All Symbol Dictionaries for Mathematical Theology#
This page provides the complete integrated listing of all formal symbols used in the matheology axiom system, drawn from their canonical sources.
PET symbols: PET Symbol Dictionary — used in ax1–ax14, th1–th4
JUB symbols: JUB Symbol Dictionary — introduced in ax15–ax25, th5–th11
PET Symbols#
Entities and Variables#
Symbol |
Name |
Meaning |
Technical context |
|---|---|---|---|
G |
God |
The distinguished divine entity |
Distinguished constant |
W |
The World |
Totality of all finite/created entities |
Distinguished constant |
\(G_n\) |
Necessary divine aspect |
The abstract, unchanging divine nature that exists in every possible world |
Component of dipolar decomposition (ax11) |
\(G_c\) |
Contingent divine aspect |
God’s concrete experience, which varies depending on which world exists |
Component of dipolar decomposition (ax11) |
\(G_c(w_i)\) |
Subworld divine experience |
God’s contingent experience specific to subworld \(w_i\) |
Functional structure added in strengthened ax11 (lines 3–4) |
\(R\) |
God’s self-knowledge |
The set of true propositions about God |
Defined so that ax12 is tautological by design; substantive work shifts to ax14 |
\(p, q\) |
Propositions |
Statements that can be true or false |
Propositional variables |
\(x, y\) |
Entities |
Parts of God or the world |
Individual variables |
\(w_i\) |
Subworld |
A part of the world W (i.e., \(w_i \leq W\)) |
Used in ax11 to index divine experience |
Relations and Predicates#
Symbol |
Name |
Meaning |
Technical context |
|---|---|---|---|
\(\leq\) |
“is part of” |
Mereological parthood: reflexive, transitive, antisymmetric |
Mereology (part-whole logic) |
\(<\) |
“is proper part of” |
\(x \leq y\) and \(y \nleq x\) (part of, but not identical to) |
Derived from \(\leq\) |
\(P(x, y)\) |
“x is present to y” |
A relation of immediate awareness or access |
Primitive relation (axiomatically introduced, not further reduced) |
\(S(x, y)\) |
“x sustains y” |
y’s continued existence depends on x |
Primitive relation |
\(\text{Pos}(\varphi)\) |
“φ is a positive property” |
A perfection in Gödel’s sense |
From Gödel’s ontological framework; listed but unused in ax1–ax14 |
\(\text{claim}(p)\) |
“p is claimed divine” |
A human claim that proposition p is divinely revealed |
Introduced in ax14 (Revelation Claims Test) |
Logical Operators#
Symbol |
Name |
Meaning |
Technical context |
|---|---|---|---|
\(\Box\) |
Necessarily |
True in every possible world |
Modal logic S5 |
\(\Diamond\) |
Possibly |
True in at least one possible world |
Modal logic S5 |
\(\forall\) |
For all |
Every entity satisfies the condition |
First-order logic (universal quantifier) |
\(\exists\) |
There exists |
At least one entity satisfies the condition |
First-order logic (existential quantifier) |
\(\exists!\) |
There exists exactly one |
Exactly one entity satisfies the condition |
First-order logic (uniqueness quantifier) |
\(\wedge\) |
And |
Both conditions hold simultaneously |
Propositional logic (conjunction) |
\(\vee\) |
Or |
At least one condition holds |
Propositional logic (disjunction) |
\(\neg\) |
Not |
The condition does not hold |
Propositional logic (negation) |
\(\rightarrow\) |
Implies / If…then |
If the first condition holds, then the second must hold |
Propositional logic (material conditional) |
\(\oplus\) |
Mereological sum |
The combination of parts into a whole |
Mereology |
\(\in\) |
Is a member of |
The element belongs to the set |
Set theory |
\(\neq\) |
Is not equal to |
The two entities are distinct |
Standard mathematics |
Note
Modal logic S5 is the system where “possibly necessary” implies “necessary.” This means the accessibility relation between possible worlds is an equivalence relation: every world can “see” every other world. S5 is the standard choice for reasoning about metaphysical necessity.
