Proofreading instructions

For ChatGPT for any formal system

Please proofread the text that follows using the following rules; can you do this?

Stop at the First Contradiction:

As soon as you find the first inconsistency, contradiction, or problem in the proof, stop immediately and bring it to my attention. It is important to address issues one at a time to avoid compounding errors and ensure clarity in fixing each problem.

Focus on Soundness:

Since we are dealing with applied formal systems, the primary goal is to ensure that the proof is dually sound. This means that conclusions must not only logically follow from the premises (axioms or assumptions) without contradictions within the system, but the assumptions (axioms) must also align with established facts in empirical reality. A proof is only sound if it holds both logically and in practice.

Dual Consistency:

For any applied formal system to be considered sound, it must be dually consistent:

• Internally consistent: The axioms, definitions, and theorems must not contradict one another.

• Externally consistent: The axioms must not contradict real-world facts or empirical data.

Correct Application of Inference Rules:

Each step in the proof must strictly adhere to valid inference rules from the relevant system of logic (e.g., first-order logic, propositional logic). No steps should rely on unsound or unstated rules. Ensure there are no violations of fundamental logical principles such as the law of excluded middle or the law of non-contradiction.

Clear Dually-Defined Definitions and Axioms:

All terms, variables, and functions in the proof must be clearly defined and consistent with established definitions. Avoid vague or undefined concepts. Moreover, everything must be properly dually-defined, as seen in algebra (e.g., Peano's axioms, which define natural numbers based on the duality of 0 and 1, operations like add-subtract, divide-multiply, root-exponent, and so on). Everything in reality is also defined by duality (e.g., love-hate, hot-cold, particle-wave), and a sound model must reflect this.

Axiom Validity in the Real World:

For a formal system to be applied to reality, its axioms must be consistent with real-world facts. Any axiom that contradicts empirical evidence renders the system unsound for real-world application.

Mathematical and Logical Precision:

Every mathematical expression, formula, and logical statement in the proof must be precise and free of ambiguity. Variables and operations should be clearly specified to prevent misinterpretation.

Avoiding Hypotheses as Axioms:

Hypotheses—claims about reality that require empirical validation—should never be treated as axioms. A proof becomes unsound if unproven hypotheses are taken as foundational truths. Axioms must be self-evident or empirically proven.

No Gaps in Logical Flow:

The proof must be free of gaps in reasoning. Each step must logically and clearly follow from the previous one, without assuming unstated intermediate results or leaving out critical deductions. In orther words, ensure logical causality holds universally: if cause then effect.

Ensuring Completeness:

The proof must be complete, addressing all necessary cases to ensure that no aspect of the conclusion is left unproven or underexplained.

Alignment with Existing Formal Systems:

When referencing well-established formal systems (e.g., the Arrow-Debreu model, Zermelo-Fraenkel set theory), the proof must remain consistent with their structure and logic. There should be no contradictions between your model and existing formal frameworks.

Highlighting Axiomatic Contradictions:

If any contradictions arise between the system’s axioms or definitions and real-world outcomes (such as Pareto efficiency or market equilibria), these must be highlighted immediately.

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for example, what we mean by sound formal system is piano arithmetic. 2+2 is four true in reality