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.
False: L-space and S-space existence are dual ZFC theorems
Statement
ZFC settles L-space and S-space existence in parallel by dual arguments: ZFC proves that an L-space exists and also proves that an S-space exists, with one proof obtained from the other by interchanging separability and Lindelöfness.
Facts & Assumptions
Given: The theories in the cited result are interpreted as separate branches. For the metamathematical nonprovability clause, assume .
L-space and S-space existence is asymmetric proves that ZFC constructs an L-space, ZFC+CH constructs a strong S-space, ZFC+PFA proves that no S-space exists, and under the displayed source-consistency assumption S-space existence is not a theorem of ZFC.
Refutation
The first half of the proposed parallel is correct: [F1] constructs an L-space in ZFC. But [F1] obtains an S-space only in the separate CH branch and obtains the incompatible conclusion that no S-space exists in the PFA branch.
Under the source-consistency assumption in Given, [F1] proves that ZFC does not prove the existence of an S-space. Hence the assertion that both existence statements are ZFC theorems is false relative to that same standard large-cardinal consistency assumption.
Nor do the actual arguments arise by a formal interchange of two words: the L-space branch uses Moore's minimal-walk colouring in ZFC, the positive S-space branch uses CH, and the negative S-space branch uses PFA. These hypotheses and conclusions cannot be conjoined, and empty or singleton spaces supply neither kind of witness. Thus both the claimed parallel ZFC status and the claimed dual proof are refuted, with no converse consistency implication asserted.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Moore, A solution to the L space problem, Theorem 1.3 and Section 7 (standard reference, not scraped)
- Hart--Kunen, Ultra Strong S-Spaces, Corollary 4.18 (standard reference, not scraped)
- Abraham, Three applications of ideal dichotomy, slides 1--4 (standard reference, not scraped)