Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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An unbounded regressive domain without a stationary fibre

Statement refuted

The assertion that every regressive map on an unbounded subset of a regular uncountable κ has a stationary constant fibre is false. Let S={α+1:α<κ} and f(α+1)=α.

Facts & Assumptions

[F1]

Regressive functions on ordinals: Regressive means f(ξ)<ξ at every nonzero domain point.

Counterexample

Given: The objects and hypotheses in the statement.

1.1

S consists of successors, is unbounded in kappa, and omits zero. The uniquely defined predecessor map satisfies f(α+1)=α<α+1, so it is regressive.

F1
2.1

Each fibre is a singleton, hence misses a club tail and is nonstationary. Also S itself misses the club of nonzero limits: these are unbounded because β+ω<κ, and closed because limits of limit ordinals are limits. Thus stationarity, the missing hypothesis of pressing down, cannot be replaced by unboundedness.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources