Theory Continuum_Not_Denumerable
section ‹Non-Denumerability of the Continuum›
theory Continuum_Not_Denumerable
imports
Complex_Main
"HOL-Library.Countable_Set"
begin
subsection ‹Abstract›
text ‹
The following document presents a proof that the Continuum is uncountable.
It is formalised in the Isabelle/Isar theorem proving system.
❙‹Theorem:› The Continuum ‹ℝ› is not denumerable. In other words, there does
not exist a function ‹f: ℕ ⇒ ℝ› such that ‹f› is surjective.
❙‹Outline:› An elegant informal proof of this result uses Cantor's
Diagonalisation argument. The proof presented here is not this one.
First we formalise some properties of closed intervals, then we prove the
Nested Interval Property. This property relies on the completeness of the
Real numbers and is the foundation for our argument. Informally it states
that an intersection of countable closed intervals (where each successive
interval is a subset of the last) is non-empty. We then assume a surjective
function ‹f: ℕ ⇒ ℝ› exists and find a real ‹x› such that ‹x› is not in the
range of ‹f› by generating a sequence of closed intervals then using the
Nested Interval Property.
›
text ‹%whitespace›
theorem real_non_denum: "∄f :: nat ⇒ real. surj f"
proof
assume "∃f::nat ⇒ real. surj f"
then obtain f :: "nat ⇒ real" where "surj f" ..
txt ‹First we construct a sequence of nested intervals, ignoring \<^term>‹range f›.›
have "a < b ⟹ ∃ka kb. ka < kb ∧ {ka..kb} ⊆ {a..b} ∧ c ∉ {ka..kb}" for a b c :: real
by (auto simp add: not_le cong: conj_cong)
(metis dense le_less_linear less_linear less_trans order_refl)
then obtain i j where ij:
"a < b ⟹ i a b c < j a b c"
"a < b ⟹ {i a b c .. j a b c} ⊆ {a .. b}"
"a < b ⟹ c ∉ {i a b c .. j a b c}"
for a b c :: real
by metis
define ivl where "ivl =
rec_nat (f 0 + 1, f 0 + 2) (λn x. (i (fst x) (snd x) (f n), j (fst x) (snd x) (f n)))"
define I where "I n = {fst (ivl n) .. snd (ivl n)}" for n
have ivl [simp]:
"ivl 0 = (f 0 + 1, f 0 + 2)"
"⋀n. ivl (Suc n) = (i (fst (ivl n)) (snd (ivl n)) (f n), j (fst (ivl n)) (snd (ivl n)) (f n))"
unfolding ivl_def by simp_all
txt ‹This is a decreasing sequence of non-empty intervals.›
have less: "fst (ivl n) < snd (ivl n)" for n
by (induct n) (auto intro!: ij)
have "decseq I"
unfolding I_def decseq_Suc_iff ivl fst_conv snd_conv
by (intro ij allI less)
txt ‹Now we apply the finite intersection property of compact sets.›
have "I 0 ∩ (⋂i. I i) ≠ {}"
proof (rule compact_imp_fip_image)
fix S :: "nat set"
assume fin: "finite S"
have "{} ⊂ I (Max (insert 0 S))"
unfolding I_def using less[of "Max (insert 0 S)"] by auto
also have "I (Max (insert 0 S)) ⊆ (⋂i∈insert 0 S. I i)"
using fin decseqD[OF ‹decseq I›, of _ "Max (insert 0 S)"]
by (auto simp: Max_ge_iff)
also have "(⋂i∈insert 0 S. I i) = I 0 ∩ (⋂i∈S. I i)"
by auto
finally show "I 0 ∩ (⋂i∈S. I i) ≠ {}"
by auto
qed (auto simp: I_def)
then obtain x where "x ∈ I n" for n
by blast
moreover from ‹surj f› obtain j where "x = f j"
by blast
ultimately have "f j ∈ I (Suc j)"
by blast
with ij(3)[OF less] show False
unfolding I_def ivl fst_conv snd_conv by auto
qed
lemma uncountable_UNIV_real: "uncountable (UNIV :: real set)"
using real_non_denum unfolding uncountable_def by auto
corollary complex_non_denum: "∄f :: nat ⇒ complex. surj f"
by (metis (full_types) Re_complex_of_real comp_surj real_non_denum surj_def)
lemma uncountable_UNIV_complex: "uncountable (UNIV :: complex set)"
using complex_non_denum unfolding uncountable_def by auto
lemma bij_betw_open_intervals:
fixes a b c d :: real
assumes "a < b" "c < d"
shows "∃f. bij_betw f {a<..<b} {c<..<d}"
proof -
define f where "f a b c d x = (d - c)/(b - a) * (x - a) + c" for a b c d x :: real
{
fix a b c d x :: real
assume *: "a < b" "c < d" "a < x" "x < b"
moreover from * have "(d - c) * (x - a) < (d - c) * (b - a)"
by (intro mult_strict_left_mono) simp_all
moreover from * have "0 < (d - c) * (x - a) / (b - a)"
by simp
ultimately have "f a b c d x < d" "c < f a b c d x"
by (simp_all add: f_def field_simps)
}
with assms have "bij_betw (f a b c d) {a<..<b} {c<..<d}"
by (intro bij_betw_byWitness[where f'="f c d a b"]) (auto simp: f_def)
then show ?thesis by auto
qed
lemma bij_betw_tan: "bij_betw tan {-pi/2<..<pi/2} UNIV"
using arctan_ubound by (intro bij_betw_byWitness[where f'=arctan]) (auto simp: arctan arctan_tan)
lemma uncountable_open_interval: "uncountable {a<..<b} ⟷ a < b" for a b :: real
proof
show "a < b" if "uncountable {a<..<b}"
using uncountable_def that by force
show "uncountable {a<..<b}" if "a < b"
proof -
obtain f where "bij_betw f {a <..< b} {-pi/2<..<pi/2}"
using bij_betw_open_intervals[OF ‹a < b›, of "-pi/2" "pi/2"] by auto
then show ?thesis
by (metis bij_betw_tan uncountable_bij_betw uncountable_UNIV_real)
qed
qed
lemma uncountable_half_open_interval_1: "uncountable {a..<b} ⟷ a < b" for a b :: real
apply auto
using atLeastLessThan_empty_iff
apply fastforce
using uncountable_open_interval [of a b]
apply (metis countable_Un_iff ivl_disj_un_singleton(3))
done
lemma uncountable_half_open_interval_2: "uncountable {a<..b} ⟷ a < b" for a b :: real
apply auto
using atLeastLessThan_empty_iff
apply fastforce
using uncountable_open_interval [of a b]
apply (metis countable_Un_iff ivl_disj_un_singleton(4))
done
lemma real_interval_avoid_countable_set:
fixes a b :: real and A :: "real set"
assumes "a < b" and "countable A"
shows "∃x∈{a<..<b}. x ∉ A"
proof -
from ‹countable A› have *: "countable (A ∩ {a<..<b})"
by auto
with ‹a < b› have "¬ countable {a<..<b}"
by (simp add: uncountable_open_interval)
with * have "A ∩ {a<..<b} ≠ {a<..<b}"
by auto
then have "A ∩ {a<..<b} ⊂ {a<..<b}"
by (intro psubsetI) auto
then have "∃x. x ∈ {a<..<b} - A ∩ {a<..<b}"
by (rule psubset_imp_ex_mem)
then show ?thesis
by auto
qed
lemma uncountable_closed_interval: "uncountable {a..b} ⟷ a < b" for a b :: real
using infinite_Icc_iff by (fastforce dest: countable_finite real_interval_avoid_countable_set)
lemma open_minus_countable:
fixes S A :: "real set"
assumes "countable A" "S ≠ {}" "open S"
shows "∃x∈S. x ∉ A"
proof -
obtain x where "x ∈ S"
using ‹S ≠ {}› by auto
then obtain e where "0 < e" "{y. dist y x < e} ⊆ S"
using ‹open S› by (auto simp: open_dist subset_eq)
moreover have "{y. dist y x < e} = {x - e <..< x + e}"
by (auto simp: dist_real_def)
ultimately have "uncountable (S - A)"
using uncountable_open_interval[of "x - e" "x + e"] ‹countable A›
by (intro uncountable_minus_countable) (auto dest: countable_subset)
then show ?thesis
unfolding uncountable_def by auto
qed
end