At-most-countable set

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

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 — An injection into the natural numbers supplies a comparison.
• Natural number — The codomain is N.
• Injectivity and distinct inputs — The certificate must be universally injective.
• Surjectivity and codomain coverage — 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.