Every nonzero natural number is a successor

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• The principle of mathematical induction

Used by

• A rational root of xᵏ = m is an integer: if k ≥ 1, m ∈ ℤ, x ∈ ℚ and xᵏ is the image of m, then x is the image of an integer Corollary
• ι(Dₙ) = ι(n) ι(Dₙ₋₁) + (-1)ⁿ for n ≥ 1, and Dₙ = (n-1)(Dₙ₋₁ + Dₙ₋₂) for n ≥ 2 Corollary
• Inside the space of eventually zero families, the linear subspace spanned by { eᵢ : i ≥ 1 } is proper and has a basis equinumerous with a basis of the whole space, so "equal dimension forces equality" fails without finite dimension Counterexample
• Isomorphism does not make an inclusion elementary Counterexample
• ⌈ m/n ⌉ for naturals m and n ≥ 1: the least q ∈ ℕ with m ≤ n q Definition
• G_δ and F_σ subsets of a topological space, agreeing with the real-line notion Definition
• The dyadic rationals of [0,1], their finite levels Dₙ, and their density in [0,1] Definition
• The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ Definition
• Zero sets and cozero sets of continuous real-valued functions Definition
• (ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1} Example
• nℤ is a subgroup of (ℤ, +) for every n ∈ ℤ, and every subgroup of (ℤ, +) has this form Example
• The forward shift on F^ℕ is injective but not surjective Example
• The Sorgenfrey line: ℝ with the half-open intervals [a,b) as a basis is strictly finer than the usual topology, is first countable, has a countable dense subset, and its sequences converge only from the right Example
• FALSE: an arbitrary intersection of open sets is open in every topological space False statement
• FALSE: every injection of a set into itself is a bijection False statement
• A finite sum is unchanged by a permutation of its index range: ∑_k<n a_π(k) = ∑_k<n aₖ for every bijection π : n → n Lemma
• A subset S ⊆ V is linearly dependent if and only if some s ∈ S lies in span(S ∖ {s}); and span(S) is already the set of linear combinations of INJECTIVE finite lists into S Lemma
• Discreteness: σ(n) is the immediate successor Lemma
• Every subset of an at most countable set is at most countable Lemma
• Finite sums re-indexed along an injection, with a zero term deleted, and concatenated; and the closure properties of linear independence: an independent list is injective and never 0_V, its sublists are independent, a list is independent exactly when it is injective with linearly independent image, and every subset of a linearly independent set is linearly independent Lemma
• If S ⊆ V is linearly independent and w ∉ span(S) then S ∪ {w} is linearly independent and span(S) ⊊ span(S ∪ {w}); and if w ∈ span(S) then span(S ∪ {w}) = span(S) Lemma
• In a finite group, every element g satisfies gⁿ = e for some natural n ≥ 1 Lemma
• The K-topology on ℝ, generated by the open intervals together with their complements of K = {1/(n+1) : n ∈ ℕ}, is T₁ and Hausdorff but not regular Lemma
• The natural numbers have no zero divisors Lemma
• The nonempty finite subsets of ℝ are exactly the listable ones Lemma
• The pigeonhole principle on ℕ Lemma
• Trichotomy of the order on ℕ Lemma
• ω is the least limit ordinal Lemma
• ≤ is a linear order on ℕ Theorem
• Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b Theorem
• Every list of mn+1 pairwise distinct reals has a strictly increasing sublist of length m+1 or a strictly decreasing sublist of length n+1 Theorem
• Every natural number and ω are cardinals, every infinite cardinal is a limit ordinal, and on the natural numbers the cardinal operations are the published finite counting operations, with | A | in the finite sense equal to | A | in the cardinal sense Theorem
• Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either Theorem
• If | A| > k| B| then every f : A → B has a fibre with more than k elements, and for nonempty B some fibre has at least ⌈ | A| / | B|⌉ elements Theorem
• If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V Theorem
• In a metric space every closed set is a zero set and a G_δ, and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal Theorem
• ℕ × ℕ ≈ ℕ Theorem
• The binomial theorem in ℝ: (x+y)ⁿ = ∑_k<n+1 iotaC(n, k) xᵏ y^ n-k Theorem
• The Euclidean algorithm: for a ∈ ℤ and b > 0 the remainder descent from (a,b) terminates, and the last nonzero remainder is gcd(a,b) Theorem
• The fundamental theorem of arithmetic: every integer n ≥ 1 is a product of primes, and the factorisation is unique up to order — if ∏_i<r pᵢ = ∏_j<s qⱼ with every pᵢ and qⱼ prime, then r = s and qᵢ = p_π(i) for some π ∈ Sym(r) Theorem

…and 3 more results.

Dependency tree · two levels

7 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)
• Set-theoretic definition of natural numbers (Wikipedia) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)
• B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)