Construct diagonal elements to prove non-enumerability in Lean 4.
npx skills add https://github.com/bneb/perqed --skill cantors-diagonalization
Please help me install this Agent Skill. Skill: cantors_diagonalization Source: https://github.com/bneb/perqed/tree/main/.agents/skills/cantors_diagonalization Command: npx skills add https://github.com/bneb/perqed --skill cantors-diagonalization
Cantor's diagonalization proves that a set has strictly greater cardinality than its power set by constructing a diagonal element that differs from every element of any proposed enumeration, thereby establishing non-enumerability. The technique also generalizes to computability arguments (e.g., no Turing machine can decide the Halting problem) and other diagonal arguments in mathematics and logic. In Lean 4 / mathlib4, Cantor.not_surjective provides the abstract result; the method typically explicitly defines a diagonal function such as d : ℕ → Bool that inverts f(n)(n) at each step.
Provide a surjection f : ℕ → S and request the diagonal element that differs from every f(n) at position n.
High-intent search queries and answers about installing and using this skill.
Cantor's diagonalization proves a set has strictly greater cardinality than its power set by constructing a diagonal element differing from every element in any proposed enumeration, establishing non-enumerability. It explicitly defines d(n) = not f(n)(n) to exhibit a counterexample.
Cantor's diagonalization applies to computability arguments such as Halting problem undecidability by constructing a diagonal element that differs from every f(n) in the n-th position. It generalizes from set-theoretic uncountability to refute Turing machine decidability.
To start a Cantor-style proof, provide a surjection f : ℕ → S and request the diagonal element that differs from every f(n) at position n. The method uses d(n) = not f(n)(n) to construct this counterexample directly.
The best way to construct a diagonal counterexample is to define d(n) = not f(n)(n), ensuring the diagonal element differs from every f(n) at index n. This directly refutes enumerability by exhibiting an element outside the proposed surjection's range.
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