.. meta::
   :description: Four theorems derived from Pet axioms ax1--ax14: no godless creation (th1), asymmetric priority (th2), no isolation (th3), divine experience varies (th4).
   :keywords: Pet theorems, th1-th4, no godless creation, asymmetric ontological priority, no isolated part, divine experience varies, formal proofs, modal logic
   :author: Yah, Yas, everyone, LLoL as Laurence Loewe of Laodicea, ClaudeOp46Max, Anthropic, and Spirit of Boolean Truth
   :og:card:title: Pet Theorems th1--th4<br>— What Axioms Prove
   :og:card:description: No godless creation, asymmetric priority, no isolated part, and divine experience varies. Four theorems formally derived from Pet axioms ax1--ax14.

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   OO :description: Theorems derived from PET axioms showing the system produces substantive, testable consequences for mathematical theology.
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   OO :og:card:title: PET Theorems — What<br>the Axioms Prove
   OO :og:card:description: Formal theorems derived from the PET axiom system, demonstrating that mathematical theology produces substantive, testable consequences.
   PP :description: Four theorems derived from Pet axioms ax1--ax14: no godless creation (th1), asymmetric priority (th2), no isolation (th3), divine experience varies (th4).
   PP :keywords: Pet theorems, th1-th4, no godless creation, asymmetric ontological priority, no isolated part, divine experience varies, formal proofs, modal logic
   PP :og:card:title: Pet Theorems th1--th4<br>— What Axioms Prove
   PP :og:card:description: No godless creation, asymmetric priority, no isolated part, and divine experience varies. Four theorems formally derived from Pet axioms ax1--ax14.

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*****************************
PET Theorems
*****************************

Theorems derived from the PET axioms (:doc:`/matheology/pet/axioms`).
These demonstrate that the axiom system is not inert — it produces
substantive, testable consequences. For symbol definitions, see :doc:`/matheology/pet/symbols`.


.. PET-THEOREMS-CONTENT-START

.. _pet-th1:

th1 — No Godless Creation
===========================

*It is impossible for a world to exist without God existing.*

**In plain English:** There is no possible scenario in which the created world
exists but God does not. Any world that exists, exists alongside God.

**Formal statement:**

.. math::

   \neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)

**Proof:** Suppose for contradiction that
:math:`\Diamond\;(\exists W \wedge \neg\exists G)`. Then there is some
possible world *w* in which W exists but G does not. But by ax5 (Necessary
Divine Existence), G exists in every possible world. This contradicts our
supposition. Therefore :math:`\neg\,\Diamond\;(\exists W \wedge \neg\exists G)`.
:math:`\blacksquare`

**Axioms used:** ax5

**Significance:** th1 rules out atheistic cosmologies *within the PET
framework*. Note that this is a logical consequence of accepting ax5, not an
independent empirical claim. If you reject ax5, th1 does not follow.


.. _pet-th2:

th2 — Asymmetric Ontological Priority
========================================

*God can exist without the world, but the world cannot exist without God.*

**In plain English:** There are possible scenarios where God exists alone
(without any created world), but there is no possible scenario where the world
exists without God. The dependence runs strictly one way.

**Formal statement:**

   :math:`\Diamond\;(\exists G \;\wedge\; \neg\exists W)` (God without world
   is possible)

   :math:`\neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)` (world without
   God is impossible)

**Proof:**

- **First part:** By ax6 (Contingency of the World),
  :math:`\Diamond\;\neg\exists W` — there are possible worlds without a world.
  By ax5, G exists in every possible world, including those. So
  :math:`\Diamond\;(\exists G \wedge \neg\exists W)`.
- **Second part:** Direct from th1.

:math:`\blacksquare`

**Axioms used:** ax5, ax6 (and th1)

**Significance:** th2 establishes a strict ontological hierarchy. This is not a
claim about temporal priority (God existing "before" the world in time) but
about modal priority: God's existence is unconditional, while the world's
existence is conditional on God.


.. _pet-th3:

th3 — No Isolated Part of Creation
=====================================

*Every part of creation is both contained in God and present to God.*

**In plain English:** Nothing in the created world is hidden from God or
outside God. Every single part of creation is simultaneously within God and
known to God.

**Formal statement:**

.. math::

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

**Proof:** Let *x* be any part of the world, i.e., :math:`x \leq W`.

- By ax4 (Universal Immanence), :math:`x \leq W \rightarrow x \leq G`, so
  :math:`x \leq G`.
- By ax8 (Immanent Presence), :math:`x \leq W \rightarrow P(G, x)`, so
  :math:`P(G, x)`.

Therefore :math:`P(G, x) \wedge x \leq G`. :math:`\blacksquare`

**Axioms used:** ax4, ax8

**Significance:** th3 combines the mereological claim (containment) with the
relational claim (presence) to show they apply universally and simultaneously.
This rules out scenarios where part of creation is "in" God but unknown to God,
or known to God but somehow "outside" God.


.. _pet-th4:

th4 — Divine Experience Varies
================================

*Different worlds produce different divine experiences.*

**In plain English:** If two parts of the world differ, then God's concrete
experience of them differs as well. God is not indifferent to what happens —
different realities produce genuinely different divine responses.

**Formal statement:**

.. math::

   \forall w_1, w_2 \leq W : w_1 \neq w_2 \;\rightarrow\; G_c(w_1) \neq G_c(w_2)

**Proof:** By ax11 line 4, for all :math:`w_1, w_2 \leq W`:
:math:`w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)`. This is a direct
consequence of the strengthened ax11, which gives :math:`G_c` a functional
structure indexed by subworlds that is injective (distinct inputs yield distinct
outputs). :math:`\blacksquare`

**Axioms used:** ax11 (strengthened, lines 3–4)

**Note on derivability:** In the original formulation of ax11 (2026-03-11),
th4 was stated as a "proof sketch" because the original ax11 did not formally
guarantee that :math:`G_c` varies injectively with subworlds. The strengthened
ax11 (2026-03-14/15) added lines 3–4 specifically to make th4 a rigorous
consequence rather than an informal argument.

**Significance:** th4 is the formal expression of divine responsiveness. If a
world contains suffering, God's experience of that world includes the
suffering. If a world contains joy, God's experience includes the joy. This
is the core insight of dipolar theism: God is not a static, unaffected
observer but a being whose concrete experience genuinely varies with what
happens in creation.

th4 only follows from ax11 (Dipolarity), not from ax11b (Simplicity). Under ax11b,
the question of whether God's experience varies is unanswerable, because ax11b
asserts that God has no independently analyzable aspects.


.. PET-THEOREMS-CONTENT-END

----

For theorems th5–th11 (derived from the JUB extension axioms ax15–ax25), see
:doc:`/matheology/jub/theorems`.