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c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib
namespace OAI
section
open Set Cardinal
namespace InfiniteMatroidCounterexample
abbrev CIndex := Cardinal.continuum.{0}.ord.ToType
lemma cindex_card : Cardinal.mk CIndex = Cardinal.continuum := Cardinal.mk_ord_toType _
lemma cindex_initial : (Cardinal.mk CIndex).ord = Ordinal.type ((· < ·) : CIndex → CIndex → Prop) := by
rw [cindex_card, Ordinal.type_toType]
lemma cindex_predecessors (i : CIndex) : Cardinal.mk (Iio i) < Cardinal.continuum := by
simpa only [cindex_card] using Cardinal.mk_Iio_lt i cindex_initial
lemma cindex_initial_segment (i : CIndex) : Cardinal.mk (Iic i) < Cardinal.continuum := by
have h : Iic i = insert i (Iio i) := by ext x; simp [le_iff_lt_or_eq, eq_comm, or_comm]
rw [h]
apply lt_of_le_of_lt Cardinal.mk_insert_le
have h0 := cindex_predecessors i
have hc : ℵ₀ < Cardinal.continuum := Cardinal.aleph0_lt_continuum
calc
_ ≤ max (Cardinal.mk (Iio i)) ℵ₀ + max (Cardinal.mk (Iio i)) ℵ₀ :=
add_le_add (le_max_left _ _) (le_trans Cardinal.one_le_aleph0 (le_max_right _ _))
_ = max (Cardinal.mk (Iio i)) ℵ₀ := Cardinal.add_eq_self (le_max_right _ _)
_ < Cardinal.continuum := max_lt h0 hc
/-- Cardinal arithmetic used at each stage; it does not assert regularity of the continuum. -/
lemma finite_union_small {α ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum)
(F : ι → Set α) (hF : ∀ i, (F i).Finite) :
Cardinal.mk (⋃ i, F i) < Cardinal.continuum := by
calc
Cardinal.mk (⋃ i, F i) ≤ Cardinal.mk ι * ⨆ i, Cardinal.mk (F i) := Cardinal.mk_iUnion_le F
_ ≤ max (Cardinal.mk ι) ℵ₀ * max (Cardinal.mk ι) ℵ₀ := by
apply mul_le_mul' (le_max_left _ _)
apply ciSup_le'
intro i
let := (hF i).fintype
exact (Cardinal.lt_aleph0_of_finite (F i)).le.trans (le_max_right _ _)
_ = max (Cardinal.mk ι) ℵ₀ := Cardinal.mul_eq_self (le_max_right _ _)
_ < Cardinal.continuum := max_lt hι Cardinal.aleph0_lt_continuum
lemma exists_avoid_finite_family {ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum)
(F : ι → Set CIndex) (hF : ∀ i, (F i).Finite) :
∃ t : CIndex, ∀ i, t ∉ F i := by
have hcard := finite_union_small hι F hF
have hn : (⋃ i, F i) ≠ Set.univ := by
intro heq
rw [heq, Cardinal.mk_univ, cindex_card] at hcard
exact lt_irrefl _ hcard
obtain ⟨t, ht⟩ := (Set.ne_univ_iff_exists_notMem _).mp hn
exact ⟨t, fun i hi ↦ ht (mem_iUnion.mpr ⟨i, hi⟩)⟩
lemma binary_card : Cardinal.mk (ℕ → Bool) = Cardinal.continuum := by
rw [Cardinal.mk_arrow]
simp only [Cardinal.mk_fintype, Fintype.card_bool, Cardinal.mk_nat, Cardinal.lift_id, Nat.cast_ofNat]
exact Cardinal.two_power_aleph0
noncomputable def cindexBinaryEquiv : CIndex ≃ (ℕ → Bool) :=
Classical.choice (Cardinal.eq.mp (cindex_card.trans binary_card.symm))
/-- Enumeration of all subsets of any countably infinite type, on the initial continuum ordinal. -/
lemma exists_set_enumeration (α : Type) [Countable α] [Infinite α] :
∃ f : CIndex → Set α, Function.Surjective f := by
have hc : Cardinal.mk (Set α) = Cardinal.continuum := by
rw [Cardinal.mk_set, Cardinal.mk_eq_aleph0]
rfl
obtain ⟨e⟩ := Cardinal.eq.mp (cindex_card.trans hc.symm)
exact ⟨e, e.surjective⟩
lemma cardinal_prod_bool_lt {ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum) :
Cardinal.mk (ι × Bool) < Cardinal.continuum := by
rw [Cardinal.mk_prod,Cardinal.lift_id,Cardinal.lift_id]
have hb : Cardinal.mk Bool ≤ ℵ₀ := (Cardinal.lt_aleph0_of_finite Bool).le
calc
_ ≤ max (Cardinal.mk ι) ℵ₀ * max (Cardinal.mk ι) ℵ₀ :=
mul_le_mul' (le_max_left _ _) (hb.trans (le_max_right _ _))
_ = max (Cardinal.mk ι) ℵ₀ := Cardinal.mul_eq_self (le_max_right _ _)
_ < Cardinal.continuum := max_lt hι Cardinal.aleph0_lt_continuum
end InfiniteMatroidCounterexample
end
end OAI
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