Left identity for addition

Depends on

• Addition of natural numbers
• The principle of mathematical induction

Used by

• 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
• (ℕ, ≤) has no maximal element: Zorn's chain hypothesis fails Counterexample
• A commutative monoid in which cancellation holds need not be a group: (ℕ, +) Counterexample
• Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ Definition
• 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
• Addition is commutative Lemma
• Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
• On ℕ the order is membership: m < n ⟺ m ∈ n Lemma
• The naturals embed in the integers Lemma
• Trichotomy of the order on ℕ Lemma
• Zero and one under multiplication Lemma
• ω is the least limit ordinal Lemma
• ≤ is a linear order on ℕ Theorem
• ℕ × ℕ ≈ ℕ Theorem
• Stay-put elimination with an explicit simulation interface Theorem
• The integers form a commutative ring Theorem
• The well-ordering principle Theorem

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

• T. Tao, Analysis I, 3rd ed., §2.2 (Addition) (standard reference, not scraped)
• Mathematical induction (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)