Gödel’s Incompleteness, Mathematical Witnesses & Peano Arithmetic

1. Gödel’s First Incompleteness Theorem

Any consistent, effectively axiomatized system that can express basic arithmetic contains true statements that cannot be proven within the system itself, and it cannot prove its own consistency from inside.

2. Mathematical Witnesses

A witness for an existential claim ∃x P(x) is a concrete value of x making P(x) true. Gödel’s result shows that some true arithmetic statements have no provable witness in the system that expresses them.

3. Peano Arithmetic Example: Goodstein’s Theorem

Goodstein’s Theorem asserts that every Goodstein sequence eventually terminates at zero. It’s purely about natural numbers, yet it cannot be proven in Peano Arithmetic (PA). A proof requires transfinite induction up to the ordinal ε0.

Z₀ = m in hereditary base–2 notation → replace base k with k+1, subtract 1 for each step → sequence always reaches 0

• Start with a number m in hereditary base–k form.
• To get the next term: replace every “k” with “k+1”, then subtract 1.
• Although elementary to state, its termination cannot be demonstrated inside PA.