authorized by the Holy Spirit within the community of faith; it is the truth of theology intra systema fidei. Veritas externa is the adequation of this discourse to divine reality: the truth of theology de Deo ipso.

The former concerns the integrity of theological grammar; the latter, the participation of that grammar in the infinite Word. The two are not opposed but ordered: the Spirit authorizes language internally so that it may participate externally in the Logos.

Formally expressed:

$$\text{<i>Auth</i>}<i>(</i>{\mathrm{<i>L</i>}}_{\mathrm{<i>t</i>}}<i>)</i><i>\Rightarrow </i>\mathrm{<i>\exists </i>}\mathrm{<i>I</i>}<i>:</i>{\mathrm{<i>L</i>}}_{\mathrm{<i>t</i>}}<i>\hookrightarrow </i>{\mathrm{<i>L</i>}}_{\mathrm{<i>\infty </i>}}$$

This states that theological truth obtains only if the Spirit establishes an interpretive inclusion of the finite theological language $L_t$ into the infinite divine discourse $L_{\mathrm{\infty }}$. This symbol (↪) indicates participatory inclusion, not the formal subset relation. It is the Spirit’s act by which finite discourse is gathered into infinite meaning.

Locus Classicus

“I am the way, and the truth, and the life.” — John 14:6

In this declaration, Christ identifies truth not as correspondence or coherence, but as personal participation in the divine Logos. Augustine reads this as meaning that “veritas non est aliquid extra Deum, sed ipse Deus veritas est” (De Trinitate VIII.4). Aquinas echoes: “Veritas est adaequatio intellectus et rei; in Deo autem idem est intellectus et res” (ST I.16.5). Therefore, for theology, truth cannot be merely formal, but it must be participatory. The finite intellect is true only insofar as it shares in the divine act of knowing.

Explicatio

On the Transcendental Nature of Truth

The transcendentals—ens, unum, verum, bonum—are convertible: whatever is, is one, true, and good.

1.  Ens signifies being under the aspect of actuality.
2. Verum signifies being under the aspect of intelligibility, that is being as capable of being known
3. Thus, truth is not something added to being; it is the luminosity of being itself.
4. Wherever there is being, there is an implicit veritas, because to be is to be capable of being understood
5. Hence, truth is a transcendental property of being.

In God, who is ipsum esse subsistens, being and intelligibility are identical; He is the verum primum, the archetype and measure of all other truths.

In creatures, truth is participatory: finite beings are true insofar as they share in divine intelligibility and correspond to the divine idea in which they are conceived.

On Internal and External Truth

a. Veritas interna refers to the integrity and coherence of a discourse or system—its syntactic and performative correctness, its internal felicity.

b. Veritas externa refers to the correspondence between what is said and what is—between sign and reality, intellect and being.

Theological language, like all human speech, can achieve felicity without necessarily achieving truth; yet without internal felicity, it could not even aim at truth.

Felicity, then, is the condition of possibility for truth, but not its ground. The ground of truth is the being of God, in whom every finite coherence finds its measure.

Theology, as language about God, must therefore transcend its own grammar: its truth lies not merely in saying well (bene dicere), but in saying what is (dicere esse). It is true when its internal felicity participates in divine external truth.

On the Participation of Theology in Divine Truth

Theology does not produce truth; it participates in it. When the theologian speaks truly of God, this occurs because divine being grants intelligibility to that speech—because the light of being shines through language. The Holy Spirit is the mediating act through which this participation occurs: the one who makes human words capable of bearing divine meaning.

Thus, every theological assertion is twofold:

• Internally: it is grammatically felicitous, coherent, and consistent.
• Externally: it is illuminated by the light of the One who is Truth itself.

In this participation, truth and being are united without confusion. Theology is not the measure of God; God is the measure of theology.

Obiectiones

Ob I. Empiricism claims that all truth must be verifiable by observation. Theological claims are unverifiable and thus have no external truth. There is only the internal coherence of theological discourse for believers using it. 

Ob II. From the cultural-linguistic standpoint, theology’s meaning arises only within the communal grammar of faith. Thus, to speak of “external truth” misunderstands language as representational rather than formative. Theology is true insofar as it performs its grammar.

Ob III. Post-modernity assumes that every discourse is self-referential such that “outside” a language game there is nothing. Hence “external truth” is a non-starter. All truth is internal to interpretation.

