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

Depends on

• Inductive set
• The natural numbers exist: a smallest inductive set
• Peano system
• The Axiom of Infinity: there is a set containing a set with no elements and closed under $y \mapsto y \cup \{y\}$
• The Axiom Schema of Separation: for each formula $\varphi$, $\forall \bar p\,\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow (z \in x \wedge \varphi(z,\bar p)))$

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
• Addition is a well-defined total operation Corollary
• Every linear subspace U of a vector space V has a complement: a linear subspace W with V = U ⊕ W 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
• For an integer p > 1: p is prime if and only if, for all integers a and b, p ∣ ab implies p ∣ a or p ∣ b Corollary
• If a prime p divides a finite product ∏_i<n aᵢ of integers then p ∣ aᵢ for some i < n; at n = 0 the product is 1 and the hypothesis cannot hold 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
• Multiplication is a well-defined total operation Corollary
• Positive integers 4ᵃ m with m≡ 7 (mod 8) need four nonzero squares Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• The extended Euclidean algorithm: the same descent produces integers x, y with ax + by = gcd(a,b), so Bézout coefficients are computed and not merely shown to exist Corollary
• {(1,0), (0,1), (1,1)} spans F² and is linearly dependent, so a spanning set need not be a basis; each of its three two-element subsets is a basis Counterexample
• A commutative monoid in which cancellation holds need not be a group: (ℕ, +) Counterexample
• A count that overcounts because the blocks are not disjoint, and exactly where the sum rule's hypothesis is spent Counterexample
• A free ultrafilter induces a finitely additive zero-one probability that is not countably additive Counterexample
• A poset with a bottom, a top and countably many incomparable middle elements has an infinite interval, so convolution of constant-one functions is not defined Counterexample
• A reflexive coequalizer of sets not preserved by Set(ℕ,-) Counterexample
• An arbitrary isotopy of arcs need not be a braid isotopy Counterexample
• Collapsing the set of naturals inside ℝ to a point gives a quotient of ℝ that is not locally compact at the collapsed point 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 bounded real-valued functions on ℕ with the supremum metric, the closed unit ball is closed and bounded and is not compact: the indicator functions of the singletons are pairwise at distance 1 Counterexample
• In the indiscrete topology every sequence converges to every point, and in the cofinite topology on an infinite set an injective sequence converges to every point 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
• Kelley's cofinite set is not closed Counterexample
• ℕ × {a,b} with the indiscrete topology on the second factor is limit point compact and not countably compact, so the hypothesis that singletons are closed is not decoration Counterexample
• ℕ with the discrete metric is bounded and is not totally bounded Counterexample
• On (0,∞) the metrics |x-y| and |1/x - 1/y| have the same topology and are not uniformly equivalent Counterexample
• On ℕ with d(m,n) = 1 + 1/(m+n) for m ≠ n the sets {n, n+1, …} are nested, closed, bounded and complete with empty intersection Counterexample
• On the positive integers the metrics |m-n| and |1/m - 1/n| both induce the discrete topology, and only the first is complete Counterexample
• Refuted: every limit ordinal has an at most countable cofinal subset — ω₁ has none, assuming countable choice Counterexample
• The first quadrant of ℝ² contains 0 and is closed under addition and is not a linear subspace, since it is not closed under multiplication by -1 Counterexample
• The functor D(X)=X⊔ X on Set is not covariantly representable Counterexample
• The induction axiom is independent of the other Peano axioms Counterexample
• The one-point compactification of discrete ℕ is not βℕ Counterexample
• The Samuel compactification map need not be a uniform embedding for the original uniformity 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

…and 279 more results.

Dependency tree · two levels

12 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

• Set-theoretic definition of natural numbers (Wikipedia) (standard reference, not scraped)
• Axiom of infinity (Wikipedia) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)
• B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)