‍Core Concepts

This article explores several pairs of opposing yet unifying concepts:

• Computability vs. Formalizability/Provability
• Consistency vs. Completeness
• Deduction vs. Induction/Intuition
• Turing Machine vs. Oracle Machine

Formal Systems and Computability

A formal system consists of axioms, inference rules, and a symbolic language, with reasoning that is mechanical and deterministic.

The Turing Machine is the ultimate abstraction of a formal system, a mathematical model of computability.

The Church–Turing thesis asserts that any intuitively computable function can be computed by a Turing Machine.

Conclusion: Deductive reasoning in formal systems and mechanical computation in Turing Machines are essentially isomorphic.

The Paradox of Consistency and Completeness

• Consistency: A system cannot simultaneously prove both P and ¬P.
• Completeness: Every proposition within the system can be proven either true or false.
  

Gödel’s incompleteness theorem shows that any consistent and sufficiently powerful system must necessarily be incomplete.

Conclusion: Consistency and completeness cannot be simultaneously achieved.

Deduction and Induction: Two Modes of Reasoning

• Deduction: Strictly follows logical rules (law of excluded middle, syllogism), ensuring consistency.
• Induction/Intuition: Relies on experience, pattern recognition, and constructive proofs, often requiring a leap beyond the system.
  

When formal systems are constrained by incompleteness, induction and intuition become sources of new axioms.

Turing’s Transcendence: Oracle and Ordinal Logic

Turing proposed the Oracle Machine, which can “instantly solve” problems unsolvable by a Turing Machine (e.g., the halting problem).

He constructed a logic system based on ordinals:

1. Start from system S₀;
2. Use the oracle to determine its unprovable true proposition G₀;
3. Add G₀ as a new axiom, obtaining system S₁;
4. Repeat iteratively, forming an ordinal-level chain of systems.
   

Logical Pathway: Induction/Intuition (Oracle) → provides new axioms → Deduction (Formal System) expands → generates new incompleteness → Oracle invoked again.

Conclusion

Formal systems and Turing Machines ensure computability and consistency, but are inevitably incomplete. Induction and oracles provide the power to transcend formal systems, allowing knowledge to continually approach completeness. The dynamic tension between the two forms the fundamental driving force of logic, computation, and the evolution of knowledge.