# The Incompleteness of Reasoning

**Author:** Zixi Li
**Date:** November 2025

---

## Abstract

We present a fundamental critique of the notion of "pure reasoning" divorced from semantic priors. Our central thesis is that any attempt to strip away semantics and retain only formal structure inevitably leads to a self-referential loop.

We establish this through **four complementary approaches**:

1. **Kantian antinomy** showing that semantic stripping is self-refuting—the operator S both depends on and negates interpretation I, creating a structural Ouroboros
2. **Turing-inspired construction** proving that computational completeness does not imply reasoning completeness
3. **Limit analysis (the Yonglin Formula)** demonstrating that all reasoning returns to its prior anchor, but the prior cannot equal its own meta-reflection (A ≠ A*)—object-level closure, meta-level rupture
4. **Self-dismantling protocol** showing that the paper can be falsified using only its own formulas, thereby proving its core claim: reasoning cannot complete itself within a single world

**Conclusion:** Either one admits *a priori* semantic anchors, or one abandons the concept of "pure reasoning" altogether. There is no third option.

---

## Key Contributions

### The Yonglin Formula

$$\lim_{n \to \infty} \Pi^{(n)}(s) = A$$

All reasoning, no matter how many steps, returns to its prior anchor in the limit. However, the prior cannot be identical to its own meta-reflection:

$$A \neq A^*$$

This yields **object-level closure with meta-level rupture**—the loop closes at one stratum but breaks at the next.

### Semantic Delamination Antinomy

The operation of semantic stripping satisfies both:
- $S \vdash I$ (depends on interpretation)
- $S \dashv I$ (negates interpretation)

This structural Ouroboros makes pure semantic stripping impossible.

### Self-Dismantling Protocol

The paper provides four simple substitutions that allow any reader to falsify its conclusions using only its own formulas. Crucially, each falsification strategy *proves* the paper's core claim:

$$\text{The paper being falsified} \implies \text{The paper is proven.}$$

---

## Epilogue: Four Priors of Reasoning

The incompleteness of reasoning embodies four conditions without which reasoning cannot exist:

1. **Knowledge** — The ontological ground (Kant)
2. **Reasoning** — The epistemological boundary (Turing)
3. **Love** — The reflexive given, that which needs no proof (Yonglin)
4. **Freedom** — The critical space for rational critique (Reader)

> *"This paper proves its own incompleteness. In doing so, it proves that incompleteness is the condition of proof."*

---

## Files

| File | Description |
|------|-------------|
| `arxiv_preprint.tex` | Full LaTeX source code |

---

## How to Compile

```bash
pdflatex arxiv_preprint.tex
pdflatex arxiv_preprint.tex  # Run twice for cross-references
```

---

## Citation

```bibtex
@misc{oz_lee_2025,
	author       = { Oz Lee },
	title        = { The_Incompleteness_of_Reasoning (Revision bccc46d) },
	year         = 2025,
	url          = { https://huggingface.co/datasets/OzTianlu/The_Incompleteness_of_Reasoning },
	doi          = { 10.57967/hf/7060 },
	publisher    = { Hugging Face }
}
```

---

## License

This work is released for academic and research purposes.

---

## Contact

For questions or discussions, please open an issue in this repository.