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–ax14.

The Convergence Result#

The most striking finding from the PET axiom project is not the axioms themselves but a cross-traditional scriptural convergence — partial, differentiated, and more interesting than a blanket agreement.

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 secular philosophy, a differentiated pattern emerged: six traditions converge strongly on a core subset of the axioms, support others with significant qualifications, and actively contest the distinctively panentheistic claims. The full evidence — including counter-evidence and out-of-context citation corrections — is presented on the PET Axioms ax1–ax14 page.

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.

What this means: The convergence is real but not uniform. The traditions agree on the structural axioms (transcendence, divine surplus, necessity, sustaining, claim-testing) but contest the distinctively panentheistic claims (containment, necessary containment, dipolarity). This distinction is itself a result: it locates where traditions genuinely agree and where the real theological work lies, rather than claiming blanket agreement.

What this does not mean: Scriptural convergence does not prove the axioms true. Nor does contestation of some axioms invalidate the system. The contested axioms are the system’s strongest proposals — the claims that push beyond established common ground into territory where traditions genuinely disagree and where formal analysis is most needed.

Confidence Tiers#

Based on independent tradition-specific review (see the scriptural review llog for full evidence), the 14 axioms fall into three confidence tiers:

Tier A — Robust convergence (Strong or Moderate support from 4+ traditions, with strong theological grounding):

  • ax2 (Transcendence) — the strongest convergence point across all traditions

  • ax3 (Divine Surplus) — universal agreement that God exceeds creation

  • ax5 (Necessary Existence) — strong from all four religious traditions

  • ax6 (Contingency of the World) — strong, with Hindu cyclical-cosmology qualification

  • ax9 (Sustaining Dependence) — the strongest relational axiom

Tier B — Supported with qualifications (Moderate support across traditions, but with tradition-specific caveats that must be acknowledged):

  • ax4 (Universal Immanence) — moderate from all traditions; citations support omnipresence, which is different from mereological containment

  • ax8 (Immanent Presence) — moderate; qualified by Judaism’s hester panim (hiding of the face) and Islam’s essence-vs-knowledge-presence debate

  • ax10 (Asymmetric Dependence) — strong from Islam (al-Ghani), moderate elsewhere; qualified by Kabbalistic itaruta de-letata and Christian kenosis

  • ax12 (Revelation Reliability) — strong from all, but tautological by design

  • ax13 (Consistency of Revelation) — strong from Islam (Quran 4:82); moderate elsewhere; complicated by Talmudic machloket and Islamic naskh

  • ax14 (Claims Test) — moderate-to-strong across all traditions; the epistemological hierarchy concern (human axioms judging revelation) is raised by all three Abrahamic traditions

Tier C — Proposed and contested (the distinctively panentheistic claims; these represent the paper’s strongest proposals, not its established results):

  • ax1 (Containment) — contested by all four religious traditions through distinct arguments: tzimtzum (Judaism), Incarnation (Christianity), tanzih (Islam), Advaita/Dvaita counter-readings (Hinduism). This is the foundational panentheistic claim and the most critical finding of the scriptural review

  • ax7 (Necessary Containment) — inherits ax1’s contestation and amplifies it by modalizing a contested claim

  • ax11 (Dipolarity) — opposed by mainstream Sunni Islam (tawhid, samad), contested by Judaism (Maimonides’ divine simplicity, anthropomorphism hermeneutic) and Christianity (Chalcedonian two-natures, impassibility tradition); Hindu nirguna/saguna distinction provides the strongest support but with Advaita and Dvaita qualifications

Structural analogy only:

The secular column maps the axiom structure to observations about reality without invoking God. The secular analogies are strongest for the epistemological axioms (ax14 parallels the scientific method) and weakest for the ontological axioms (ax1, ax5, ax7, ax8 — where the secular analog is either too generic or equivocates between abstract and concrete existence).

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 Fork: Dipolarity vs. “Divine Simplicity”#

This is the most important choice point in the axiom system. ax11 (Dipolarity) and ax11b (Divine Simplicity) are mutually exclusive alternatives that lead to fundamentally different theological consequences.

ax11 (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 (Divine Simplicity) says God has no proper parts that are independent of each other.

The name “Divine Simplicity” is arguably misleading. What ax11b 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 (Dipolarity) is more tractable because it decomposes God into analyzable aspects — making it possible to derive theorems like th4 (Divine Experience Varies) and to reason about how God’s relationship to the world actually works.

ax11b sits in tension with ax1 + ax3: If the world is in God (ax1) and something beyond the world is also in God (ax3), then God appears to have distinguishable parts — which ax11b 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–ax14. 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. \(\forall\varphi\;(\text{Pos}(\varphi) \rightarrow \varphi(G))\) — God has every positive property.

Combined with ax12–ax14, this could be powerful but also dangerous. Benzmüller and Woltzenlogel Paleo showed that Gödel’s original axioms (which include a version of ax15) entail modal collapse: everything that is true is necessarily true, eliminating contingency entirely. This would contradict ax6.

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, or ax5, or ax12. 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, ax9) 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.