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Mathlib.SetTheory.Cardinal.Continuum

Cardinality of continuum #

In this file we define Cardinal.continuum (notation: 𝔠, localized in Cardinal) to be 2 ^ ℵ₀. We also prove some simp lemmas about cardinal arithmetic involving 𝔠.

Notation #

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def Cardinal.continuum :
Cardinal.{u}

Cardinality of the continuum.

Equations
Instances For
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    def Cardinal.term𝔠 :
    Lean.ParserDescr

    Cardinality of the continuum.

    Equations
    Instances For
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      @[simp]
      theorem Cardinal.two_power_aleph0 :
      2 ^ aleph0 = continuum
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      @[simp]
      theorem Cardinal.lift_continuum :
      lift.{v, u_1} continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_le_lift {c : Cardinal.{u}} :
      continuum ≤ lift.{v, u} c ↔ continuum ≤ c
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      @[simp]
      theorem Cardinal.lift_le_continuum {c : Cardinal.{u}} :
      lift.{v, u} c ≤ continuum ↔ c ≤ continuum
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      @[simp]
      theorem Cardinal.continuum_lt_lift {c : Cardinal.{u}} :
      continuum < lift.{v, u} c ↔ continuum < c
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      @[simp]
      theorem Cardinal.lift_lt_continuum {c : Cardinal.{u}} :
      lift.{v, u} c < continuum ↔ c < continuum

      Inequalities #

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      theorem Cardinal.aleph0_lt_continuum :
      aleph0 < continuum
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      theorem Cardinal.aleph0_le_continuum :
      aleph0 ≤ continuum
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      @[simp]
      theorem Cardinal.beth_one :
      beth 1 = continuum
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      theorem Cardinal.nat_lt_continuum (n : ℕ) :
      ↑n < continuum
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      theorem Cardinal.mk_set_nat :
      mk (Set ℕ) = continuum
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      theorem Cardinal.continuum_pos :
      0 < continuum
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      theorem Cardinal.continuum_ne_zero :
      continuum ≠ 0
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      theorem Cardinal.aleph_one_le_continuum :
      aleph 1 ≤ continuum
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      @[simp]
      theorem Cardinal.continuum_toNat :
      toNat continuum = 0
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      @[simp]
      theorem Cardinal.continuum_toENat :
      toENat continuum = ⊤

      Addition #

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      @[simp]
      theorem Cardinal.aleph0_add_continuum :
      aleph0 + continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_add_aleph0 :
      continuum + aleph0 = continuum
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      @[simp]
      theorem Cardinal.continuum_add_self :
      continuum + continuum = continuum
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      @[simp]
      theorem Cardinal.nat_add_continuum (n : ℕ) :
      ↑n + continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_add_nat (n : ℕ) :
      continuum + ↑n = continuum
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      @[simp]
      theorem Cardinal.ofNat_add_continuum {n : ℕ} [n.AtLeastTwo] :
      OfNat.ofNat n + continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_add_ofNat {n : ℕ} [n.AtLeastTwo] :
      continuum + OfNat.ofNat n = continuum

      Multiplication #

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      @[simp]
      theorem Cardinal.continuum_mul_self :
      continuum * continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_mul_aleph0 :
      continuum * aleph0 = continuum
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      @[simp]
      theorem Cardinal.aleph0_mul_continuum :
      aleph0 * continuum = continuum
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      @[simp]
      theorem Cardinal.nat_mul_continuum {n : ℕ} (hn : n ≠ 0) :
      ↑n * continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_mul_nat {n : ℕ} (hn : n ≠ 0) :
      continuum * ↑n = continuum
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      @[simp]
      theorem Cardinal.ofNat_mul_continuum {n : ℕ} [n.AtLeastTwo] :
      OfNat.ofNat n * continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_mul_ofNat {n : ℕ} [n.AtLeastTwo] :
      continuum * OfNat.ofNat n = continuum

      Power #

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      @[simp]
      theorem Cardinal.aleph0_power_aleph0 :
      aleph0 ^ aleph0 = continuum
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      @[simp]
      theorem Cardinal.nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) :
      ↑n ^ aleph0 = continuum
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      @[simp]
      theorem Cardinal.continuum_power_aleph0 :
      continuum ^ aleph0 = continuum
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      theorem Cardinal.power_aleph0_of_le_continuum {x : Cardinal.{u_1}} (h₁ : 2 ≤ x) (h₂ : x ≤ continuum) :
      x ^ aleph0 = continuum