PET Theorems#
Theorems derived from the PET axioms (PET Axioms ax1–ax14). These demonstrate that the axiom system is not inert — it produces substantive, testable consequences. For symbol definitions, see PET Symbol Dictionary.
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:
Proof: Suppose for contradiction that \(\Diamond\;(\exists W \wedge \neg\exists G)\). 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 \(\neg\,\Diamond\;(\exists W \wedge \neg\exists G)\). \(\blacksquare\)
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.
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:
\(\Diamond\;(\exists G \;\wedge\; \neg\exists W)\) (God without world is possible)
\(\neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)\) (world without God is impossible)
Proof:
First part: By ax6 (Contingency of the World), \(\Diamond\;\neg\exists W\) — there are possible worlds without a world. By ax5, G exists in every possible world, including those. So \(\Diamond\;(\exists G \wedge \neg\exists W)\).
Second part: Direct from th1.
\(\blacksquare\)
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.
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:
Proof: Let x be any part of the world, i.e., \(x \leq W\).
By ax4 (Universal Immanence), \(x \leq W \rightarrow x \leq G\), so \(x \leq G\).
By ax8 (Immanent Presence), \(x \leq W \rightarrow P(G, x)\), so \(P(G, x)\).
Therefore \(P(G, x) \wedge x \leq G\). \(\blacksquare\)
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.
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:
Proof: By ax11 line 4, for all \(w_1, w_2 \leq W\): \(w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)\). This is a direct consequence of the strengthened ax11, which gives \(G_c\) a functional structure indexed by subworlds that is injective (distinct inputs yield distinct outputs). \(\blacksquare\)
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 \(G_c\) 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.
For theorems th5–th11 (derived from the JUB extension axioms ax15–ax25), see JUB Theorems — th5–th11.