Ob IV. Barthians held that God’s revelation is self-grounded and free, and that appeal to participation or adequation cannot verify it. Truth exists only in the event of revelation, it is not tied to ontology.

Ob V. Contemporary analytic thinking holds that model-theoretic analogies fail for theology. There is no definable model of God; hence talk of inclusion Lt​↪L∞​ is metaphorical and lacks formal content.

Responsiones

Ad I. Verificationism mistakes the order of reality for the order of appearance. Theology is not an empirical but a participatory science: it knows by union, not by observation. External truth in theology is not sensory correspondence but ontological inclusion in the act of God.

Ad II. The Church’s grammar is indeed formative, yet its form is the Spirit’s work, not a human construct. The Spirit’s authorship makes the grammar porous to transcendence; hence, its truth cannot be merely communal but is grounded in the divine speech that precedes the Church.

Ad III. Postmodern closure presupposes the very transcendence it denies. The internal system’s finitude points beyond itself to the infinite that constitutes it, just as, by the Löwenheim–Skolem principle, any consistent system admits higher interpretations. The finite theological discourse testifies by its very limitation to the necessity of the divine meta-language.

Ad IV. Revelation is not opposed to participation but presupposes it. God’s free act of self-disclosure is the mode in which creatures participate in divine truth. To say revelation alone grounds truth is already to affirm that truth has external reality in Deo ipso.

Ad V. While theology cannot construct a formal model of God, the analogy holds analogically: God is the modelus sui sermonis, the reality to which divine discourse is perfectly adequate. Finite theology participates in that adequation by the Spirit. Thus, the inclusion ${\mathrm{<br>}}_{}$ is not formal but real: it signifies the Spirit’s act of joining human speech to the eternal Word.

Nota 

The distinction between internal and external truth in theology mirrors the structure of revelation itself. Veritas interna designates the truth of faith—the Spirit’s authorization of discourse within the divine economy of speech, and veritas externa names the correspondence of that discourse to divine reality as such. These are not two truths, but two perspectives upon one act. The internal truth is the participation of the believer in the Word, while the external truth is the participation of the Word in the world.

Within the sphere of veritas interna, felicity and faith coincide: the statement “Jesus is Lord” is true because it is spoken in the Spirit. Within veritas externa, that same statement is true because the incarnate Word is objectively Lord of all. The Spirit assures, the Word grounds, and the Father unites these two horizons in the single act of truth.

Thus, theology’s truth is not reducible to logic nor to experience; it is a relation of participation. Language, illumined by the Spirit, shares in the ontological act of the Word and so becomes both performative and correspondent. The finite utterance is true when it is gathered into the divine discourse that both causes and completes its meaning.

Determinatio

1. Truth is a transcendental property of being: verum est ens in quantum cognoscibile.

2. Being and intelligibility are coextensive; full being entails full intelligibility.

3. In God, being and knowing are one: esse et intelligere sunt idem.

4. Every finite act of truth is participatory, grounded in the infinite act of self-identical being.

5. Theology’s internal coherence (felicity) depends upon its participation in external divine truth.

6. Therefore, theology is true because it shares in the self-identical fullness of being, in whom to be and to be known are one.

7. Veritas interna is the pneumatological authorization of theological discourse, its faithfulness, coherence, and integrity within the Spirit’s grammar.

8. Veritas externa is the Christological participation of that discourse in the divine Logos, the ontological adequation by which the Word that words also constitutes what is.

9. The two are ordered: the Spirit perfects language internally so that it may correspond externally to the Word.

10. Finite discourse, like a logical system, cannot ground its own truth; it requires inclusion in the infinite speech of God.

11. Therefore, theological truth is neither merely communal nor purely propositional but participatory, It is rather the inclusion of finite utterance in infinite meaning.

Hence we conclude: Veritas interna sine externa est infidelis; veritas externa sine interna est muta. Only when the Spirit authorizes and the Logos fulfills does theology speak the truth.

Postscriptum Modernum

Gödel’s incompleteness theorems and the Löwenheim–Skolem results together illuminate the formal necessity of theological participation.
Gödel showed that any sufficiently rich, consistent formal system contains truths that cannot be proven within it; there are statements true in the system but not demonstrable by it. The Löwenheim–Skolem theorems, conversely, reveal that no formal language uniquely determines its own model, for even first-order theories with infinite models admit both smaller and larger interpretations.

Taken together, these findings expose a deep structural fact: no finite system can secure its own truth. Consistency does not entail completeness and satisfaction within does not entail adequation without. Hence, every coherent finite language gestures beyond itself toward a meta-language or an infinite frame in which its truth is grounded.

Theology mirrors this logic. The finite L_t of human discourse may be internally consistent—Spiritually felicitous—but its truth as about God depends upon participation in the infinite L_∞ of the divine Logos. The incompleteness of reason is not its defect but its vocation: it is the mark of the finite’s openness to the Infinite.

Thus, what logic demonstrates negatively—that no system can prove itself complete—theology confesses positively: Finite speech becomes true only when the Word that words gathers it into the world that worlds. In this gathering, internal felicity becomes external truth; the Spirit’s authorization becomes the Logos’s fulfillment.

Transitus ad Disputationem XXXI

If the Spirit authorizes theology’s internal felicity and the Logos grounds its external truth, what is the nature of the concept that mediates between them? The next disputation investigates the structure of human conceptuality itself: its finitude, its schematism, and its completion in the real Word. 

We proceed to Disputatio XXXI: De Conceptuali Schematismo et Verbo Reali, and ask how human concepts, limited by finitude, become vessels of infinite meaning, and how the Real Word transforms thought itself into a mode of divine speech.

Disputatio XXIX: De Paradoxo Löwenheim–Skolemiano

On the Löwenheim–Skolem Paradox

Quaeritur

Quaeritur utrum systema formale possit determinare extensionem suam propriam, an vero iuxta theoremata Löwenheim et Skolem omnis ordo formalis habeat multitudinem modelorum diversae magnitudinis, unde sequitur relativitas veritatis ad modelum, et necessitas fontis veritatis externi.

It is asked whether a formal system can determine its own proper extension, or whether, according to the theorems of Löwenheim and Skolem, every formal order admits a plurality of models of different sizes, from which follows the relativity of truth to a model and the necessity of a source of truth external to the system.

Thesis

The Löwenheim–Skolem theorems reveal that no consistent formal theory can uniquely fix the structure of its universe, for every such theory possesses models of varying magnitude and scope. Hence, truth within a model (internal truth) differs from truth about the model (external truth). This formal distinction corresponds analogically to the theological distinction between felicity and truth: the first internal to theology’s discourse, the second dependent on divine reality beyond it.

Locus Classicus

“For now we see through a glass, darkly; but then face to face.” — 1 Corinthians 13:12

Augustine comments (De Trinitate XV.8): “Nondum est species, sed per speculum et in aenigmate.” Thomas Aquinas (ST I.12.11) interprets this as the difference between cognitio viatoris and cognitio comprehensoris, the knowledge of the pilgrim and the knowledge of the blessed. The former is mediate and partial, and the latter direct and complete. So too in logic: truth within a system (per speculum) and truth from beyond the system (facie ad faciem) are distinct orders of knowing. The formal result mirrors the metaphysical condition of creaturely understanding.

Explicatio

I. The Discovery

Between 1915 and 1920, Leopold Löwenheim and Thoralf Skolem, working independently, established two theorems foundational for modern model theory.

1. The Downward Löwenheim–Skolem Theorem: If a first-order theory has an infinite model, then it has a countable model.

2. The Upward Löwenheim–Skolem Theorem: If a theory has a model of some infinite size, then for every larger cardinal number, it also has a model of that larger size.

Together, these results imply that no first-order theory with an infinite model can control the cardinality of its universe. A theory formulated in a finite language cannot uniquely determine the size or structure of the reality it describes.

II. The Skolem Paradox

The most striking consequence arises when these theorems are applied to set theory itself, the very discipline designed to describe infinite sets. Zermelo–Fraenkel set theory (ZF) proves the existence of uncountable sets such as the set of real numbers. Yet, by the Downward Löwenheim–Skolem Theorem, ZF has a countable model, a model in which, from an external perspective, all its “uncountable” sets are actually countable!

This tension is called the Skolem Paradox. It reveals that statements true within a model (“there exists an uncountable set”) need not correspond to what is true about that model from outside it. Accordingly, the model cannot see its own countability, and its internal truth diverges from external truth.

III. Philosophical Meaning

The Skolem Paradox formally demonstrates the relativity of truth to the level of discourse. What is “true in a model” depends on the interpretation supplied from outside the model. A system cannot guarantee that its own truths are absolute; they are true within a given structure, not simpliciter.