JUB Symbols#
Entities and Variables#
Symbol |
Name |
Meaning |
Technical context |
|---|---|---|---|
H |
Humanity |
The set of all human agents |
Distinguished set |
\(h, h^*\) |
Human agent |
An individual human; \(h^*\) is the one with maximal causal influence at time t (ax19) |
Individual variable |
\(D_f\) |
Forced domain |
Choices constrained by physics, coercion, or circumstance; not subject to moral evaluation in the innovation theodicy |
Domain partition |
\(D_{\text{free}}\) |
Free domain |
Choices where humans possess genuine capacity to select among alternatives; the domain of moral responsibility |
Domain partition |
\(D_{\text{inno}}\) |
Innovation subdomain |
The critical subset of \(D_{\text{free}}\) where novel solutions, creative acts, and innovation occur; \(D_{\text{inno}} \subseteq D_{\text{free}}\) |
Domain partition |
\(W_{\text{earth}}\) |
Earth |
The delegated domain of human governance (ax16) |
Subworld constant |
\(W_{\text{future}}\) |
Future trajectory |
The space of future outcomes over which causal influence is measured |
Used in ax19, th6 |
\(W_{\text{physics}}\) |
Physical law substrate |
The physical laws sustained by God (ax9) as precondition for agency |
Used in th10 |
\(O_{\text{genuine}}\) |
Genuine outputs |
The class of outputs (care, insight, innovation) that require free choice for full quality (ax23) |
Used in ax23 |
\(E\) |
Innovation economy |
An economic system operating under ax24 life-trifecta conditions |
Used in ax25 |
\(R\) |
Recalibration |
A periodic Jubilee reset mechanism that redistributes accumulated concentration (ax25) |
Used in ax25 |
\(i\) |
Innovation |
Any creative act, solution, or systemic change subject to the life-trifecta criterion |
Used in ax24, th8 |
Relations and Predicates#
Symbol |
Name |
Meaning |
Introduced in |
|---|---|---|---|
\(\text{can-choose}(h, a, s)\) |
Can choose |
Agent h can choose action a in situation s |
ax15 |
\(\text{Agency}(H)\) |
Genuine agency |
Humans possess real capacity to choose among alternatives |
ax15 |
\(\text{Delegated}(G, H, D)\) |
Delegation |
God has delegated authority over domain D to humanity |
ax16 |
\(\text{PrimaryResponsible}(H, o)\) |
Primary responsibility |
Humans are the primary responsible agents for outcome o |
ax16 |
\(\text{Guide}(G, h)\) |
Divine guidance |
God provides guidance to human h |
ax17 |
\(\text{Force}(G, h)\) |
Divine coercion |
God compels human h (negated in ax17) |
ax17 |
\(\text{Responsible}(x, D)\) |
Moral responsibility |
Entity x bears moral responsibility for outcomes in domain D |
ax18 |
\(\text{MaxCausalInfluence}(h, t, W)\) |
Maximum causal influence |
Agent h has strictly maximal causal influence at time t over future world W |
ax19 |
\(\text{CausalInfluence}(h, t, W)\) |
Causal influence |
The degree of causal influence agent h exerts at time t |
ax19 |
\(\text{Willing}(h)\) |
Willingness |
Agent h voluntarily accepts a responsibility |
ax20, ax21 |
\(\text{Accepts}(h, R)\) |
Acceptance |
Agent h accepts role or responsibility R |
ax20 |
\(\text{TranslatorRole}(G, H, W)\) |
Translator/Mediator role |
Permanent role translating between God’s optimal solutions and humanity’s understanding |
ax21 |
\(G_n\text{-valuation}(G_c(w))\) |
Divine valuation |
God’s necessary nature evaluates contingent experience states |
ax22 |
\(\text{Stable}(i)\) |
Stability |
Innovation i is structurally stable |
ax24 |
\(\text{Extensible}(i)\) |
Extensibility |
Innovation i can be extended without breakdown |
ax24 |
\(\text{LifeFriendly}(i)\) |
Life-friendliness |
Innovation i returns more than it extracts |
ax24 |
\(\text{Lasting}(i)\) |
Lasting innovation |
Innovation i satisfies all three life-trifecta cords |
ax24 |
\(\text{BABL-attractor}(i)\) |
BABL attractor |
Innovation i is on the self-destructive trajectory (Blindly Assuming Blind Leveraging) |
th8 |