Axioms — Expert View (All Models)#

This compiled view presents all 25 axioms of mathematical theology. Content is extracted from the canonical PoR sources. For the beginner-friendly version, see Axioms — Beginner-Friendly Overview.

PoR sources:

PET Axioms ax1–ax14 — a1–a14 (PET foundation)
JUB Axioms — ax15–ax25 (Group VI: Agency & Delegation) — a15–a25 (JUB extension)

Navigation: a1 · a2 · a3 · a4 · a5 · a6 · a7 · a8 · a9 · a10 · a11 · a12 · a13 · a14 · a15 · a16 · a17 · a18 · a19 · a20 · a21 · a22 · a23 · a24 · a25

All 25 Axioms at a Glance#
id	Title	Summary
a1	Containment	The world is part of God.
a2	Transcendence	God is not part of the world.
a3	Divine Surplus	There is something in God that is not in the world.
a4	Universal Immanence	Every part of the world is in God.
a5	Necessary Divine Existence	God necessarily exists.
a6	Contingency of the World	The world’s existence is contingent — it might or might not exist.
a7	Necessary Containment	If any world exists, it is necessarily in God.
a8	Immanent Presence	God is present to every part of the world.
a9	Sustaining Dependence	If the world exists, God sustains it.
a10	Asymmetric Dependence	The world does not sustain God.
a11	Divine Structure (Dipolarity)	God has both an unchanging nature and a world-responsive experience.
a12	Revelation Reliability	God’s self-knowledge is true.
a13	Consistency of Revelation	God’s self-knowledge contains no contradictions.
a14	Revelation Claims Test	Human claims about divine revelation must be mutually consistent and consistent with a1–a13.
a15	Human Genuine Agency	Humans possess real capacity to choose among alternatives within D_free.
a16	Delegation of Dominion	God has granted humans governance authority over the Earth.
a17	Non-Coercive Guidance	God guides (but does not force) humans within D_free and D_inno.
a18	Responsibility Localization	Moral responsibility for outcomes rests with the agent, not the delegator.
a19	Probabilistic Causal Concentration	At any time t, there exists a unique h* with strictly maximal causal influence.
a20	Transient Volunteer	God seeks humans who will voluntarily accept specific D_inno responsibilities.
a21	Permanent Mediator	God seeks one human willing to permanently translate between divine optimality and human understanding.
a22	Divine Preference for Genuine Love	God values genuine, freely-chosen love above compelled obedience.
a23	Freedom-Quality Superiority	Genuine care, insight, and innovation are qualitatively inferior when compelled.
a24	Life-Trifecta of Lasting Innovation	Innovation lasts iff simultaneously stable, extensible, and life-friendly.
a25	Jubilee Recalibration	Innovation economies need periodic recalibration to prevent BABL-cascade.

Group I — Mereological Core (PET)#

These axioms encode the distinctive panentheistic claim and distinguish it from both pantheism (\(G = W\)) and classical theism (which denies a1).

a1 — Containment#

PoR source

The world is a mereological part of God.

Everything that exists in the created world is contained within God. The world exists inside God, not separate from God.

Formal statement:

\[W \leq G\]

Context: The mereological parthood relation \(\leq\) asserts that W is a part of G. This is the foundational panentheistic claim — the “en” (in) of pan-en-theism. Combined with a2, it means God strictly exceeds the world (\(W < G\), proper parthood).

Logic: Mereology (CEM — Classical Extensional Mereology); Modal logic S5

Reasoning: a1 defines the fundamental parthood relation. Combined with a2 (\(G \nleq W\)), it yields proper parthood \(W < G\). This single pair distinguishes panentheism from pantheism (\(G = W\)).

Limit: The mereological relation assumes a specific logic (CEM). Alternative mereologies (e.g., non-extensional) may yield different structural results.

a2 — Transcendence#

PoR source

God is not reducible to the world.

While the world is in God, God is not reducible to the world. God exceeds creation — you cannot capture all of God by examining all of the world.

Formal statement:

\[G \nleq W\]

Context: Together with a1, this yields \(W < G\) (proper parthood): God strictly exceeds the world. This single axiom pair (a1 + a2) is what distinguishes panentheism from pantheism (which asserts \(G = W\)).

