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Formal Proof of Extended Modal Realism (EMR) by Necessity
Definitions
- Let U be the universe, the totality of all empirical facts.
- Let M(U) be a formal model of the universe: a set of propositions or axioms that, in principle, accounts for all truths about U.
- Let PSR be the Principle of Sufficient Reason: Every fact or truth has a sufficient explanation.
- Let EMR be Extended Modal Realism: All logically possible worlds exist — the actual world is just one among them.
Step 1: The Universe as a Formal System of Itself
P1.1 Any theory that purports to explain all truths about the universe must include a model M(U) that expresses those truths formally.
P1.2 If M(U) is sufficiently expressive (e.g., includes arithmetic), consistent, and recursively enumerable, then Gödel’s First Incompleteness Theorem applies.
C1 Therefore, M(U) is incomplete: there exist true statements about U that are unprovable within M(U).
Step 2: Rejection of Strong Empiricism
P2.1 Strong empiricism asserts that all truths are derivable from empirical observation (i.e., from within M(U)).
P2.2 But from C1, some truths about U are not derivable from within any empirical model.
C2 Therefore, strong empiricism is false.
Step 3: Refutation of Brutalism
P3.1 A brutalist response to C1 is to treat Gödelian truths as brute — true but without explanation.
P3.2 Suppose we add such a brute proposition to M(U) to form M′(U).
P3.3 By Gödel, M′(U) is also incomplete — a new unprovable truth G(M′) emerges.
P3.4 This process iterates infinitely. There is no final complete model reachable by brute supplementation.
C3 Brutalism fails to halt explanatory incompleteness and degenerates into an infinite series of ungrounded assumptions.
Step 4: PSR is the Only Stable Principle of Explanation
P4.1 PSR demands that all truths, including foundational ones, have sufficient reason.
P4.2 Empiricism fails to explain all truths (C2); brutalism fails to explain any truths coherently (C3).
C4 Therefore, PSR is the only principle that preserves rational explanation in metaphysics.
Step 5: Axiom-Based Theories Violate PSR
P5.1 All finite metaphysical systems (e.g., theism, materialism, idealism, structural realism) begin with a finite set of axioms.
P5.2 Axioms are, by definition, unexplained assumptions within a system.
P5.3 By P4.1, PSR rejects any unexplained assumptions.
C5 Therefore, all axiom-based theories of reality violate PSR and are necessarily false if PSR is true.
Step 6: Only EMR Contains No Axioms
P6.1 Extended Modal Realism (EMR) asserts that all logically possible worlds exist.
P6.2 EMR assumes no privileged world, no arbitrary structure, no selected laws — every possibility is realized.
P6.3 Therefore, EMR posits no unexplained exclusions or brute facts — it has no axioms.
C6 EMR satisfies the PSR fully.
Step 7: EMR is the Only Metaphysical Theory Consistent with PSR
P7.1 From C4, PSR is the only stable principle of explanation.
P7.2 From C5, all axiom-based theories violate PSR.
P7.3 From C6, EMR is the only theory that satisfies PSR.
C7 Therefore, EMR is the only metaphysical theory that satisfies the PSR.
Final Conclusion
If the Principle of Sufficient Reason is true, then Extended Modal Realism is necessarily true.
All other theories of reality are either incomplete, arbitrary, or incoherent.
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