# At-most-countable set

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite.

mko-at-most-countable-set · definition · v0.10.0



## Mathematical statement

This slice uses complete finite-set certificates or universal injectivity certificates from integer affine encodings. A failed encoding refutes that particular witness, not countability of the set.





## Representative formula (exact TeX)

A(X)





## Assumptions


- Finite sets are listed completely; the symbolic family has explicit integer parameters and a universal injection formula.
- Finite, countably infinite and at most countable are kept distinct.





## Symbol scope


- A — at-most-countable certificate
- X — set





## Intuition, definition, and misconceptions

Every finite set is at most countable.

A nonzero slope gives a universal integer affine-injection certificate.

Failure of one witness does not prove uncountability.

At most countable does not mean countably infinite, and a collision in one proposed encoding does not establish uncountability.





## Adopted derivation guide

[A universal code for three natural-number tracks](https://ocme.evemisslab.com/en/derivations/derivation-mko-at-most-countable-set-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) — An injection into the natural numbers supplies a comparison.
- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — The codomain is N.
- [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/mko/mko-injective/) — The certificate must be universally injective.
- [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/mko/mko-surjective/) — Countable infinitude needs additional bijection or surjection obligations.





## Verification boundary

Only complete finite certificates or explicit universal injections are accepted; no general uncountability claim is asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add countable infinitude, diagonal arguments and countable unions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/at-most-countable-set.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-at-most-countable-set/index.md)