$\le$ is a linear order on $\mathbb{N}$

Depends on

• Order on the natural numbers
• Addition of natural numbers
• Left identity for addition
• Addition is associative
• Addition is commutative
• Addition is cancellative
• Every nonzero natural number is a successor
• Trichotomy of the order on $\mathbb{N}$
• The von Neumann naturals form a Peano system

Used by

• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value Corollary
• (ℕ, ≤) has no maximal element: Zorn's chain hypothesis fails Counterexample
• A free ultrafilter induces a finitely additive zero-one probability that is not countably additive 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
• Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence Counterexample
• Transfinite induction fails if the least element is exempted from the hypothesis Counterexample
• A summability (Toeplitz) matrix, the transformed sequence yₙ = ∑ₖ c_n,k xₖ, and regularity Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise Definition
• Well-order and well-ordered set Definition
• A finite coordinate path and a bounded nonrectifiable path Example
• A positive sequence making all three inequalities of the ratio-to-root chain strict Example
• An explicit well-ordering of the integers: 0≺1≺-1≺2≺-2≺⋯ Example
• Changing the first hundred terms leaves convergence to 0 and boundedness intact Example
• Deleting one point of a countably infinite set leaves a set equinumerous with it Example
• F^ℕ is a vector space and the eventually zero families form a linear subspace of it that is the span of the standard unit families Example
• Five finite AM-GM proofs, with the same sharp equality case Example
• Five finite proofs of Bernoulli, with rational exponents and the quadratic refinement Example
• Lexicographic orders: {0,1}×ℕ and ℕ×ℕ Example
• min is a choice function on ℕ Example
• 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
• The block sequence 1/1; 1/2, 2/2; 1/3, 2/3, 3/3; … has subsequential limit set exactly [0,1] Example
• The standard unit families eₖ ∈ F^ℕ form a basis of the linear subspace of eventually zero families: an explicit infinite basis, built with no choice principle Example
• The subsets of ℕ containing a tail form the Fréchet filter, and it is proper and not an ultrafilter Example
• FALSE, once the ultrafilter lemma is available: every ultrafilter is principal False statement
• FALSE: every bounded sequence converges False statement
• FALSE: limits preserve strict inequalities False statement
• A Cauchy sequence with a convergent subsequence converges, to that subsequence’s limit Lemma
• A real coefficient list of degree d has at most d real roots Lemma
• A sequence has at most one limit Lemma
• A strictly increasing index map satisfies nₖ ≥ k Lemma
• An ordered field with the least-upper-bound property has the nested interval property and is Archimedean Lemma
• Basic bounding and dominating relations Lemma
• Cauchy completeness plus the Archimedean property imply the monotone convergence property Lemma
• Completeness of the sup metric on finite-state value functions Lemma
• Convergence depends only on the tail Lemma
• Countable Union of Compact Null Sets Is Null Lemma
• Every Cauchy sequence of reals is bounded Lemma

…and 53 more results.

Dependency tree · two levels

16 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., §2.1-2.3 (Peano axioms, recursion, arithmetic) (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)