Strong (complete) induction

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• Order on the natural numbers
• The well-ordering principle
• Trichotomy of the order on $\mathbb{N}$

Used by

• Converse of Lagrange for finite abelian groups: every divisor occurs as a subgroup order Corollary
• The 2-adic absolute value gives an ultrametric on ℚ, in which every triangle is isosceles and every point of a ball is a centre Example
• A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls Lemma
• Every nonzero nonunit polynomial over a field factors into irreducible polynomials Lemma
• Every subset of an at most countable set is at most countable Lemma
• For positive integers a and b: a ∣ b if and only if vₚ(a) ≤ vₚ(b) for every prime p Lemma
• If every diagonal value of an incidence function is a unit, recursive interval formulas construct both a left and a right convolution inverse Lemma
• Unique parsing of finite syntax Lemma
• A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials Theorem
• A degree-n polynomial has a splitting field spanned over F by at most n! explicit root monomials Theorem
• A maximal-order cyclic subgroup splits off a finite abelian p-group Theorem
• An arithmetic function has a Dirichlet inverse exactly when its value at 1 is nonzero Theorem
• Bₙ is exactly the set of words of length 2n over {texttt(,texttt)} in which every prefix has at least as many texttt( as texttt) and the totals are equal Theorem
• Cauchy's theorem for finite abelian groups Theorem
• Dirichlet convolution preserves multiplicativity, and multiplicative inverses stay multiplicative Theorem
• Division by a monic polynomial over a commutative ring Theorem
• Every finite abelian p-group is a direct product of cyclic p-groups Theorem
• Every integer n ≥ 1 is a finite product of primes: there are r ∈ ℕ and a list p : r → ℤ of primes with n = ∏_i<r pᵢ, the case n = 1 being the empty product Theorem
• Every nontrivial finite abelian group is an internal direct product of indecomposable subgroups Theorem
• Every nonzero polynomial over a field has a splitting field Theorem
• Galois orbits classify simple modules after splitting base change Theorem
• ℕ × ℕ ≈ ℕ Theorem
• On a finite chain, the Möbius function is 1 on the diagonal, -1 on covers and 0 on longer intervals Theorem
• Structural induction and recursion on syntax Theorem
• Sylow I: every finite group has a Sylow p-subgroup Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Mathematical induction (Wikipedia) (standard reference, not scraped)
• Well-ordering principle (Wikipedia) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)