On the Löwenheim–Skolem Paradox
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.
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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.
The Downward Löwenheim–Skolem Theorem: If a first-order theory has an infinite model, then it has a countable model.
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
Objiectio I. Logical postivism 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.
Objiectio 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.
Objiectio 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.
Objiectio 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.
Objiectio 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.
Responses
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.
Determinatio
From the foregoing it is determined that:
The Löwenheim–Skolem Theorems reveal that every formal system admits models of many sizes. Internal coherence does not yield external uniqueness.
Truth within a model (veritas interna) and truth about a model (veritas externa) are formally distinct. The former depends on the latter for interpretation.
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.
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.
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.
