Order on the natural numbers

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• Addition of natural numbers

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
• Every nondegenerate interval of ℝ is uncountable Corollary
• Every nonzero integer n is u ∏_i<r pᵢ with u ∈ {1,-1} and every pᵢ prime; u and r are determined by n, and the list is determined up to a permutation Corollary
• If V = ⨁_i<n Uᵢ with every Uᵢ finite-dimensional, then V is finite-dimensional and dim_F V = ∑_i<n dim_F Uᵢ; in particular dim_F(U ⊕ W) = dim_F U + dim_F W Corollary
• If V has a spanning set with n elements, then every linearly independent subset of V is finite with at most n elements; in particular V has no linearly independent subset equinumerous with ℕ Corollary
• lvertP(A)| = 2^| A| for finite A Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 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
• (ℕ, ≤) has no maximal element: Zorn's chain hypothesis fails Counterexample
• A complete domain is necessary for sequential uniform boundedness Counterexample
• A free ultrafilter induces a finitely additive zero-one probability that is not countably additive Counterexample
• A list of six distinct reals with no strictly increasing sublist of length four and no strictly decreasing sublist of length three Counterexample
• A progressive map with no fixed point, on a poset that is not chain-complete Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• Counting-measure tails decrease to the empty set while every term has infinite measure Counterexample
• If 1 were admitted as a prime, uniqueness would fail: 6 = 2 · 3 = 1 · 2 · 3 = 1 · 1 · 2 · 3, lists of different lengths that no permutation matches Counterexample
• In the multiplicative monoid H = {1, 4, 7, 10, …} of positive integers one more than a multiple of 3, the element 100 has two genuinely different factorisations into irreducibles, 4 · 25 and 10 · 10 Counterexample
• 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
• Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence Counterexample
• The standard unit families { eᵢ : i ∈ ℕ } are linearly independent in F^ℕ but do not span it: the constant family 1_F is not a finite linear combination of them Counterexample
• ⌈ m/n ⌉ for naturals m and n ≥ 1: the least q ∈ ℕ with m ≤ n q Definition
• A finite list of reals, and its strictly increasing and strictly decreasing sublists Definition
• Common multiple, and the least common multiple lcm(a,b), taken to be 0 when a = 0 or b = 0 Definition
• Compositions and weak compositions of a natural number into a fixed number of parts Definition
• Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ Definition
• Finite, countably infinite, countable, uncountable Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• Partition of [a,b] as a finite strictly increasing list a = t₀ < t₁ < … < tₙ = b, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions Definition
• Rooted trees of finite sequences, levels, branches, and finite branching, with ordered finite successor sets Definition
• Sequences of reals: bounded, eventually, frequently, tails, subsequences Definition
• Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field Definition
• Shattering and the Vapnik–Chervonenkis dimension of a set family Definition
• The cardinality | A| of a finite set Definition
• The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise 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
• The multinomial coefficient C(n, k₀,…,kₘ₋₁) as the number of ordered partitions of an n-set into blocks of prescribed sizes Definition
• The order |G| of a finite group and the order ord(g) of an element, with ord(g) = ∞ when no positive power of g is the identity Definition
• The p-adic valuation vₚ(a) of a nonzero integer: the greatest k ∈ ℕ with pᵏ ∣ a Definition
• The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ| Definition

…and 122 more results.

Dependency tree · two levels

9 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

• T. Tao, Analysis I, 3rd ed., §2.2 (Ordering of the naturals) (standard reference, not scraped)
• Set-theoretic definition of natural numbers (Wikipedia) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)