Limit: a2 alone does not guarantee divine surplus (there could be nothing beyond the world in God). a3 strengthens this.

a3 — Divine Surplus#

PoR source

There exists genuine divine content beyond creation.

God’s transcendence is not merely formal — God has actual content beyond the created world. There are aspects of God that no examination of the world alone could reveal.

Formal statement:

\[\exists x\;(x \leq G \;\wedge\; \neg(x \leq W))\]

Context: This strengthens a2. Without a3, a1 + a2 could be satisfied trivially. a3 asserts there is genuine divine content beyond creation.

Limit: a3 asserts existence of surplus but does not specify its nature.

a4 — Universal Immanence#

PoR source

Every part of creation is within God.

There is no part of creation that is “outside” God. Every atom, every person, every corner of the universe — all are within God. This is the pan (all) of pan-en-theism.

Formal statement:

\[\forall x\;(x \leq W \;\rightarrow\; x \leq G)\]

Context: This follows trivially from a1 by transitivity of \(\leq\), but it is stated explicitly because it carries theological weight. It is the universally quantified version of a1: not just the world as a whole, but every individual part.

Limit: a4 is logically derivable from a1 via transitivity. Its axiom status is justified by theological weight, not logical independence.

Group II — Modal Axioms (PET)#

These axioms use modal logic to distinguish between what is necessary (true in every possible world) and what is contingent (true in some worlds but not others).

a5 — Necessary Divine Existence#

PoR source

God exists necessarily and uniquely.

In every possible scenario, in every conceivable way reality could be arranged, God exists and is unique. God’s existence is not an accident — it could not have been otherwise.

Formal statement:

\[\Box\;\exists!\,G\]

Context: The \(\Box\) (necessarily) operator combined with \(\exists!\) (there exists exactly one) asserts that God’s existence is a necessary truth, not a contingent fact.

Limit: Assumes S5 modal logic. The necessity claim is framework-dependent.

a6 — Contingency of the World#

PoR source

The world’s existence is not necessary.

Unlike God, the world did not have to exist. There are possible scenarios with no created world at all, and possible scenarios with one. Creation is a contingent fact, not a necessary truth.

Formal statement:

\[\Diamond\;\exists W \;\wedge\; \Diamond\;\neg\exists W\]

Context: Both \(\Diamond\;\exists W\) (a world is possible) and \(\Diamond\;\neg\exists W\) (no world is possible) are asserted. This makes the world’s existence genuinely contingent.

Limit: Does not specify whether the world’s contingency is temporal (began) or modal (might not have existed). Both readings are consistent.

a7 — Necessary Containment#

PoR source

Any possible world is necessarily within God.

The containment of the world in God is not accidental. In every possible scenario where creation exists, creation is within God. There is no possible form of creation that is external to God.

Formal statement:

\[\Box\;(\exists W \;\rightarrow\; W \leq G)\]

Context: This modalizes a1. a1 says the world is in God; a7 says it must be in God — there is no possible world where creation exists outside of God. This is a strong panentheistic commitment.

Limit: Depends on S5 modal logic. In weaker modal systems, the necessity claim may not hold.

Group III — Relational Axioms (PET)#

These go beyond mereological containment to assert that God does not merely contain the world but is actively present to and sustains it.

a8 — Immanent Presence#

PoR source

God is present to every part of the world.

It is not enough that the world is inside God. A box contains its contents without being aware of them. a8 asserts that God is intimately present to — aware of, in contact with — every single part of creation. This rules out deistic panentheism (God contains but ignores).

Formal statement:

\[\forall x\;(x \leq W \;\rightarrow\; P(G,\, x))\]

Context: The presence predicate \(P(G, x)\) goes beyond mereological containment. A container holds its contents; a8 asserts God is aware of and in contact with every part.

Limit: The presence predicate \(P\) is introduced but not fully formalized. Its semantics need further specification.

a9 — Sustaining Dependence#

PoR source

The world depends on God for continued existence.

