PET Axioms ax1–ax14#

The 14 axioms of pan-en-theistic mathematical theology, organized in 5 modular groups. Each axiom is presented with a plain-English reading, a formal statement, an explanation, and scriptural/philosophical support from six independent traditions.

This is version iv_LLoL_PPv1r1p1_2026m03d14 — the strengthened formulation with ax11 lines 3–4 and reformulated ax12–ax14.

For the symbol dictionary, see PET Symbol Dictionary. For derived theorems, see PET Theorems. For caveats and open questions, see PET Axioms — Discussions and Caveats.

Formal Framework#

Mereology (part-whole logic) combined with modal logic S5 (necessity/possibility). Panentheism’s core claim is about a containment relation (“all is in God, but God exceeds all”), making mereology the natural foundation. S5 provides the modal operators for reasoning about what is necessary vs. contingent.

Group I — Mereological Core#

These axioms encode the distinctive panentheistic claim and distinguish it from both pantheism (\(G = W\)) and classical theism (which denies ax1).

ax1 — Containment#

The world is part of God.

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

Formal statement:

\[W \leq G\]

Explanation: 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 ax2, it means God strictly exceeds the world (\(W < G\), proper parthood).

Scriptural and philosophical support:

  • Torah: Deut 4:39 (“God in heaven above and earth beneath”)

  • Prophets & Writings: 1 Kings 8:27 (“heaven cannot contain you”); Ein Sof (“the Infinite”) in Kabbalah

  • Gospel (Jesus): “I am in the Father and the Father is in me” (Jn 14:10); “the kingdom of God is within you” (Lk 17:21)

  • Christian (wider): Acts 17:28 (“in him we live and move”); Col 1:17

  • Islamic: Quran 2:115 (“wherever you turn, there is the Face of God”); wahdat al-wujud (“unity of existence” — all being is in God)

  • Hindu: “All this is Brahman” (Chandogya Up. 3.14.1); Vishishtadvaita: world is God’s body

  • Secular: We are parts of a whole that exceeds any part

ax2 — Transcendence#

God is not part of the world.

In plain English: 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:

\[G \nleq W\]

Explanation: Together with ax1, this yields \(W < G\) (proper parthood): God strictly exceeds the world. This single axiom pair (ax1 + ax2) is what distinguishes panentheism from pantheism (which asserts \(G = W\)).

Scriptural and philosophical support:

  • Torah: Exod 33:20 (“you cannot see my face and live”); Deut 4:15 (“you saw no form”)

  • Prophets & Writings: 1 Kings 8:27 (“cannot contain”); Isa 55:8–9 (“my ways higher than your ways”)

  • Gospel (Jesus): “The Father is greater than I” (Jn 14:28); “My Father’s house has many rooms” (Jn 14:2)

  • Christian (wider): Eph 4:6 (“above all”)

  • Islamic: Quran 57:3 (al-Zahir = “the Manifest/Outward”); tanzih (“beyond comparison” — core Islamic transcendence)

  • Hindu: Neti neti — “not this, not this” (Brihad. Up. 2.3.6): Brahman exceeds all description

  • Secular: The universe exceeds any observer’s complete knowledge

ax3 — Divine Surplus#

There is something in God that is not in the world.

In plain English: 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:

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

Explanation: This strengthens ax2. Without ax3, ax1 + ax2 could be satisfied trivially. ax3 asserts there is genuine divine content beyond creation.

Scriptural and philosophical support:

  • Torah: Exod 33:18–23 (Moses sees God’s “back” but not face — aspects beyond creation’s access); Deut 29:29 (“the secret things belong to the LORD”)

  • Prophets & Writings: Job 11:7–9 (“can you fathom the mysteries of God?”); Isa 40:28 (“his understanding no one can fathom”)

  • Gospel (Jesus): “No one knows the Father except the Son” (Mt 11:27); “I have much more to say to you, more than you can now bear” (Jn 16:12)

  • Christian (wider): 1 Cor 2:9 (“no eye has seen, no ear has heard what God has prepared”); Rom 11:33 (“how unsearchable his judgments”)

  • Islamic: Quran 31:27 (“if all trees were pens and the ocean ink, God’s words would not be exhausted”) — God’s content exceeds all creation

  • Hindu: Gita 10:40–42 (“there is no end to my divine manifestations… I support this cosmos with a mere fragment of Myself”)

  • Secular: Observable reality may be a fraction of total reality; mathematical structures exist beyond physical instantiation

ax4 — Universal Immanence#

Every part of the world is in God.

