Principal ordinals

We define principal or indecomposable ordinals, and we prove the standard properties about them.

Main definitions and results

• Principal: A principal or indecomposable ordinal under some binary operation. We include 0 and any other typically excluded edge cases for simplicity.
• not_bddAbove_principal: Principal ordinals (under any operation) are unbounded.
• principal_add_iff_zero_or_omega0_opow: The main characterization theorem for additive principal ordinals.
• principal_mul_iff_le_two_or_omega0_opow_opow: The main characterization theorem for multiplicative principal ordinals.

TODO

• Prove that exponential principal ordinals are 0, 1, 2, ω, or epsilon numbers, i.e. fixed points of fun x ↦ ω ^ x.

Principal ordinals

def Ordinal.Principal (op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}) (o : Ordinal.{u_1}) : Prop

An ordinal o is said to be principal or indecomposable under an operation when the set of ordinals less than it is closed under that operation. In standard mathematical usage, this term is almost exclusively used for additive and multiplicative principal ordinals.

For simplicity, we break usual convention and regard 0 as principal.

Equations

• Ordinal.Principal op o = ∀ ⦃a b : Ordinal.{?u.8}⦄, a < o → b < o → op a b < o

theorem Ordinal.principal_swap_iff {o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} : Principal (Function.swap op) o ↔ Principal op o

theorem Ordinal.not_principal_iff {o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} : ¬Principal op o ↔ ∃ a < o, ∃ b < o, o ≤ op a b

theorem Ordinal.principal_iff_of_monotone {o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} (h₁ : ∀ (a : Ordinal.{u}), Monotone (op a)) (h₂ : ∀ (a : Ordinal.{u}), Monotone (Function.swap op a)) : Principal op o ↔ ∀ a < o, op a a < o

theorem Ordinal.not_principal_iff_of_monotone {o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} (h₁ : ∀ (a : Ordinal.{u}), Monotone (op a)) (h₂ : ∀ (a : Ordinal.{u}), Monotone (Function.swap op a)) : ¬Principal op o ↔ ∃ a < o, o ≤ op a a

theorem Ordinal.principal_zero {op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}} : Principal op 0

@[simp]

theorem Ordinal.principal_one_iff {op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}} : Principal op 1 ↔ op 0 0 = 0

theorem Ordinal.Principal.iterate_lt {a o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} (hao : a < o) (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o

theorem Ordinal.op_eq_self_of_principal {a o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} (hao : a < o) (H : IsNormal (op a)) (ho : Principal op o) (ho' : o.IsLimit) : op a o = o

theorem Ordinal.nfp_le_of_principal {a o : Ordinal.{u}} {op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}} (hao : a < o) (ho : Principal op o) : nfp (op a) a ≤ o

Principal ordinals are unbounded

theorem Ordinal.not_bddAbove_principal (op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}) : ¬BddAbove {o : Ordinal.{u_1} | Principal op o}

Principal ordinals under any operation are unbounded.

@[deprecated "No deprecation message was provided." (since := "2024-10-11")]

theorem Ordinal.principal_nfp_blsub₂ (op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}) (o : Ordinal.{u}) : Principal op (nfp (fun (o' : Ordinal.{u}) => o'.blsub₂ o' fun (a : Ordinal.{u}) (x : a < o') (b : Ordinal.{u}) (x : b < o') => op a b) o)

@[deprecated "No deprecation message was provided." (since := "2024-10-11")]

theorem Ordinal.unbounded_principal (op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}) : Set.Unbounded (fun (x1 x2 : Ordinal.{u_1}) => x1 < x2) {o : Ordinal.{u_1} | Principal op o}

Additive principal ordinals

theorem Ordinal.principal_add_one : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) 1

theorem Ordinal.principal_add_of_le_one {o : Ordinal.{u}} (ho : o ≤ 1) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o

theorem Ordinal.isLimit_of_principal_add {o : Ordinal.{u}} (ho₁ : 1 < o) (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o) : o.IsLimit

@[deprecated Ordinal.isLimit_of_principal_add (since := "2024-10-16")]

theorem Ordinal.principal_add_isLimit {o : Ordinal.{u}} (ho₁ : 1 < o) (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o) : o.IsLimit

Alias of Ordinal.isLimit_of_principal_add.

theorem Ordinal.principal_add_iff_add_left_eq_self {o : Ordinal.{u}} : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o ↔ ∀ a < o, a + o = o

theorem Ordinal.exists_lt_add_of_not_principal_add {a : Ordinal.{u}} (ha : ¬Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) a) : ∃ b < a, ∃ c < a, b + c = a

theorem Ordinal.principal_add_iff_add_lt_ne_self {a : Ordinal.{u}} : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) a ↔ ∀ b < a, ∀ c < a, b + c ≠ a

theorem Ordinal.principal_add_omega0 : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) omega0

theorem Ordinal.add_omega0_opow {a b : Ordinal.{u}} (h : a < omega0 ^ b) : a + omega0 ^ b = omega0 ^ b

theorem Ordinal.principal_add_omega0_opow (o : Ordinal.{u_1}) : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) (omega0 ^ o)

theorem Ordinal.principal_add_iff_zero_or_omega0_opow {o : Ordinal.{u}} : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o ↔ o = 0 ∨ o ∈ Set.range fun (x : Ordinal.{u}) => omega0 ^ x

