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Question 15841: A Failed Encoding Does Not Prove Uncountability

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0030-15841 · q-20261014-advance-20261031-r01-at-most-countable-set-0091

Original question and givens

Declared evidence_mode=complete_finite_list, certificate_parameter=16. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

  • G001 — Declared evidence_mode=complete_finite_list, certificate_parameter=16. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

registered DECLARED evidence mode=complete_finite_list;numeric certificate parameter=16

After

qualifying declared certificate flag=Yes;classification=at_most_countable

Conditions: none

This audits the declared complete or universal certificate mode;the numeric parameter does not construct or verify an unspecified set.A failed encoding refutes that witness,not countability itself.At most countable does not mean countably infinite.

Step identifiers and dependencies

Rule: countability_declared_evidence_mode_not_unspecified_set_construction · Uses: G001

Reverse checks

  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(complete_finite_list, 16, Yes, at_most_countable)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: eb7f403085fbd6786a8adfaaf516c41a7d1f75cae23da3fda4fccc37d392c8e3
  • candidate record SHA-256: 634a4b6a4437ccf48a32b2df79eafd45c24ebf4e87698d8f34b765329ea871aa
  • semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
  • question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON