(* Title: HOL/Analysis/Continuum_Not_Denumerable.thy Author: Benjamin Porter, Monash University, NICTA, 2005 Author: Johannes Hölzl, TU München *) section ‹Non-Denumerability of the Continuum› theory Continuum_Not_Denumerable imports Complex_Main "HOL-Library.Countable_Set" begin subsection✐‹tag unimportant› ‹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✐‹tag important› ‹%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 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 apply (rule iffI) apply (metis atLeastAtMost_singleton atLeastatMost_empty countable_finite finite.emptyI finite_insert linorder_neqE_linordered_idom) using real_interval_avoid_countable_set by fastforce 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