# Formal Proof of Extended Modal Realism (EMR) by Necessity

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## 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.*

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## 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)**.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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**.

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## 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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