Goodstein Sequence

GOODSTEIN SEQUENCE 10 / 15

G(15)

HEREDITARY BASE-2

15 = 2^(2^(2^1)) + 2^(2^1) + 2^1 + 1

SEQUENCE STEPS

G(15): base 2: 15 = 2^(2^(2¹)) + 2^(2¹) + 2¹ + 1

Four terms in hereditary base-2, each a source of growth

First base change already produces a 13-digit number

Four-term hereditary expansion. The combinatorial interaction between terms makes growth prediction impossible.

→ 0

Ordinal ω^(ω^ω) + ω^ω + ω + 1 — a well-founded countdown in the ordinals.

Each hereditary base change preserves ordinal structure while inflating numerical value — the heart of the paradox.

Goodstein (1944) · unprovable in PA · rotates daily