# Question 15880: A Complete Finite List Gives an At-Most-Countable Certificate

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0030-15880 · q-20261014-advance-20261031-r01-at-most-countable-set-0005



## Original question and givens

Complete nonempty duplicate-free finite integer list elements=[5, -6, 48] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].


- G001 — Complete nonempty duplicate-free finite integer list elements=[5, -6, 48] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].





## 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.


- [mko-at-most-countable-set](https://ocme.evemisslab.com/en/mko/mko-at-most-countable-set/)





## Step-by-step derivation

S001 · impliesBefore

DECLARED complete finite integer set in original list order=[5, -6, 48];length=3

After

nonempty duplicate-free registered family;original order retained;empty-set countability not denied

Conditions: none

Completeness is the declared set premise,not inferred for an unknown universe from samples.Keep order and reject duplicates rather than sorting or deduplicating.The family's empty-list rejection is an interface limit,not denial of empty-set countability.

Step identifiers and dependenciesRule: countable_complete_finite_list_premise_not_sampled_subset · Uses: G001





S002 · impliesBefore

original element=5;unique original position=0

After

injection pair element_to_position=[5, 0]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependenciesRule: finite_countability_injection_direction · Uses: G001, S001





S003 · impliesBefore

original element=-6;unique original position=1

After

injection pair element_to_position=[-6, 1]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependenciesRule: finite_countability_injection_direction · Uses: G001, S002





S004 · impliesBefore

original element=48;unique original position=2

After

injection pair element_to_position=[48, 2]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependenciesRule: finite_countability_injection_direction · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([5, -6, 48], [[5, 0], [-6, 1], [48, 2]], Yes, finite_implies_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: 62fbb65f594d04aea330a374d17afedcb20dd5eb620ee84a82167a033de802b2
- candidate record SHA-256: 994cc8bc067cc82d00a9f2a52f3d346893b8016a63855268d941a86f23249d4a
- semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
- question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0005.json)