In plain English: 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:

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

Explanation: This follows trivially from ax1 by transitivity of \(\leq\), but it is stated explicitly because it carries theological weight. It is the universally quantified version of ax1: not just the world as a whole, but every individual part.

Scriptural and philosophical support:

  • Torah: Deut 4:39 (“in heaven above and on earth beneath — there is no other”); Gen 28:16 (Jacob: “God is in this place and I did not know”)

  • Prophets & Writings: Ps 139:7–10 (“where can I flee?”); Jer 23:24 (“do I not fill heaven and earth?”)

  • Gospel (Jesus): “I am with you always, to the end of the age” (Mt 28:20)

  • Christian (wider): Eph 4:6 (“in all”)

  • Islamic: Quran 57:4 (“He is with you wherever you are”); 50:16 (“closer than the jugular vein”)

  • Hindu: Tat tvam asi — “Thou art That” (Chandogya Up. 6.8.7): the divine is in everything

  • Secular: Every part of nature is subject to the same fundamental laws

Group II — Modal Axioms#

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).

ax5 — Necessary Divine Existence#

God necessarily exists.

In plain English: 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:

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

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

Scriptural and philosophical support:

  • Torah: Exod 3:14 (“I AM WHO I AM” — Ehyeh asher Ehyeh)

  • Prophets & Writings: Ps 90:2 (“from everlasting to everlasting, you are God”)

  • Gospel (Jesus): “Before Abraham was, I AM” (Jn 8:58)

  • Christian (wider): Rev 1:8 (“who is and was and is to come”)

  • Islamic: Quran 28:88 (“everything will perish except His Face” — only God is permanent)

  • Hindu: Brahman is Sat (Being itself) — existence is Brahman’s essential nature

  • Secular: Something exists necessarily (laws of logic cannot not-hold)

ax6 — Contingency of the World#

The world’s existence is contingent — it might or might not exist.

In plain English: 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:

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

Explanation: 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.

Scriptural and philosophical support:

  • Torah: Gen 1:1 (“In the beginning God created” — creation is a contingent act; before it, no world)

  • Prophets & Writings: Ps 90:2 (“before the mountains were born” — God predates the world); Ps 102:25–26 (“they will perish, but you remain”)

  • Gospel (Jesus): “Heaven and earth will pass away, but my words will not” (Mk 13:31); “the glory I had with you before the world began” (Jn 17:5)

  • Christian (wider): Heb 1:10–12 (“they will perish… you roll them up like a garment”); Rev 21:1 (“first heaven and earth had passed away”)

  • Islamic: Quran 21:104 (“We will fold the heaven like the folding of a scroll”); 14:48 (“the earth will be replaced by another earth”)

  • Hindu: Cyclic creation/destruction in kalpas: Gita 8:17–19 (“when Brahma’s day dawns, all come forth; when his night falls, they dissolve”)

  • Secular: The universe began (Big Bang) and may end (heat death); physical constants could in principle have been different

ax7 — Necessary Containment#

If any world exists, it is necessarily in God.

In plain English: 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:

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

Explanation: This modalizes ax1. ax1 says the world is in God; ax7 says it must be in God — there is no possible world where creation exists outside of God. This is a strong panentheistic commitment.

Scriptural and philosophical support:

  • Torah: Gen 1:1 + Deut 4:39 together: God created all, there is no other — any world must be within God’s domain

  • Prophets & Writings: Isa 66:1 (“heaven is my throne, earth my footstool” — wherever creation is, it is within God)

  • Gospel (Jesus): “All things were made through him, and without him nothing was made” (Jn 1:3) — any possible creation goes through God

  • Christian (wider): Acts 17:28 (stated as universal principle); Col 1:16 (“all things created through him and for him”)

  • Islamic: Quran 39:67 (“the whole earth will be in His grip”) — even eschatologically, creation is in God’s hand

  • Hindu: Gita 9:4 (“By Me, in My unmanifested form, this entire universe is pervaded”)

  • Secular: Any subsystem is necessarily embedded in its containing system

Group III — Relational Axioms#

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

ax8 — Immanent Presence#

God is present to every part of the world.

In plain English: It is not enough that the world is inside God. A box contains its contents without being aware of them. ax8 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:

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

Scriptural and philosophical support:

  • Torah: Exod 3:2–5 (burning bush — God present in ordinary matter); Deut 31:6 (“He will never leave you nor forsake you”)

  • Prophets & Writings: Ps 139:7–10 (“where can I flee from your presence?”); Isa 57:15 (“I dwell in the high and holy place AND ALSO with the humble”); leit atar panuy (“no place is devoid of Him” — Hasidic maxim)

  • Gospel (Jesus): “Whatever you did for the least of these, you did for me” (Mt 25:40); “where two or three gather, there am I” (Mt 18:20)

  • Christian (wider): Col 1:17 (“in him all things hold together”); essence–energies doctrine (Gregory Palamas, 14th c.)

  • Islamic: Quran 50:16 (“closer than the jugular vein”); 57:4 (“with you wherever you are”)

  • Hindu: “The Lord dwells in the hearts of all beings” (Gita 18:61); “I am the Self in the hearts of all creatures” (Gita 10:20)

  • Secular: Natural law operates everywhere without exception

ax9 — Sustaining Dependence#

If the world exists, God sustains it.

In plain English: 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:

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

Scriptural and philosophical support:

  • Torah: Deut 8:3 (“man lives by every word from God’s mouth”); Gen 2:7 (God breathes life into being)

  • Prophets & Writings: Neh 9:6 (“you preserve them all”)

  • Gospel (Jesus): “Apart from me you can do nothing” (Jn 15:5)

  • Christian (wider): Col 1:17; Heb 1:3 (“sustains all things by his powerful word”)

  • Islamic: Quran 35:41 (“God holds the heavens and earth lest they cease”)

  • Hindu: “I support the entire cosmos with a mere fragment of Myself” (Gita 10:42)

  • Secular: Continued existence of structures depends on underlying processes (conservation laws)

ax10 — Asymmetric Dependence#

The world does not sustain God.

In plain English: The dependence runs strictly one way. God sustains the world (ax9), but the world does not sustain God. God’s existence does not depend on the world in any way.

Formal statement:

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

Explanation: Together with ax9, this encodes a strict ontological asymmetry. God can exist without the world (th2); the world cannot exist without God. This distinguishes PET from process-theology variants where God genuinely needs the world.

Scriptural and philosophical support:

  • Torah: Exod 19:5 (“the whole earth is mine”); Gen 1:1 — God existed before creation, needing nothing

  • Prophets & Writings: Ps 50:10–12 (“every animal is mine… if I were hungry I would not tell you”); Isa 40:17 (“nations are as nothing before him”)

  • Gospel (Jesus): “My kingdom is not of this world” (Jn 18:36); “Do you think I cannot call on my Father for twelve legions of angels?” (Mt 26:53)

  • Christian (wider): Acts 17:25 (“not served by human hands, as if he needed anything, since he himself gives everyone life”)

  • Islamic: Quran 29:6 + 35:15 (al-Ghani = “the Self-Sufficient” — “you are in need of God; God is free of need”)

  • Hindu: Gita 9:4–5 (“All beings exist in Me, but I do not dwell in them; My Self sustains all beings”)

  • Secular: Fundamental laws don’t depend on what they govern

Group IV — Divine Nature#

ax11 — Divine Structure (Dipolarity)#

God has both an unchanging nature and a world-responsive experience.

