12.1 Formal Arithmetic and Gödel Numbering

Formal arithmetic builds on Peano axioms, defining natural numbers and operations. It establishes key properties like consistency and completeness, using axioms and inference rules to derive theorems. This foundation is crucial for understanding mathematical logic.

Gödel numbering uniquely encodes mathematical statements, enabling self-reference in arithmetic. This concept plays a vital role in metamathematics, facilitating proofs of important theorems like Gödel's Incompleteness Theorems. It's a powerful tool for exploring formal systems.

Foundations of Formal Arithmetic

Properties of formal arithmetic

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• Formal arithmetic builds axiomatic system for natural numbers rooted in Peano axioms
• Key properties ensure consistency prevents contradictions completeness covers all true statements decidability determines truth value of statements
• Axioms of formal arithmetic define successor function $S(n) = n + 1$ establish rules for addition $a + 0 = a$ and $a + S(b) = S(a + b)$ set foundation for multiplication $a × 0 = 0$ and $a × S(b) = (a × b) + a$
• Inference rules like modus ponens $(A \rightarrow B, A \vdash B)$ and universal generalization $(\vdash \phi(x) \Rightarrow \vdash \forall x \phi(x))$ allow derivation of new theorems

Concept of Gödel numbering

• Gödel numbering creates unique encoding of mathematical statements enables arithmetic to self-reference
• Role in metamathematics facilitates self-reference in formal systems crucial for proving incompleteness theorems (Gödel's First and Second Incompleteness Theorems)
• Metamathematical concepts explored through Gödel numbering include provability within a system consistency absence of contradictions completeness ability to prove all true statements

Gödel Numbering Systems and Applications

Assignment of Gödel numbers

• Prime factorization method assigns unique primes to symbols (1 → 2, + → 3, = → 5) uses exponents for symbol positions $(1 + 1 = 2 \rightarrow 2^1 × 3^2 × 5^3 × 7^4)$
• Sequence coding utilizes $\beta$-function $\beta(a,b,i) = rem(a, 1 + (1+i)×b)$ and Cantor pairing function $\pi(a,b) = \frac{1}{2}(a+b)(a+b+1)+b$ for efficient encoding
• Encoding formulas starts with atomic formulas $(x = y \rightarrow g(x) = g(y))$ builds up to compound formulas $(\forall x (x > 0 \rightarrow \exists y (x = y^2)))$
• Encoding proofs represents sequences of formula Gödel numbers allows verification of proof validity

Construction of Gödel numbering systems

• Steps to create Gödel numbering system:
  1. Define symbol set (variables, constants, operators)
  2. Assign unique integers to symbols (a → 1, b → 2, + → 3)
  3. Choose encoding method (prime factorization, polynomial encoding)
• Efficient systems minimize number size (use smaller primes) ensure unique decodability (no ambiguity in decoding)
• Extending to complex expressions handles variables with placeholder numbers $(x \rightarrow 11, y \rightarrow 13)$ encodes quantifiers $(\forall \rightarrow 17, \exists \rightarrow 19)$
• Verifying system properties checks injective mapping (one-to-one correspondence) confirms effective computability (can be calculated algorithmically)