Discreteness: $\sigma (n)$ is the immediate successor

Depends on

• Order on the natural numbers
• Addition of natural numbers
• Every nonzero natural number is a successor
• Left successor law for addition
• Addition is cancellative
• Addition is commutative
• The von Neumann naturals form a Peano system

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 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
• Positive integers 4ᵃ m with m≡ 7 (mod 8) need four nonzero squares Corollary
• A free ultrafilter induces a finitely additive zero-one probability that is not countably additive Counterexample
• A nonnegative non-monotone sequence for which ∑ aₖ and ∑ 2ᵏ a_2ᵏ behave differently 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
• ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric Counterexample
• Compositions and weak compositions of a natural number into a fixed number of parts Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• The p-adic valuation vₚ(a) of a nonzero integer: the greatest k ∈ ℕ with pᵏ ∣ a Definition
• 2 is prime, and it is the only even prime: every even integer n > 2 is composite Example
• Ceiling ⊣ inclusion ⊣ floor: an adjoint triple between (ℝ,≤) and (ℤ,≤) Example
• Consecutive Fibonacci numbers are coprime, and for every m ∈ ℕ the Euclidean algorithm on (Fₘ₊₃, Fₘ₊₂) takes exactly m+1 divisions, with quotient 1 in the first m of them and quotient 2 in the last Example
• Eisenstein proves xⁿ-2 irreducible over ℚ for every positive n Example
• For every n ∈ ℕ there are n consecutive composite integers: with N := ∏_j<n(j+2), each of N+2, …, N+n+1 is composite Example
• No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary Example
• Reduction modulo 2 proves x³+17x+391 irreducible over ℚ 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 Cauchy sequences of rationals form a commutative ring that is not an integral domain: two eventually-constant sequences with disjoint supports multiply to zero Example
• The polynomial x²-2 is irreducible over ℚ Example
• The subsets of ℕ containing a tail form the Fréchet filter, and it is proper and not an ultrafilter Example
• The trigonometry-free oscillator ψ(x) = inf_n ∈ ℤ |x - n| is well defined and attained at a nearest integer, takes values in [0, 1/2], vanishes exactly on ℤ, equals 1/2 at half-integers, and is 1-periodic Example
• FALSE, once the ultrafilter lemma is available: every ultrafilter is principal False statement
• FALSE: continuity from above needs no finiteness hypothesis False statement
• FALSE: every Fermat number 2^2ⁿ + 1 is prime False statement
• FALSE: for all sets A and B with B having at least two elements, A × B is strictly larger than A False statement
• FALSE: for every finite list p₀, …, pₙ₋₁ of distinct primes, p₀ ⋯ pₙ₋₁ + 1 is prime False statement
• FALSE: limits preserve strict inequalities False statement
• FALSE: n² + n + 41 is prime for every natural number n False statement
• (ℤ, ·, 1) is a commutative monoid whose group of units is {1, -1}; equivalently u ∣ 1 holds exactly for u = 1 and u = -1 Lemma
• 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 strictly increasing index map satisfies nₖ ≥ k Lemma
• A translation-invariant Borel measure giving the unit cube measure one gives each generation-k dyadic cube measure 2⁻ᵏⁿ Lemma
• Bolzano-Weierstrass alone forces the Archimedean property, so it needs no separate Archimedean hypothesis Lemma
• Every integer n > 1 has a prime divisor; indeed the least divisor of n that exceeds 1 is prime Lemma
• Every real sequence has a monotone subsequence (the peak / rising-sun lemma) Lemma
• Every subgroup of (ℤ, +) is ⟨ n ⟩ = nℤ for exactly one natural number n Lemma

…and 28 more results.

Dependency tree · two levels

14 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)
• Well-ordering principle (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)