In plain English: 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. God’s experience of a world with suffering differs from God’s experience of a world without it, even though the essential divine nature remains constant.

Formal statement (strengthened, 4 lines):

  1. \(G = G_n \oplus G_c\) — God is the union of both aspects

  2. \(\Box\;\exists\, G_n\) — the necessary divine nature exists in every possible world

  3. \(G_c = \bigoplus\{G_c(w_i) \mid w_i \leq W\}\) — the contingent aspect is composed of subworld-indexed experiences

  4. \(\forall w_1, w_2 \leq W : w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)\) — distinct subworlds yield distinct divine experiences

Lines 3–4 were added (2026-03-14/15) to give \(G_c\) internal structure, making theorem th4 (Divine Experience Varies) formally derivable.

Scriptural and philosophical support:

  • Torah: Necessary: Exod 3:14 (“I AM” — eternal, unchanging). Contingent: Exod 32:14 (“the LORD relented”); Gen 6:6 (“the LORD regretted… his heart was deeply troubled”)

  • Prophets & Writings: Necessary: Mal 3:6 (“I the LORD do not change”). Contingent: Jer 18:7–10 (God changes plans based on nations’ behavior); Hosea 11:8 (“my heart is changed within me; all my compassion is aroused”)

  • Gospel (Jesus): Necessary: “Before Abraham was, I AM” (Jn 8:58). Contingent: “Jesus wept” (Jn 11:35); Lk 15:7 (“more rejoicing in heaven over one sinner who repents”)

  • Christian (wider): Necessary: Heb 13:8 (“Jesus Christ the same yesterday, today, forever”). Contingent: Phil 2:7 (“emptied himself, taking the form of a servant”)

  • Islamic: Necessary: Quran 57:3 (“the First and the Last”). Contingent: 2:186 (“I respond to the supplicant when he calls” — al-Mujib = “the Responsive”). The 99 names include both transcendent (al-Quddus = “the Holy”) and relational (ar-Rahman = “the Merciful”)

  • Hindu: The most explicit tradition. Nirguna Brahman (without qualities — necessary aspect) and saguna Brahman / Ishvara (with qualities — contingent, relational aspect). This is literally the same concept as dipolarity.

  • Secular: Fundamental laws are invariant (necessary), but their manifestation varies with initial conditions (contingent)

Alternative — ax11b (Divine Simplicity):

G has no proper parts that are independent of each other.

ax11b is the classical theism alternative. It sits in tension with ax1 + ax3. See PET Axioms — Discussions and Caveats for analysis of the ax11/ax11b fork.

Group V — Revelation Bridge#

These axioms are optional extensions that bridge from philosophical theology to revealed theology. They define a rigorous methodology for incorporating scriptural claims into the formal system.

ax12 — Revelation Reliability#

God’s self-knowledge is true.

In plain English: R is defined as the set of true propositions about God — God’s own self-knowledge. Since R consists of true propositions by definition, ax12 is tautological by design. This is deliberate: the substantive work of testing claims shifts to ax14 (Revelation Claims Test).

Formal statement:

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

Scriptural and philosophical support:

  • Torah: Num 23:19 (“God is not a man that he should lie”); Deut 18:22 (test of true prophecy — God’s actual words are reliable)

  • Prophets & Writings: Ps 119:160 (“all your words are true”); Isa 55:11 (“my word will not return to me empty”)

  • Gospel (Jesus): “Your word is truth” (Jn 17:17); “Heaven and earth will pass away, but my words will not” (Mk 13:31); “I am the way, the truth, and the life” (Jn 14:6)

  • Christian (wider): 2 Tim 3:16 (“all Scripture is God-breathed”); Heb 6:18 (“it is impossible for God to lie”)

  • Islamic: Quran 4:122 (“who is more truthful than God in statement?”); 10:94 (the Quran confirms previous scriptures)

  • Hindu: The Vedas are apaurusheya (not of human origin) — self-evident truth; Gita 4:1–3 (Krishna reveals ancient eternal yoga)

  • Secular: Self-knowledge (accurate self-model) is definitionally true when the model matches reality — tautological by design

ax13 — Consistency of Revelation#

God’s self-knowledge contains no contradictions.

In plain English: 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 (not actually true about God).

Formal statement:

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

Scriptural and philosophical support:

  • Torah: Deut 32:4 (“He is the Rock, his works are perfect… faithful God who does no wrong”) — perfection implies no contradiction

  • Prophets & Writings: Ps 18:30 (“God’s way is perfect; the LORD’s word is flawless”)

  • Gospel (Jesus): “A house divided against itself cannot stand” (Mk 3:25) — applied to God’s kingdom, implies internal consistency; “Scripture cannot be broken” (Jn 10:35)

  • Christian (wider): 1 Cor 14:33 (“God is not a God of disorder/confusion”); Jas 1:17 (“no variation or shadow due to change”)

  • Islamic: Quran 4:82 (“If it had been from other than God, they would have found within it much contradiction”) — the key verse: internal consistency as proof of divine origin

  • Hindu: Brahman is sat-chit-ananda (being-consciousness-bliss) — perfect unity implies no contradiction; Vedas considered internally harmonious

  • Secular: Any consistent framework must be free of internal contradiction (Hilbert’s program); logical consistency is the minimum standard

ax14 — Revelation Claims Test#

Human claims about divine revelation must be mutually consistent and consistent with ax1–ax13.

In plain English: This is the most 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 ax1–ax13.

Formal statement (reformulated, 2 lines):

  1. \(\forall p, q : \text{claim}(p) \wedge \text{claim}(q) \rightarrow \neg(p \wedge q \rightarrow \text{contradiction})\)

  2. \(\forall p : \text{claim}(p) \rightarrow \neg\exists q \in \{A1\text{–}A13\} : (p \wedge q \rightarrow \text{contradiction})\)

Key design features:

  • No self-reference: ax14 references ax1–ax13, not ax1–ax14. This avoids the circularity of a test that tests itself.

  • The claim(p) predicate: Distinguishes between God’s actual self-knowledge (R, which is true by ax12) and human claims about what is in R. Humans can be wrong about what God has revealed; ax14 provides a method for detecting such errors.

  • Testable method: Given two traditions that make conflicting claims, ax14 asks: can both claims be true simultaneously without contradiction? If not, at least one claim is not genuinely divine. This pinpoints where traditions actually disagree vs. where they merely think they disagree.

Scriptural and philosophical support:

  • Torah: Deut 13:1–3 (even a sign-working prophet must be tested against known truth); Deut 18:21–22 (“if what a prophet proclaims does not take place, that is a message the LORD has not spoken”)

  • Prophets & Writings: 1 Kings 22:19–23 (Micaiah vs. 400 false prophets — conflicting claims must be tested); Isa 8:20 (“to the law and the testimony! If they do not speak according to this word, they have no light”)

  • Gospel (Jesus): “By their fruits you shall know them” (Mt 7:16–20); “Beware of false prophets” (Mt 7:15); “Many will come in my name… do not believe them” (Mk 13:5–6)

  • Christian (wider): 1 Thess 5:21 (“test everything; hold fast what is good”); 1 Jn 4:1 (“test the spirits”); Gal 1:8 (“even if an angel preaches a different gospel, let them be accursed”)

  • Islamic: Quran 4:82 (test for contradiction as proof of origin); 2:111 (“produce your proof if you are truthful”); 17:81 (“truth has come and falsehood has departed”)

  • Hindu: Nyaya school: pramana (valid means of knowledge) must be tested; Gita 4:34 (“inquire… the wise who have seen truth will instruct you”)

  • Secular: Falsifiability (Popper); peer review; the scientific method IS a claims test — any claim must withstand scrutiny against established knowledge

For axioms ax15–ax25 (Group VI — Agency & Delegation), which extend PET into the Jubilee-based innovation economy, see JUB Axioms — ax15–ax25 (Group VI: Agency & Delegation).



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