PET Axioms — Discussions and Caveats#

Honest assessment of the axiom system’s strengths, limitations, open questions, and the surprisingly strong scriptural convergence result. For the axioms themselves, see PET Axioms ax1_A1–ax14_A14.

The Convergence Result#

The most striking finding from the PET axiom project is not the axioms themselves but the cross-traditional scriptural convergence.

When the 14 axioms were checked against the scriptures of Judaism (Torah + Prophets & Writings), the direct teachings of Jesus (Gospel), wider Christianity, Islam, Hinduism, and even secular philosophy, a consistent pattern emerged: all six perspectives independently support the same formal structure for the relationship between God (or ultimate reality) and the world.

This was not expected. The axioms were constructed from panentheistic philosophy (mereology + modal logic), not from scriptural exegesis. The scriptural check was performed after the axioms were defined, as a test of whether the formal structure matches what the traditions actually say.

The convergence is strongest for the core axioms (ax1_A1, ax2_A2, ax4_A4, ax5_A5, ax8_A8, ax9_A9) and extends with varying degrees of directness to all 14 axioms, including the methodological axioms ax12_A12–ax14_A14. The full evidence is presented on the PET Axioms ax1_A1–ax14_A14 page under each axiom’s “Scriptural and philosophical support” section.

What this means: If the convergence holds up under expert review, it suggests that the theological disagreements driving real-world conflict between Abrahamic traditions are not about the structure of God’s relationship to the world — where the scriptures agree — but about other claims layered on top of that structure. The axiom system provides a formal tool for distinguishing where traditions genuinely disagree from where they merely think they disagree.

What this does not mean: Scriptural convergence does not prove the axioms true. It demonstrates internal consistency across traditions, which is a necessary but not sufficient condition for truth.

Confidence Levels#

  • High confidence (direct, mainstream scriptural readings): ax1_A1, ax2_A2, ax4_A4, ax5_A5, ax8_A8, ax9_A9, ax13_A13, ax14_A14

  • Good confidence (solid, sometimes requires combining verses): ax3_A3, ax6_A6, ax7_A7, ax10_A10, ax12_A12

  • Interesting but contested: ax11_A11 (Dipolarity) — the Torah references to God relenting/regretting are read as anthropomorphisms by classical theologians, as genuine divine change by process theologians. The Hindu nirguna/saguna distinction is the most explicit and least contested source for ax11_A11.

  • Structural analogy only: The secular column maps the axiom structure to observations about reality without invoking God. This is by design: the pattern is recognizable even without theism.

Mereology May Be Too “Spatial”#

The part-whole relation (\(\leq\)) carries physical connotations. When we say “the world is part of God,” readers may imagine spatial containment — God as a cosmic container with the world inside.

A category-theoretic approach could be more abstract and less misleading: God as an ambient category, the world as a subcategory with an inclusion functor that is not essentially surjective. This would capture the same logical structure without the spatial metaphor.

However, mereology was chosen deliberately for accessibility. The axioms are meant to be readable by theologians, diplomats, and educated laypeople, not only by mathematicians. “Part of” is intuitive in a way that “inclusion functor” is not. The formal properties (reflexive, transitive, antisymmetric) are stated explicitly to prevent misreadings.

The ax11_A11 Fork: Dipolarity vs. “Divine Simplicity”#

This is the most important choice point in the axiom system. ax11_A11 (Dipolarity) and ax11b_A11b (Divine Simplicity) are mutually exclusive alternatives that lead to fundamentally different theological consequences.

ax11_A11 (Dipolarity) says God has a necessary aspect \(G_n\) (unchanging divine nature) and a contingent aspect \(G_c\) (God’s concrete experience, which varies with the world). This comes from Hartshorne’s dipolar theism and captures the idea that God is affected by the world.

ax11b_A11b (Divine Simplicity) says God has no proper parts that are independent of each other.

The name “Divine Simplicity” is arguably misleading. What ax11b_A11b actually asserts is that everything in God depends on everything else — making God maximally entangled from a structural standpoint. The only thing that becomes “simple” is the human task of thinking about God: since no aspect can be isolated or analyzed independently, the conclusion is effectively “don’t try to understand God’s internal structure, because God is incomprehensible.” This is the classical apophatic theology position.

By contrast, ax11_A11 (Dipolarity) is more tractable because it decomposes God into analyzable aspects — making it possible to derive theorems like th4_T4 (Divine Experience Varies) and to reason about how God’s relationship to the world actually works.

ax11b_A11b sits in tension with ax1_A1 + ax3_A3: If the world is in God (ax1_A1) and something beyond the world is also in God (ax3_A3), then God appears to have distinguishable parts — which ax11b_A11b denies. Resolving this tension is one of the deepest problems in panentheistic theology.

Tradition alignment: Classical theists (Thomists, much of Islamic and Jewish philosophical tradition) insist on simplicity. Process theologians and most panentheists insist on dipolarity. The axiom system is modular here: explore consequences of each branch separately.

The Pos(φ) Predicate: Present but Unused#

The primitive term \(\text{Pos}(\varphi)\) (“φ is a positive property,” in Gödel’s sense of a perfection) is listed in the PET Symbol Dictionary table but is not used in any of ax1_A1–ax14_A14. It would become relevant if the system were extended to derive God’s specific attributes (omniscience, goodness, omnipotence, etc.) rather than just the containment structure.

A possible extension:

ax15_A15. \(\forall\varphi\;(\text{Pos}(\varphi) \rightarrow \varphi(G))\) — God has every positive property.

Combined with ax12_A12–ax14_A14, this could be powerful but also dangerous. Benzmüller and Woltzenlogel Paleo showed that Gödel’s original axioms (which include a version of ax15_A15) entail modal collapse: everything that is true is necessarily true, eliminating contingency entirely. This would contradict ax6_A6.

Any extension using Pos(φ) must therefore be carefully checked against the existing axioms for consistency.

Formalization Does Not Equal Truth#

As with Gödel’s ontological proof, the rigor of this axiom system is in the deduction, not in the axioms. Anyone can reject ax1_A1, or ax5_A5, or ax12_A12. The value is in making explicit what follows from what — and in discovering whether a given theological position is even internally consistent.

The axiom system is designed to be critiqued, not believed. If you reject an axiom, the system shows you exactly which consequences you also lose. If you accept an axiom, the system shows you what else you are committed to.

What Is Missing#

The current 14-axiom system says nothing about:

  • The problem of evil — how to reconcile God’s presence/sustaining (ax8_A8, ax9_A9) with suffering in the world

  • Free will — whether and how creaturely freedom operates within divine containment

  • Time and eternity — whether God experiences time, and how modal logic relates to temporal logic

  • Personal vs. impersonal divinity — whether God is a person, a force, or something else entirely

  • Trinitarian structure — Christian-specific internal structure of God

  • Prophetic succession — Islamic-specific claims about the seal of prophecy

  • Messianic claims — claims across traditions about specific individuals

Each of these would require additional axioms and likely additional formal machinery. They represent natural directions for extending the PET system.

TELES migration report (2026m04d04)

Mechanical identifier migration applied to this file. All axiom/theorem text references were migrated from short form (e.g., A15) to compound form (e.g., ax15_A15) as part of the matheology compound naming operation. Both forms refer to the same formal object. The old form survives as the suffix to ensure consistency with the oldest records; the new form adds a temporary-status prefix. Forward-facing pages use brief form (ax15) only. See TELES Axiom/Theorem Compound Naming — Execution Prompt for the complete mapping table and DD b12 — Legacy Naming for PET/JUB Axioms and Theorems for the permanent reference.