Alphabeta MathHow statement and proof provenance work
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.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The transfinite subway argument
Example
In ZFC a train stops at every . At most countably many passengers board at each stop; a passenger boards once and stays until disembarking. At every stop with a nonempty arrival, at least one passenger disembarks before boarding occurs. Then the empty-arrival stops contain a club, and no passenger remains aboard through all stops after boarding below .
Facts & Assumptions
Fodor’s pressing-down lemma: A regressive map on a stationary domain has a stationary constant fibre.
Basic stationary-set calculus: Stationary sets on a regular uncountable cardinal are unbounded and have full cardinality.
Verification
Given: The objects and hypotheses in the statement.
Let S be the nonempty-arrival stops. Zero is not in S. If S were stationary, choose one departing passenger at each of its stops, and send that stop to the chosen passenger's boarding index. This is strictly below the arrival stop because departures precede new boarding. Pressing down gives a stationary set of such stops with one common boarding index beta.
The selected passengers at distinct stops are distinct: each leaves only once and cannot reboard. The stationary fibre has cardinality aleph-one, although at most countably many passengers boarded at beta, a contradiction. Hence S is nonstationary, and its complement contains a club by definition of nonstationarity.
A passenger remaining after boarding at beta would make every arrival after beta nonempty. That would exclude empty arrivals on an entire tail, contradicting the unbounded club of empty arrivals.
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
- Vasey, Example 15.2, pp.82–83 (standard reference, not scraped)