Philosophically, this means that finitude entails perspectivality. No finite framework can encompass all possible interpretations of its own symbols. Every internal horizon is bounded by a greater horizon of meaning. The finite world’s intelligibility, therefore, is not exhausted by its own immanent relations but opens onto a transcendent ground that “models” it from beyond.

IV. Theological Analogy

Here we reach the theological analogue: The relation between internal and external truth in logic mirrors the relation between theological felicity and theological truth.

• Felicity (in Austin’s and later theological sense) denotes statements that function properly within the authorized discourse of theology, e.g., confessional utterances, liturgical speech, or inspired proclamation.

• Truth refers to correspondence between theological discourse and divine reality itself.

As the formal system cannot secure external truth by internal means, theology cannot verify divine truth by linguistic coherence alone. It must depend on the Spirit, the “external source” who bridges internal felicity and external reality. In this way, the Löwenheim–Skolem results offer a formal reflection of pneumatological mediation.

Obiectiones

Ob I. Logical positivism supposes that the Löwenheim–Skolem theorems concern formal semantics only. They tell us nothing about metaphysics or theology. To interpret them as analogies of divine truth is poetic, not logical.

Ob II. Nominalism decries that the distinction between internal and external truth is artificial. All truth is internal to a framework; there is no standpoint outside language or model.

Ob III. Postmodern relativism argues that since every theory has multiple models, there is no absolute truth. The theorems confirm that meaning is plural, not that there is an external ground.

Ob IV. Rationalistic metaphysicians hold that if external truth is required, then finite knowledge becomes impossible. We can only know within a system; appealing to an Infinite ground destroys epistemic closure.

Ob V. Finally, cautious theology itself declares that to identify divine reality with a “meta-model” risks subordinating revelation to logic. However, God is not a semantic extension but a personal will.

Responsiones

Ad I. The interpretation is analogical, not literal. Logic reveals structural truths about expression and interpretation that parallel ontological relations between being and its ground. Analogy discloses order without confusion.

Ad II. To deny any standpoint beyond a system is self-refuting, for the assertion itself pretends to transcend the system it describes.The very recognition of frameworks implies an external perspective.

Ad III. The multiplicity of models does not entail relativism but dependence. That truth is manifold within systems implies that there must exist a unifying act among them. Otherwise, plurality becomes unintelligible.

Ad IV. Appeal to the Infinite does not abolish finite knowledge but secures it. Just as the meta-theory is the condition for model-theory’s truth, so too the external ground is the condition for internal intelligibility. 

Ad V. God is not a meta-model but the living ground of truth itself.
The analogy is formal, not an ontological identity. It shows that even logic intimates the same structure that theology names personally as Word and Spirit.

Nota

The Löwenheim–Skolem paradox is more than a logical curiosity. It is a parable of theology’s condition. The paradox teaches that a system’s internal truth may remain intact even when interpreted within a domain smaller than its intention. Thus, the same structure that claims to describe an uncountable reality can find a countable model in which all its sentences are true. This means that while within the model, truth is preserved, from the standpoint of the greater reality, that very truth is inadequate.

So too with theology: the veritas interna of faith—truth within the system of grace—is coherent and authorized by the Spirit, yet it cannot exhaust the veritas externa of the infinite God it confesses. While the Church’s language can constitute a faithful model, it cannot articulate the totality of divine reality. This is not error but mercy. Accordingly, incomprehensibility is the form of participation. To speak truly of God is to speak within a model sustained by God Himself, while knowing that the truth spoken exceeds the model that bears it.

Hence the paradox reveals the theological structure of all knowledge of God: truth in the model depends on truth about the model, just as immanence depends on transcendence. The Spirit guarantees the coherence of the inner discourse while the Logos grounds its correspondence to reality. Their unity is the event of revelation: the infinite entering the finite without ceasing to be infinite.

In this way, Löwenheim and Skolem unwittingly formalized a metaphysical law: The finite can truly mirror the infinite only because the infinite freely indwells the finite.

Determinatio

From the foregoing it is determined that:

1. The Löwenheim–Skolem Theorems reveal that every formal system admits models of many sizes. Internal coherence does not yield external uniqueness.

2. Truth within a model (veritas interna) and truth about a model (veritas externa) are formally distinct. The former depends on the latter for interpretation.

3. The finite order, therefore, mirrors the condition that it is self-consistent yet semantically open. Thus, its meaning cannot be secured from within but requires reference to a transcendent source.

4. Theologically, this formal relation corresponds to the distinction between felicity and truth. Theology’s internal felicity (Spirit-authorized speech) depends upon external truth (the reality of God) which the Spirit mediates.

5. Hence, the Löwenheim–Skolem Paradox becomes a logical parable of participation. The finite model cannot perceive its own countability. In the same manner, creation cannot grasp its own dependence. Both require an Infinite perspective in which their truths are integrated and completed.

Therefore, the multiplicity of models is not chaos but a sign, an index of the Infinite intelligibility that sustains every finite order of meaning.
What logic calls the meta-model, theology calls the Logos; what logic calls interpretation from without, theology names the act of the Spiritus Veritatis.

Transitus ad Disputationem XXX

The Skolem paradox discloses the difference between truth within and truth about a system—between felicity and correspondence.Theology must interpret this difference in light of the Spirit’s authorization and the Logos’s adequation. 

Hence the next inquiry, Disputatio XXX: De Veritate Interna et Externa Theologiae, examines how finite theological language can be both internally faithful and externally true of God.

Disputatio XXVIII De Systemate Incompleto et Veritatis Factore Infinito

On the Incomplete System and the Infinite Truthmaker

Quaeritur

Quaeritur utrum systema finitum, si sit consistent, possit continere veritatem suam propriam, an vero, iuxta theoremata incompleti Gödeliana, omnis ordo finitus necessario referat ad veritatis fontem extra se—ad infinitum veritatis factorem.

It is asked whether a finite system, if consistent, can contain its own truth, or whether—according to Gödel’s incompleteness theorems—every finite order must necessarily refer to a source of truth beyond itself, to an infinite truthmaker.

Thesis

Gödel’s incompleteness results demonstrate formally what metaphysics has long intuited: The finite cannot ground the totality of its own truth. Every consistent formal system sufficient for arithmetic contains true statements it cannot prove. Hence, truth exceeds derivation, and the complete explanation of truth demands participation in something transcending the finite system.

Locus Classicus

“Great is our Lord, and abundant in power; his understanding is infinite.”
— Psalm 147:5

Aquinas comments: “Intellectus divinus est infinitus, quia adaequat veritatem ipsius Dei, quae est infinitum esse.” (STI.14.6.) The divine intellect alone comprehends all truth as being identical to being. Human or finite systems of reason, by contrast, express truth participatively, that is, as reflections of the infinite intellect. Thus, the logic of finitude corresponds to the metaphysics of participation.

Explicatio

I. The Context of Gödel’s Discovery

In 1931, 25 year-old Kurt Gödel, an Austrian logician, published “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (Monatshefte für Mathematik und Physik 38, 1931). His goal was to investigate the limits of formal systems such as the Principia Mathematica by Whitehead and Russell, which sought to derive all mathematical truths from a finite set of axioms through mechanical rules of deduction.

To understand the significance of this, we must review some key notions. A formal system may be thought of as a rigorously defined language governed by rules. While its syntactic component consists of symbols and derivations its semantic component is concerned with truth and meaning about numbers or other entities to which it refers. For such a system to be a satisfactory foundation of mathematics, it must have two crucial properties:

1. Consistency: No contradiction can be derived within the system.

2. Completeness: Every true statement expressible in the system can be derived from the system's axioms.

Gödel’s work proved that these two properties cannot coexist in any finite system capable of expressing arithmetic.

II. Gödel’s First Incompleteness Theorem

Gödel showed in this proof how to assign to each formula and proof a numerical code, a process that is now called Gödel numbering. By this ingenious device, statements about formulas could become statements about numbers. He then constructed a sentence G that effectively says of itself,

“This statement is not provable within this system.”

If the system is consistent, it cannot prove G, for to do so would render it inconsistent, that is, it would prove a falsehood. Yet if the system is consistent, G is in fact true, since its unprovability makes the assertion it contains correct. Hence, G is true but unprovable within the system. The upshot of this is this: No consistent, sufficiently expressive finite system can be complete. Simply put, there will always exist true propositions that escape its derivations.

III. Gödel’s Second Incompleteness Theorem

Gödel then proved a deeper corollary, that no consistent system can prove its own consistency. But to show that its axioms are non-contradictory, one must appeal to a meta-system, to a higher language standing outside the system itself. Hence, every finite logical order depends on another for its assurance of truth and coherence.