The world does not keep itself in existence. Its continued existence depends on God’s active sustaining. If God withdrew sustaining, the world would cease to exist.

Formal statement:

\[\Box\;(\exists W \;\rightarrow\; S(G,\, W))\]

Context: The sustaining predicate \(S(G, W)\) asserts an active relationship. The world does not persist by inertia; it persists because God sustains it.

Limit: The sustaining predicate \(S\) is introduced but not formally defined. Its relationship to mereological parthood needs specification.

a10 — Asymmetric Dependence#

PoR source

The world does not sustain God.

The dependence runs strictly one way. God sustains the world (a9), but the world does not sustain God. God’s existence does not depend on the world in any way.

Formal statement:

\[\neg\, S(W,\, G)\]

Context: Together with a9, this encodes a strict ontological asymmetry. God can exist without the world (t2); the world cannot exist without God. This distinguishes PET from process-theology variants where God genuinely needs the world.

Limit: Process theology disagrees with this axiom, asserting mutual dependence.

Group IV — Divine Nature (PET)#

a11 — Divine Structure (Dipolarity)#

PoR source

God has both unchanging nature and world-responsive experience.

God has two aspects. One aspect (the necessary divine nature, \(G_n\)) never changes — it is the same in every possible scenario. The other aspect (the contingent divine experience, \(G_c\)) varies depending on what happens in the world.

Formal statement:

Strengthened, 4 lines:

\(G = G_n \oplus G_c\)
\(\Box\;\exists\, G_n\)
\(G_c = \bigoplus\{G_c(w_i) \mid w_i \leq W\}\)
\(\forall w_1, w_2 \leq W : w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)\)

Lines 3–4 were added (2026-03-14/15) to give \(G_c\) internal structure, making theorem t4 (Divine Experience Varies) formally derivable.

Context: The dipolar structure \(G = G_n \oplus G_c\) parallels Whitehead’s primordial/consequent nature distinction and the Hindu nirguna/saguna Brahman distinction. Lines 3–4 add subworld-indexed structure to the contingent pole.

Limit: The a11/a11’ fork (dipolarity vs. divine simplicity) is the deepest point of theological disagreement. Classical theists will reject a11 in favor of a11’. See PET Axioms — Discussions and Caveats.

Group V — Revelation Bridge (PET)#

These axioms are optional extensions bridging from philosophical theology to revealed theology.

a12 — Revelation Reliability#

PoR source

God’s self-knowledge is true by definition.

R is defined as the set of true propositions about God — God’s own self-knowledge. Since R consists of true propositions by definition, a12 is tautological by design. The substantive work of testing claims shifts to a14 (Revelation Claims Test).

Formal statement:

Let \(R = \{p \mid p \text{ is true about } G\}\). For all \(p \in R\): \(p\) is true.

Context: a12 is deliberately tautological. The point is to establish the concept of R (God’s self-knowledge) so that a13 and a14 can build on it. The substantive work happens in a14.

Limit: Tautological by design. Does not assess any specific human claim about revelation.

a13 — Consistency of Revelation#

PoR source

God’s self-knowledge contains no contradictions.

The true propositions about God are mutually consistent. You cannot derive a contradiction from what is genuinely true about God. If two claims appear to contradict each other, at least one of them is not actually in R.

Formal statement:

\[\neg\exists\;\text{contradiction within } R\]

Context: a13 establishes the internal consistency of R as a formal property. Combined with a14, it provides a method for detecting human error in revelation claims.

Limit: Does not address paraconsistent logic approaches that tolerate controlled contradictions.

a14 — Revelation Claims Test#

PoR source

Human claims about revelation must pass consistency checks.

This is a powerful and practical axiom. It provides a formal method for testing whether a proposition that someone claims is divinely revealed actually passes two consistency checks: (1) it must not contradict any other claimed revelation, and (2) it must not contradict axioms a1–a13.

