PARIS-HARRINGTON
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Relatively Large Sets
n=4 · k=3-subsets · 2 colors
SCENARIO
A set S is 'relatively large' if |S| ≥ min(S). So {5,7,9,12,14} is relatively large (5 elements ≥ 5) but {5,7,9} is not (3 elements < 5).
RAMSEY CONTEXT
Without the largeness condition, Ramsey's theorem for this case is easily provable in PA using induction.
PA ANALYSIS
This innocent-seeming extra condition — |S| ≥ min(S) — transforms a provable-in-PA statement into an unprovable one.
The largeness condition forces the homogeneous set to be so large that its existence can only be proved using transfinite induction beyond ε₀.
True but unprovable in PA · ε₀ boundary · rotates daily