IV. Philosophical Significance

Gödel’s theorems thus reveal a structural transcendence of truth over formal expression. They are not merely mathematical curiosities but demonstrations of a universal condition of finitude, that truth always surpasses the framework that tries to contain it. Every closed system that seeks to explain itself without remainder either collapses into contradiction or appeals to a higher order.

Metaphysically, this mirrors the ancient insight that the finite requires the infinite as its truthmaker. The correspondence between logical form and ontological order is not accidental but structural: just as a formal system needs a meta-system to ground its truth, so the finite world needs a transcendent act of being to ground its reality.

What Gödel discovered formally, metaphysics already discerned existentially: veritas non est intra ordinem finitum nisi per participationem veritatis infiniti.

Obiectiones

Ob I. Formalists like David Hilbert hold that the incompleteness theorems apply only to mathematical systems, not to reality. They concern symbols and proofs, not the metaphysical order of being.

Ob II. Scientific empiricism argues that science does not need to be “complete” in this logical sense. Explanatory power depends on observation, not on formal derivation. Thus, Gödel’s results have no bearing on physical intelligibility.

Ob III. Reductive naturalists claim that the analogy between formal systems and the finite world is metaphorical, and thus to move from logical incompleteness to ontological dependence is an illicit category jump.

Ob IV. Skeptics of many kinds opine that Gödel’s theorem requires arithmetic within a system, and that nature is not a formal calculus. Accordingly, it is meaningless to say that the universe is “incomplete” in the Gödelian sense.

Ob V. The cautious theologian claims that appealing to Gödel to prove divine necessity risks confusing logic with revelation. God’s infinity is not a corollary of syntax but a matter of faith.

Responsiones

Ad I. Gödel’s theorems indeed concern formal systems, yet they express a universal relation between expression and truth. Wherever truth is represented within a finite structure, that structure cannot exhaust it. The logical limit mirrors an ontological condition.

Ad II. Scientific explanation presupposes coherence and consistency within its theories. Gödel shows that such coherence cannot be self-guaranteed; it must be received from a higher frame. Hence, the dependence of empirical science on deeper intelligibility is reinforced, not diminished.

Ad III. The analogy is legitimate when carefully drawn. Formal systems model the relation of expression to truth; the finite world models the relation of being to its source. In both, self-sufficiency proves impossible; participation becomes the only path to completeness.

Ad IV. The universe is not a calculus, yet our reason reflects its structure through logic.To say that the world is “Gödelian” is not to mathematize it but to recognize that finitude, even in its most abstract forms, cannot close upon itself.

Ad V. The appeal to Gödel is not a theological proof but a formal analogy. It illuminates by example what theology asserts by revelation: that all truth in the finite is truth by participation in the Infinite Word.

Nota

Gödel’s theorem exposes not merely a boundary of formal systems but a metaphysical structure, for the finite, in order to remain consistent, must remain open to what it cannot contain. Incompleteness is thus not a defect but the mark of dependence. The object system’s unprovable truths are signs of an order beyond itself, an order upon which its very coherence rests.

In theology, this structure mirrors creation’s relation to its Creator. The creature is a consistent finite system whose truth is guaranteed only by participation in the infinite. Every finite logos, to be true, must be grounded in a Logos that transcends it; every rational discourse presupposes an unspoken act that makes discourse possible.

Hence Gödel’s discovery becomes a theological axiom: truth cannot be self-enclosed. Simply put, there must exist an actus essendi veritatis, an infinite truthmaker, by whom the finite is both comprehensible and incomplete. Logical incompleteness is thus a formal echo of the metaphysical participation of the finite in the divine, and the incompleteness of the finite itself. It reveals that closure is illusion, and openness to transcendence is the very condition of truth.

Determinatio

From the foregoing it is determined that:

1. Gödel’s incompleteness theorems formally demonstrate the incapacity of the finite for self-completion. Every consistent system depends upon truths it cannot contain and upon a meta-system it cannot itself generate.

2. Truth transcends formal derivation. Just as no calculus can produce all truths of arithmetic, no finite ontology can account for its own intelligibility.

3. Consistency requires transcendence. The assurance that a system is non-contradictory always arises from a higher standpoint.
   Ontologically, this implies that the finite world’s coherence depends on an Infinite act of being.

4. The Infinite functions as the universal truthmaker. The meta-system for logic corresponds analogically to the Creator for creation: the necessary being in whom all contingent truths are grounded and from whom their coherence flows.