Formal statement:

Reformulated, 2 lines:

\(\forall p, q : \text{claim}(p) \wedge \text{claim}(q) \rightarrow \neg(p \wedge q \rightarrow \text{contradiction})\)
\(\forall p : \text{claim}(p) \rightarrow \neg\exists q \in \{a1\text{--}a13\} : (p \wedge q \rightarrow \text{contradiction})\)

Key design features:

No self-reference: a14 references a1–a13, not a1–a14.
The claim(p) predicate: Distinguishes between God’s actual self-knowledge (R) and human claims about what is in R.
Testable method: Given two traditions with conflicting claims, a14 asks: can both claims be true simultaneously without contradiction?

Context: a14 is the practical workhorse of the revelation bridge. It shifts the burden from “is this true?” to “is this consistent?” — a tractable question.

Limit: a14 detects inconsistency but does not establish truth. Passing a14 is necessary but not sufficient for a claim to be in R.

Group VI — Agency & Delegation (JUB)#

These axioms extend the PET foundation to cover human agency, divine delegation, and the conditions under which innovation failure produces evil.

Domain definitions:

D_f (forced domain): choices constrained by physics, coercion, or circumstance.
D_free (free domain): choices where humans possess genuine capacity to select among alternatives.
D_inno (innovation subdomain, \(D_{\text{inno}} \subseteq D_{\text{free}}\)): the critical subset where novel solutions, creative acts, and innovation occur.

a15 — Human Genuine Agency#

PoR source

Humans possess real capacity to choose within D_free.

Within a defined domain of free choices, humans can genuinely select among at least two alternatives. This is not a polite fiction — it is the load-bearing axiom of the entire theodicy. Denial is performatively self-refuting: the act of denying one’s agency is itself an exercise of agency.

Formal statement:

\[\begin{split}\forall h \in H,\; \forall s \in D_{\text{free}} :\; & \exists a_1 \neq a_2 \;\text{such that} \\ & \text{can-choose}(h, a_1, s) \;\wedge\; \text{can-choose}(h, a_2, s)\end{split}\]

Context: D_f is explicitly excluded. The innovation subdomain D_inno (a subset of D_free) is where the critical theodicy question concentrates: whether to pursue innovation toward the flourishing of others when the capacity to do so exists.

Limit: The D_f/D_free boundary remains a formal demarcation problem. The compatibilism robustness note shows the practical argument survives under multiple accounts of agency.

a16 — Delegation of Dominion#

PoR source

God has granted humans governance authority over Earth.

God did not merely create humans and place them in the world — God entrusted them with genuine authority over it. This delegation is real, not nominal: God does not routinely override human decisions within D_free/D_inno.

Formal statement:

\[\begin{split}& \text{Delegated}(G, H, W_{\text{earth}}) \\ & \wedge\; \forall\, \text{outcome}\; o \;\text{in}\; W_{\text{earth}} : \text{PrimaryResponsible}(H, o)\end{split}\]

Context: Together with a15, this resolves the “puppet-master” objection. If humans were instruments (puppets), a16 would not transfer responsibility. But because humans are genuine agents (a15), delegation transfers primary responsibility to the delegate.

Limit: The chain-of-command objection (delegator retains responsibility) is addressed by a15 (genuine agents, not instruments) but not formally proven.

a17 — Non-Coercive Guidance#

PoR source

God guides but does not force humans.

God provides guidance — invitations, hints, opportunities, the “still small voice” — but does not compel. The non-coercion is a principled choice (explained by a22), not a power limitation.

Formal statement:

\[\begin{split}\forall h \in H :\; & \text{Guide}(G, h) \;\wedge\; \neg\text{Force}(G, h) \\ & \quad [\text{within } D_{\text{free}} \text{ and } D_{\text{inno}}]\end{split}\]

Context: a17 is scoped specifically to D_free/D_inno. Within D_f (the forced domain), God maintains physics — this is a precondition for agency, not a form of guidance or coercion. The domain split resolves the objection that God is responsible for physics-based harm.

Limit: Does not address miraculous intervention (D_f-level divine action). The domain split handles this but the boundary remains imprecise.

a18 — Responsibility Localization#

PoR source

Moral responsibility rests with the delegated agent.

Given that humans have genuine agency (a15), have been delegated authority (a16), receive non-coercive guidance (a17), and are not forced (a17), responsibility for outcomes in the delegated domain rests with the human agents. This is the formal core of the innovation theodicy.

