Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
How 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.

Under choice, c(X) is a well-defined cardinal

Statement

Assuming choice, c(X) is a well-defined cardinal.

Facts & Assumptions

Given: A topological space X and cellularity as in Under choice, Lindelöf degree L(X) and cellularity c(X) as raw cardinal functions.

[L1]

Under choice every family has a cardinality (The well-ordering theorem, A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used).

[L2]

Cardinals are initial ordinals, a set of ordinals has union as its least upper bound, and mutual injections give a bijection (Cardinal (initial ordinal) and cardinality, Basic closure properties of ordinals, The Schröder-Bernstein theorem).

Proof

technique · direct
1.1

Each cellular family is a subfamily of the topology, so its cardinality is bounded by the cardinality of the topology.

given
2.1

Let K be the set of cardinalities of cellular families and put δ=⋃K. By [L2], δ is the least ordinal upper bound of K. It is a cardinal: if β<δ and β≈δ, choose κ∈K with β<κ≤δ; then β⪯κ⪯δ≈β, so [L2] yields β≈κ, contrary to κ being a cardinal. Hence δ=sup⁡K=c(X).

step 1.1L1L2∎

Depends on

Used by

Cited to discharge well-definedness by Under choice, Lindelöf degree L(X) and cellularity c(X) as raw cardinal functions.

Dependency tree · two levels

31 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