5. Therefore, Gödel’s result, though mathematical in form, reveals a metaphysical truth: the finite is intelligible only by participation in the Infinite. The world’s incompleteness is not deficiency but sign — a structural openness to the Infinite intellect whose understanding is unbounded.

Hence, the incompleteness of systems becomes a formal witness within reason to the metaphysical participation of all truth in God — in quo sunt omnes thesauri sapientiae et scientiae absconditi (Colossians 2:3).

Transitus ad Disputationem XXIX

If every finite order requires an infinite truthmaker, how can finite language and models still signify truly? The following disputation explores the paradox of internal and external truth uncovered by the Löwenheim–Skolem theorem, showing how the structure of theology mirrors the relation between truth in a model and truth about it. 

We turn, therefore, to Disputatio XXIX: De Paradoxo Löwenheim–Skolemiano, wherein we examine how truth within a model and truth about that model diverge, and how this divergence reveals the theological relation between faith’s internal coherence and the infinite reality of God.

Disputatio XXVII De Essentiis Dispositionalibus

On Dispositional Essences

Quaeritur

Quaeritur utrum necessitas legum naturae possit sufficienter explicari per essentias dispositionales ipsarum rerum, an vero talis explicatio tandem recidat in naturalem essentialismum sine fundamento ontologico, qui rursus ad participationem infiniti redigitur.

It is asked whether the necessity of natural laws can be adequately explained by the dispositional essences of things themselves, or whether such an explanation ultimately collapses into an unfounded natural essentialism that once again requires participation in the Infinite.

Thesis

Dispositional essentialism seeks to ground the laws of nature in the intrinsic powers of entities. Accordingly, each thing, by virtue of what it is, behaves as it does. Laws are thus expressions of essence, not external constraints. Yet finite essence itself requires grounding for its actuality and coordination. Therefore, the appeal to dispositional essences displaces but does not resolve the need for an infinite ground of law.

Locus Classicus

“In him we live and move and have our being.” — Acts 17:28

The early Fathers, having read Paul in light of Hellenic metaphysics, interpreted this as a declaration that all powers and movements within creation presuppose divine causality. For example, Basil of Caesarea (Hexaemeron I.5) taught that “every natural power is the gift of divine energy,” and Aquinas affirmed that “omnis operatio naturae est actus Dei in natura” (ST I.105.5). Thus, even when power is intrinsic to a creature, its being and operation participate in the act of the Creator.

Explicatio

Dispositional essentialism emerged in late twentieth-century metaphysics as a reaction against both Humean regularity and Armstrong’s relational realism. This view aims to secure necessity without appeal to external laws or transcendent governance. Philosophers such as Brian Ellis, C. B. Martin, and Stephen Mumford argued that laws do not govern things from without but that the flow from within from the very essences or natures of entities. Accordingly, an electron repels another not because a law commands it, but because repulsion belongs to its nature. The behavior is thus essential and not contingent.

In this view, every natural property is dispositional; it is defined by its powers and tendencies. To possess a charge, mass, or spin just is to manifest appropriate dispositions under suitable conditions. Laws of nature are thus derivative descriptions of the necessary behaviors of these dispositional essences. Therefore, there are no separate laws or external principles, but only powers whose exercise constitutes the order of nature.

This approach elegantly restores necessity to the finite without invoking extrinsic governance. But the question remains: Whence the unity of this system of powers? If every essence carries its own necessity, what guarantees the coherence of those necessities across the totality of the world? Why do distinct powers not conflict or dissolve into chaos? While the finite essence, to be actual, must exist and operate within a coherent totality of being, that totality cannot itself be one of the powers. Rather, it must be the condition of their coexistence and harmony.

Hence, while dispositional essentialism succeeds in moving the locus of necessity inward—from external law to internal essence—it fails to remove the need for ontological participation. While essence, in so far as it is essence, is an intelligible structure of being, powers, however intrinsic, can only be participatory modes of a deeper enabling act.

Obiectiones

Ob I. According to Ellis in 2001, the essence of each natural kind explains its behavior. Thus, no further metaphysical foundation is required, and to demand more is to mistake explanation for regression.

Ob II. Martin argued in 2008 that dispositions constitute causal grounds for their manifestations. Since power is primitive and self-explanatory, the world’s order is the network of powers acting according to their natures.

Ob III. Mumford in 2004 argued that laws are supervenient on dispositional essences, and hence add nothing ontologically to them. Thus, the finite order is self-sufficient so long as it consists of stable powers and their mutual tendencies.