The main characterization theorem for additive principal ordinals.

theorem Ordinal.principal_add_opow_of_principal_add {a : Ordinal.{u_1}} (ha : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) a) (b : Ordinal.{u_1}) : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) (a ^ b)

@[deprecated Ordinal.principal_add_opow_of_principal_add (since := "2024-10-16")]

theorem Ordinal.opow_principal_add_of_principal_add {a : Ordinal.{u_1}} (ha : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) a) (b : Ordinal.{u_1}) : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 + x2) (a ^ b)

Alias of Ordinal.principal_add_opow_of_principal_add.

theorem Ordinal.add_absorp {a b c : Ordinal.{u}} (h₁ : a < omega0 ^ b) (h₂ : omega0 ^ b ≤ c) : a + c = c

theorem Ordinal.principal_add_mul_of_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1) (hb : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) b) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) (a * b)

@[deprecated Ordinal.principal_add_mul_of_principal_add (since := "2024-10-16")]

theorem Ordinal.mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1) (hb : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) b) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) (a * b)

Alias of Ordinal.principal_add_mul_of_principal_add.

Multiplicative principal ordinals

theorem Ordinal.principal_mul_one : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 * x2) 1

theorem Ordinal.principal_mul_two : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 * x2) 2

theorem Ordinal.principal_mul_of_le_two {o : Ordinal.{u}} (ho : o ≤ 2) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o

theorem Ordinal.principal_add_of_principal_mul {o : Ordinal.{u}} (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o) (ho₂ : o ≠ 2) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o

theorem Ordinal.isLimit_of_principal_mul {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o) : o.IsLimit

@[deprecated Ordinal.isLimit_of_principal_mul (since := "2024-10-16")]

theorem Ordinal.principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o) : o.IsLimit

Alias of Ordinal.isLimit_of_principal_mul.

theorem Ordinal.principal_mul_iff_mul_left_eq {o : Ordinal.{u}} : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o ↔ ∀ (a : Ordinal.{u}), 0 < a → a < o → a * o = o

theorem Ordinal.principal_mul_omega0 : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 * x2) omega0

theorem Ordinal.mul_omega0 {a : Ordinal.{u}} (a0 : 0 < a) (ha : a < omega0) : a * omega0 = omega0

theorem Ordinal.natCast_mul_omega0 {n : ℕ} (hn : 0 < n) : ↑n * omega0 = omega0

theorem Ordinal.mul_lt_omega0_opow {a b c : Ordinal.{u}} (c0 : 0 < c) (ha : a < omega0 ^ c) (hb : b < omega0) : a * b < omega0 ^ c

theorem Ordinal.mul_omega0_opow_opow {a b : Ordinal.{u}} (a0 : 0 < a) (h : a < omega0 ^ omega0 ^ b) : a * omega0 ^ omega0 ^ b = omega0 ^ omega0 ^ b

theorem Ordinal.principal_mul_omega0_opow_opow (o : Ordinal.{u_1}) : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 * x2) (omega0 ^ omega0 ^ o)

theorem Ordinal.principal_add_of_principal_mul_opow {b o : Ordinal.{u}} (hb : 1 < b) (ho : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) (b ^ o)) : Principal (fun (x1 x2 : Ordinal.{u}) => x1 + x2) o

theorem Ordinal.principal_mul_iff_le_two_or_omega0_opow_opow {o : Ordinal.{u}} : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) o ↔ o ≤ 2 ∨ o ∈ Set.range fun (x : Ordinal.{u}) => omega0 ^ omega0 ^ x

The main characterization theorem for multiplicative principal ordinals.

theorem Ordinal.mul_omega0_dvd {a : Ordinal.{u}} (a0 : 0 < a) (ha : a < omega0) {b : Ordinal.{u}} : omega0 ∣ b → a * b = b

theorem Ordinal.mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (fun (x1 x2 : Ordinal.{u}) => x1 * x2) b) (hb₂ : 2 < b) : a * b = b ^ Order.succ (log b a)

Exponential principal ordinals

theorem Ordinal.principal_opow_omega0 : Principal (fun (x1 x2 : Ordinal.{u_1}) => x1 ^ x2) omega0

theorem Ordinal.opow_omega0 {a : Ordinal.{u}} (a1 : 1 < a) (h : a < omega0) : a ^ omega0 = omega0

theorem Ordinal.natCast_opow_omega0 {n : ℕ} (hn : 1 < n) : ↑n ^ omega0 = omega0