Axioms of the natural numbers

∀a ∈ N ∀b ∈ N a + b = c ∈ N

∃0 ∈ N ∀a ∈ N a + 0 = 0 + a = a

6 ∀a ∈ N ∃ − a ∈ N a + (−a) = (−a) + a = 0

∀a ∈ N ∀b ∈ N a + b = b + a

∀a ∈ N ∀b ∈ N∀c ∈ N (a + b) + c = a + (b + c)

∀a ∈ N ∀b ∈ N a · b = c ∈ N

∃1 ∈ N ∀a ∈ N a · 1 = 1 · a = a

6 ∀a ∈ N ∋ a 6= 0 ∃a−1 ∈ N a · a−1 = a

−1 · a = 1

∀a ∈ N ∀b ∈ N a · b = b · a

∀a ∈ N ∀b ∈ N∀c ∈ N (a · b) · c = a · (b · c)

∀a ∈ N ∀b ∈ N∀c ∈ N a · (b + c) = a · b + a · c

Axioms of the rational numbers

∀a ∈ Q ∀b ∈ Q a + b = c ∈ Q

∃0 ∈ Q ∀a ∈ Q a + 0 = 0 + a = a

∀a ∈ Q ∃ − a ∈ Q a + (−a) = (−a) + a = 0

∀a ∈ Q ∀b ∈ Q a + b = b + a

∀a ∈ Q ∀b ∈ Q∀c ∈ N (a + b) + c = a + (b + c)

∀a ∈ Q ∀b ∈ Q a · b = c ∈ Q

∃1 ∈ Q ∀a ∈ Q a · 1 = 1 · a = a

∀a ∈ Q ∋ a 6= 0 ∃a−1 ∈ Q a · a−1 = a

−1 · a = 1

∀a ∈ Q ∀b ∈ Q a · b = b · a

∀a ∈ Q ∀b ∈ Q∀c ∈ Q (a · b) · c = a · (b · c)

∀a ∈ Q ∀b ∈ Q∀c ∈ Q a · (b + c) = a · b + a · c

Axioms of the real numbers

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∀a ∈ R ∀b ∈ R a + b = c ∈ R

∃0 ∈ R ∀a ∈ R a + 0 = 0 + a = a

∀a ∈ R ∃ − a ∈ R a + (−a) = (−a) + a = 0

∀a ∈ R ∀b ∈ R a + b = b + a

∀a ∈ R ∀b ∈ R∀c ∈ R (a + b) + c = a + (b + c)

∀a ∈ R ∀b ∈ R a · b = c ∈ R

∃1 ∈ R ∀a ∈ R a · 1 = 1 · a = a

∀a ∈ R ∋ a 6= 0 ∃a−1 ∈ N a · a−1 = a

−1 · a = 1

∀a ∈ R ∀b ∈ R a · b = b · a

∀a ∈ R ∀b ∈ R∀c ∈ R (a · b) · c = a · (b · c)

∀a ∈ R ∀b ∈ R∀c ∈ R a · (b + c) = a · b + a · c

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