Formal statement:

\[\begin{split}& \text{Delegated}(G,H,D) \;\wedge\; \text{Agency}(H) \\ & \wedge\; \text{Guide}(G,H) \;\wedge\; \neg\text{Force}(G,H) \\ & \quad \rightarrow\; \text{Responsible}(H, \text{outcomes in } D) \\ & \quad \wedge\; \neg\text{Responsible}(G, \text{outcomes in } D)\end{split}\]

Context: This may be a theorem rather than an axiom, pending formal grounding of the moral responsibility predicate. It is stated explicitly for clarity: the four premises mechanically produce the theodicy conclusion.

Limit: The D_f/D_free boundary for poverty cases needs further formal work (capabilities theory: Sen 1999, Nussbaum 2011). See Con-D.2.9.

a19 — Probabilistic Causal Concentration#

PoR source

A unique individual has maximal causal influence at each moment.

At any given moment, one person’s choices have more causal influence on the future trajectory of the world than anyone else’s. This person (h*) stands at the top of the “leviathan chain.” The uniqueness claim is the null hypothesis: exact equivalence in effect is measure-zero.

Formal statement:

\[\begin{split}\forall t\; \exists!\, h^* \in H :\; & \text{MaxCausalInfluence}(h^*, t, W_{\text{future}}) \\ & \wedge\; \forall h \neq h^* : \\ & \quad \text{CausalInfluence}(h, t, W_{\text{future}}) < \text{CausalInfluence}(h^*, t, W_{\text{future}})\end{split}\]

Context: The scalar projection is performed by Reality itself. Causal influence appears multi-dimensional, but civilization has only one future. The realized trajectory is a single path through the infinite-dimensional space of possibilities. Along this path, each person’s cumulative influence is a well-defined (if humanly uncomputable) scalar — just as evolutionary fitness projects multi-dimensional organism traits onto a scalar.

Limit: Epistemic identification of h* is NOT claimed. a19 is an ontological claim (h* exists), not an epistemic one (we can identify h*). The fitness analogy is structural, not superficial.

a20 — Transient Volunteer#

PoR source

God seeks willing humans for moment-specific responsibilities.

At each critical moment, God is actively looking for a human willing to step into a specific innovation responsibility. The emphasis is on voluntarily — consistent with a17 and a22. “Transient” because the call is moment-specific, not permanent.

Formal statement:

\[\begin{split}\exists\; & \text{voluntary-invitation from } G \text{ to } H_{\text{candidates}} \text{ at each } t, \\ & \text{seeking } h \in H : \text{Willing}(h) \\ & \wedge\; \text{Accepts}\bigl(h,\; \text{Responsible}(h, D_{\text{inno}}, t)\bigr)\end{split}\]

Context: Multiple transient volunteers may be sought simultaneously for different D_inno responsibilities. The burning bush pattern (Exod 3:3–10) is the archetype.

Limit: Does not specify the mechanism of divine invitation.

a21 — Permanent Mediator#

PoR source

God seeks one willing permanent translator between divine and human.

Beyond the moment-specific calls of a20, God seeks one person willing to permanently accept the role of translating between what God knows is optimal and what humanity currently understands. The output quality depends on it being freely chosen — a compelled translator cannot produce genuine translation.

Formal statement:

\[\begin{split}\exists\; & \text{voluntary-invitation from } G \text{ seeking } h \in H : \\ & \text{Willing}(h) \\ & \wedge\; \text{Permanent}\bigl(h,\; \text{TranslatorRole}(G, H, W_{\text{future}})\bigr) \\ & \wedge\; \text{OutputQuality depends on } \neg\text{Force}(G, h)\end{split}\]

Context: a21 is the individual permanent analog of Quran 33:72 (amanah). a20 handles transient individual calls; a21 handles the permanent mediation role.

Limit: The singularity of the permanent role is a strong claim. Multiple sequential mediators may be the historical pattern.

a22 — Divine Preference for Genuine Love#

PoR source

God’s necessary nature values freely-chosen care above forced compliance.

