# Question 4817: Natural Closed-Interval Cardinality

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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.

R0008-04817 · q-20261008-advance-20261011-r01-natural-number-0067



## Original question and givens

N includes 0. lo=1, hi=16. Find |{n∈N: lo≤n≤hi}|, including endpoints.


- G001 — N includes 0. lo=1, hi=16. Find |{n∈N: lo≤n≤hi}|, including endpoints.





## 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-natural-number](https://ocme.evemisslab.com/en/mko/mko-natural-number/)





## Step-by-step derivation

S001 · impliesBefore

N={0,1,2,...}

After

0 belongs;negative and nonintegral rational values do not

Conditions: none

This convention includes0 in N;distinguish membership,rational values,partial predecessors and closed intervals.

Step identifiers and dependenciesRule: natural_zero_convention · Uses: G001





S002 · calculationBefore

inclusive interval endpoint difference := sub(16, 1)

After

inclusive interval endpoint difference = 15

Conditions: none

Subtract the lower endpoint from the upper endpoint.

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





S003 · calculationBefore

inclusive natural interval size := sum(15, 1)

After

inclusive natural interval size = 16

Conditions: none

The closed interval contains both endpoints,so add1.

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





S004 · impliesBefore

lo=1;hi=16

After

natural interval size=16

Conditions: none

If the upper endpoint is below the lower,the set is empty with size0;do not swap endpoints to manufacture another set.

Step identifiers and dependenciesRule: ordered_natural_interval_boundary · 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 — Recompute inclusive natural interval size with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute inclusive interval endpoint difference with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

16

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: cf23e78c340d6caa6c85a6ecae2ec7571984e6a4b7ccd794916fa5c6da92260a
- candidate record SHA-256: cc4324f3bf284ba1be702bb6bb685c40068a1023555e0e31840742405ee94d55
- semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
- question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r01-natural-number-0067.json)