Ob IV. Naturalistic Metaphysics claims that to appeal to an Infinite act is unnecessary duplication. If dispositions suffice for explanation, positing divine participation is a metaphysical surplus.

Ob V. Sometimes the theological tradition assumed that grounding the powers of things directly in the infinite may risk erasing natural causality. But the integrity of secondary causes requires that creatures possess genuine powers of their own.

Responsiones

Ad I. While essence may explain behavior, it does not explain existence. To say “the electron repels because it is its nature to repel” still leaves unasked why such a nature exists at all. Essence is formal cause and being is act. The latter cannot be derived from the former without reference to a self-sufficient act of existence.

Ad II. Power cannot be self-explanatory, for power is always power to act.The actuality of its exercise depends on a larger order within which it operates. Without a unifying act of being, powers remain mere potentialities without coherence.

Ad III. Supervenience explains correspondence but not causation. That laws supervene on essences tells us that essence and law covary, not why such correlation obtains. The dependence relation itself requires grounding.

Ad IV. Appealing to the infinite is not an additional move but a natural completion in the order of explanation.The Infinite is not another entity among the powers but the act in which all finite essences receive their actuality and unity. Without such an act, the multiplicity of powers lacks ontological coherence.

Ad V. Participation does not abolish finite agency but founds it. Creatures possess true powers because the infinite communicates actuality to them. Their independence as secondary causes is secured by the divine act that continuously sustains them in being.

Nota

Dispositional essentialism rightly perceives that the necessity of nature arises from within things themselves. It holds that each being acts according to what it is, and thus, its tendencies are not imposed from without but flow from its essence. The very intelligibility of this insight, however, betrays its limit, for the intrinsic power of a thing explains its manner of acting, not its capacity to act at all. Accordingly, the essence that disposes toward activity still requires an act that gives it existence and coherence.

Hence, the metaphysical question beneath dispositional essentialism is not why things act as they do, but why there are things capable of acting at all. To say that the stone falls because it has mass, or that the charge repels because it is charged, presupposes the ontological act by which stone and charge subsist. Clearly, while the essence disposes, only the act sustains.

The theological transformation of this view is participation. On this view, every finite power is a communicated potency; it receives from the Infinite Act not only its existence but its coordination with all others. The unity of law in the world, that is, the harmony among dispositions, is thus the reflection of the divine unity that gathers all powers into a single order of being. Nature’s lawfulness is the shadow of grace: finite essences cooperate because they share in one act of creation.

Dispositional essentialism, therefore, contains a veiled confession:
to affirm inner necessity is already to acknowledge the immanence of the divine act within creation. The Spirit is the bond that makes powers conspire toward intelligibility, and the Logos is the act through which each essence becomes dispositional at all. Necessity, properly understood, is participation in the divine constancy by which all things are held in being.

Determinatio

From the foregoing it is determined that:

1. Dispositional essentialism internalizes necessity but does not abolish dependence. Thus, finite essences are intelligible structures whose actuality presupposes a unifying act of being.

2. The unity of natural order cannot arise from a plurality of isolated powers. Coordination among dispositions requires an ontological ground transcending them.

3. Essence without act is impotent. The existence and operation of every power presuppose an act that is not itself one power among others. They thus presuppose an infinite act of being.

4. The participation of finite essences in the Infinite corresponds to the metaphysical structure of creation. As Augustine said, “Omne bonum quod habet creatura, habet participando” (De Diversis Quaestionibus 83.46). Powers are real and finite, and their actuality is participatory.

5. Hence, dispositional essentialism, though the most promising finite account, nonetheless points beyond itself. Its truth lies not in rejecting participation but in clarifying the mode of it: each finite power is a share in the creative act that sustains and orders all powers.

Therefore, the necessity of natural law is neither imposed from without nor self-generated from within. It arises from the participation of dispositional essences in the infinite act of being, in the Word through whom all powers subsist and in the Spirit who continuously actualizes their operation.

Transitus ad Disputationem XXVIII

The unity of nature cannot be secured from within the multiplicity of powers. The next disputation therefore asks whether finite systems, even when internally coherent, can ever be complete in themselves. 

We proceed to Disputatio XXVIII: De Systemate Incompleto et Veritatis Factore Infinito, in which the Gödelian structure of dependence reveals that every finite necessity presupposes an infinite act of truth.