This is not merely a preference — it is part of God’s necessary nature (Gn). It explains why God uses non-coercive guidance (a17) when having the power to compel: compelled care produces a qualitatively different divine experience (Gc) that the necessary divine nature (Gn) values less. Forced love is not love, and God knows the difference.

Formal statement:

\[\begin{split}\forall w_1, w_2 \leq W \;\text{where}\; & w_1 = \text{freely-chosen-care}, \\ & w_2 = \text{forced-compliance}, \\ & \text{physical}(w_1) = \text{physical}(w_2) : \\[6pt] & G_n\text{-valuation}(G_c(w_1)) > G_n\text{-valuation}(G_c(w_2))\end{split}\]

Context: This axiom connects to the a11 dipolar structure. Gc is affected by the quality of human response — freely-chosen love produces a qualitatively different Gc state than forced compliance, even when the physical actions are identical.

Limit: Assumes God’s Gn includes a valuation ranking. This is a substantive metaphysical claim about divine nature.

a23 — Freedom-Quality Superiority#

PoR source

Genuine care and innovation require freedom to achieve full quality.

This is the empirical backbone of a22’s explanation. Genuine care, creative insight, and lasting innovation are qualitatively degraded or destroyed by compulsion. A compelled factory worker can produce identical widgets; a compelled poet cannot produce genuine poetry.

Formal statement:

\[\forall o \in O_{\text{genuine}} : \text{quality}(o \mid \text{compelled}) < \text{quality}(o \mid \text{freely-chosen})\]

Context: Self-Determination Theory (Ryan & Deci, 2000) provides the secular empirical backbone: autonomously motivated behavior produces higher quality outcomes, greater creativity, more durable results, and better wellbeing.

Limit: Applies specifically to outputs requiring creativity, empathy, or genuine engagement — not to all outputs.

a24 — Life-Trifecta of Lasting Innovation#

PoR source

Innovation lasts iff stable, extensible, and life-friendly simultaneously.

Three properties must be satisfied simultaneously — violating any one places the innovation on the BABL attractor (Blindly Assuming Blind Leveraging), leading to eventual collapse. Three formulations: ethical (Gentle/Kind/Reasonable), engineering (Stable/Extensible/Life-friendly), negative (not oversimplifying / not overcomplicating / not overreaching).

Formal statement:

\[\text{Lasting}(i) \;\leftrightarrow\; \text{Stable}(i) \;\wedge\; \text{Extensible}(i) \;\wedge\; \text{LifeFriendly}(i)\]

Context: BABL (Blindly Assuming Blind Leveraging) is the self-destructive attractor. The river of life (Ezek 47, Rev 22:1–2) is the positive attractor; the millstone/sea of destruction (Mt 18:6) is BABL. Bezalel (Exod 31:2–5) is the positive Torah archetype.

Limit: The predicates Stable, Extensible, LifeFriendly lack formal semantics. The 7TrackRole model is identified as the formalization path but not yet executed.

a25 — Jubilee Recalibration#

PoR source

Innovation economies need periodic recalibration to prevent BABL-cascade.

Even initially life-trifecta-compliant economies accumulate concentration over time. Without periodic recalibration, the life-friendly cord is eventually violated. The biblical Jubilee year (Lev 25, Deut 15) is the Torah instantiation. a25 synthesizes capitalism’s insight (incentive structures between rounds) with communism’s insight (redistribution mechanism), avoiding either’s fatal flaw.

Formal statement:

\[\begin{split}\forall\; & \text{innovation economy } E \text{ under a24} : \\ \exists\; & \text{recalibration } R : \\ & \text{periodic}(R) \;\wedge\; \text{redistributes}(R, E) \\ & \wedge\; \text{resets}(R, E) \;\rightarrow\; \text{sustainable}(E)\end{split}\]

Context: The Jubilee System preserves property rights and incentives between rounds (capitalism’s contribution) while resetting accumulated advantages at each round (communism’s contribution). Neither ideology alone satisfies the life-trifecta.

Limit: Cross-traditional support requires qualification — only Torah directly supports the periodic reset mechanism. Arrow’s impossibility constrains but does not prohibit Jubilee System design (see Con-E.2.11).

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