The fundamental number sets

0 := ∅ = {} Zero 1 = {∅} One 𝔹 = {⊥, ⊤}

ℕ = {1,2,3,...} ℤ = {...,-1,0,1,...} ℚ = { p/q | p,q ∈ ℤ, q ≠ 0 } ℝ ℂ

ℕᐩ = ℕ {0} ℕ₀ = ℕ ⋃ {0} ℕ = ℤᐩ ℙ = { p ∈ ℕ | ∀x ∈ ℕ. x ∉ {1,p} -> x ∤ p } = primes ℤ⁻ = ℤ ℕ₀

𝕎 = ℕ ⋃ {0} ℕᐩ = ℕ {0} ℤ = ℤ⁻ ⋃ {0} ⋃ ℕᐩ

ℚ = ℤ ⋃ 𝔽 ℚ = fromList [ p % q | p <- [1..], q <- [1..] ] :: Set Integer

ℝ = ℚ ⋃ 𝕀 ℝ = 𝕋 ⋃ 𝔸 ℝ = ℝ⁻ ⋃ {0} ⋃ ℝᐩ ℂ = ℝ ⋃ 𝕁

• subset-of relations are intuitively: ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ

• however, unintuitively: ℕ = ℤ = ℚ < ℝ < ℂ

ℕᐩ ℕ⁻ ℕ﹢ ℕ˗ ℤᐩ ℤ⁻ ℤ﹢ ℤ˗ ℚᐩ ℚ⁻ ℚ﹢ ℚ˗ ℝᐩ ℝ⁻ ℝ﹢ ℝ˗ ℂᐩ ℂ⁻ ℂ﹢ ℂ˗

𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 𝕄 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤ​