Quelle  TAO_99_Paradox.thy

theory9dox
imports TAO_9_PLM q_1] by simp
begin

section‹= x) in v]"

(*<*)
locale Paradox id_act_3equiv_lr] java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
begin
(*>*)

text
\label{TAO_Paradox_paradox}
Under the additional assumption that expressions of the form
@{term "(\      assu [\^bo>\<>\
\emph{proper maps}, for which ‹🚫
the theory becomes inconsistent.
\<close>

subsection‹Auxiliary Lemmas›thus "  \^>=x in v]" sing id_act3[equivrl] byfast

 lemma exe_impl_exists:
java.lang.NullPointerException
 proof (rule " x x<> \^bold>→x)in v]"
 fix φ :: "ν==>ν==>o" and x :: ν and v :: i
 assume "[((\λx . \∀ p . p \→ p),\ιy . φ y x) in v]"
 hence "[\∃y. \Aφ y x & (\∀z. \Aφ z x \→ z = y)
 & ((\λx . \∀ p . p \→ p), y^P) in v]"
 using nec_russell_axiom[equiv_lr] SimpleExOrEnc.intros by auto
 then obtain y where
java.lang.NullPointerException
java.lang.NullPointerException
 by (rule Instantiate)
 hence "[\Aφ y x & (\∀z. \Aφ z x \→ z = y) in v]"
 using "&E" by blast
 hence "[\∃y . \Aφ y x & (\∀z. \Aφ z x \→ z = y) in v]"
 by (rule existential)
 thus "[\∃!y. \Aφ y x in v]"
 unfolding exists_unique_def by simp
 next
 fix φ :: "ν==>ν==>o" and x :: ν and v :: i
 assume "[\∃!y. \Aφ y x in v]"
 hence "[\∃y. \Aφ y x & (\∀z. \Aφ z x \→ z = y) in v]"
 unfolding exists_unique_def by simp
 then obtain y where
 "[\Aφ y x & (\∀z. \Aφ z x \→ z = y) in v]"
java.lang.NullPointerException
 moreover have "[((\λx . \∀ p . p \→ p),y^P) in v]"
 apply (rule beta_C_meta_1[equiv_rl])
 apply show_proper
 by PLM_solver
 ultimately have "[\Aφ y x & (\∀
java.lang.NullPointerException
java.lang.NullPointerException
 hence "[\∃ y . \Aφ y x & (\open
java.lang.NullPointerException
 by (rule existential)
java.lang.NullPointerException
 using nec_russell_axiom[equiv_rl]
 SimpleExOrEnc.intros by aud bac w
 qed

 lemma exists_unique_actual_equiv:
 "[(bold🚫
 proof (rule "\≡I"; rule CP)
 fix x v
 let ?φ = "λ y x. y = x & ψ (x^P)"
 assume "[\∃!y. \A?φ y x in v]"
 hence "[\∃α. \A?φ α x & (\∀β. \A?φ β x \→ β = α) in v]"
 unfolding exists_unique_def by simp
 then obtain α where
 "[\A?φ α x & (\∀β. \A?φ β x \→ β = α) in v]"
 by (rule Instantiate)
 hence "[\A(α = x & ψ (x^P)) in v]"
 using "&E" by blast
 thus "[\A(ψ (x^P)) in v]"
 using Act_Basic_2[equiv_lr] "&E" by blast
 next
 fix x v
 let ?φ = "λ y x. y = x & ψ (x^P)"
 assume 1: "[\Aψ (x^P) in v]"
 have "[x = x in v]"
 using id_eq_1[where 'a=ν] by simp
 hence "[\A(x = x) in v]"
 using id_act_3[equiv_lr] by fast
 hence "[\A(x = x & ψ (x^P)) in v]"
 using 1 Act_Basic_2[equiv_rl] "&I" by blast
 hence "[\A?φ x x in v]"
 by simp
 moreover have "[\∀β. \A?φ β x \→ β = x in v]"
 proof (rule "\∀I"; rule CP)
 fix β
 assume "[\A?φ β
java.lang.NullPointerException
 using Act_Basic_2[equiv_lr] "&E" by fast
 thus "[β = x in v]" using id_act_3[equiv_rl] by fast
 qed
 ultimately have "[\A?φ x x & (\∀β. \<And <>y
 using "&I" by fast
 hence "[\∃α. \A?φ α x & (\∀β. \A?φ β x \→ β = α) in v]"
 by (rule existential)
 thus "[\∃!y. (>x . \< x
 unfolding exists_unique_def by simp
 qed

 ‹Fake $\beta$-Conversion using Description Backdoor›
 ‹
 \label{TAO_Paradox_description_backdoor}
 ›
 
 definition backdoor where
 "backdoor ≡ λ ψ . \λx . ((\λx . \∀ p . p \→ p), \ιy . y = x & ψ (x^P))"
 
 lemma fake_beta:
 assumes "∧ G φ. IsProperInX (λx . (G,\ιy . φ y x))"
 shows "[( r CP)
 proof (rule "\≡I"; rule CP)
 assume "[(backdoor ψ,x^P) in v]"
java.lang.NullPointerException
 using beta_C_meta_1[equiv_lr, OF assms]
 unfolding backdoor_def identity_ν_def by fast
 hence "[\∃!y. \A (y [<>\y >))
 using exe_impl_exists[equiv_lr] by fast
 thus "[\Aψ (x^P) in v]"
 using exists_unique_actual_equiv[equiv_lr] by blast
 next
 assume "[\Aψ (x^P) in v]"
 hence "[\∃!y. \A (y = x & ψ (x^P)) in v]"
 using exists_unique_actual_equiv[equiv_rl] by blast
 hence "[(\λx. \∀p. p \→ p,\ιy. y = x & ψ (x^P)) in v]"
 using exe_impl_exists[equiv_rl] by fast
 thus "[(backdoor ψ,x^P) in v]"
 using beta_C_meta_1[equiv_rl, OF assms]
 unfolding backdoor_def unfolding identity_ν_def by fast
 qed

 lemma fake_beta_act:
 assumes "∧ G φ. IsProperInX (λx . (G,\ιy . φ y x))"
 shows "[(backdoor (λ x . ψ x), x^P) \≡ ψ (x^P) in dw]"
 using fake_beta[OF assms]
 logic_actual[necessitation_averse_axiom_instance]
 intro_elim_6_e by blast

 ‹

 \  \<nudef
 \label{TAO_Paradox_russell-paradox}
 ›
 
 lemma paradox:
 assumes "∧ G φexi>!y.\<^><
 shows "False"
 proof -
 obtain K where K_def:
 "K = backdoor (λ x . \∃ F . {x,F} & \¬(F,x))" by auto
 have "[\∃x. (A!,x^P) & (\∀F. {x^P,F} \≡ (F = K)) in dw]"
 using A_objects[axiom_instance] by fast
 then obtain x where x_prop:
java.lang.NullPointerException
 by (rule Instantiate)
 {
 assume "[(K,x^P) in dw]"
java.lang.NullPointerException
 unfolding K_def using fake_beta_act[OF assms, equiv_lr]
 by blast
 then obtain F where F_def:
 "[nex
 hence "[F "
 using x_prop[conj2, THEN "\∀E"[where β=F], equiv_lr]
 "&E" unfolding K_def by blast
 hence "[\¬(K,x^P) in dw]"
 using laxiodeduc,dedu]
 F_def[conj2] by fast
 }
 hence 1: "[\¬(K,x^P) in dw]"
 using reductio_aa_1 by blast
 hence "[\¬(\∃ F . {x^P,F} & \¬
 using fake_beta_act[OF assms,
 THEN oth_classhe "\lparr🚫
 equiv_lr]
 unfolding K_def by blast
 hence "[\∀ F . {x^P,F} \→ (F,x^P) in dw]"
 apply - unfolding exists_def by PLM_solver
 moreover have "[{x^P,K} in dw]"
 using x_prop[conj2, THEN "\∀E"[where β=K], equiv_rl]
 id_eq_1 by blast
 ultimately have "[(K,x^P) in dw]"
 using "\∀E" vdash_properties_10 by blast
 hence "∧φ. [φ in dw]"
 using raa_cor_2 1 by blast
 thus "False" using Semantics.T4 by auto
 qed

 ‹Original Version of the Paradox›

 ‹
 \label{TAO_Paradox_original-paradox}
 Originally the paradox was discovered using the following
 construction based on the comprehension theorem for relations
 without the explicit construction of the description backdoor
 and the resulting fake-‹β›-conversion.
 ›
 
 lemma assumes "∧ G φ. IsProperInX (λx . (G,\ιy . φ y x))"
java.lang.NullPointerException
 proof (rule "\∀I")
 fix H
java.lang.NullPointerException
 obtain φ where φ_def: "φ = (λ y x . (y^P) = x & {x,H})" by auto
 have "[\∃F. \◻( backdoor_def unfolding identity_ν_def by fast
 using relations_1[OF assms] by simp
 hence 1: "[\∃F. \◻(\∀x. (F,x^P) \≡ (\∃!y . \Aφ y (x^P))) in v]"
 apply - apply (PLM_subst_method
 "λ x . (?G,\ιy . φ y (x^P))" "λ x . (\∃!y. \Aφ
 using exe_impl_exists by auto
java.lang.NullPointerException
 by (rule Instantiate)
 moreover have 2: "∧ v x . [(\∃!y . \Aφ y (x^P)) \≡ {x^P,H} in v]"
 proof (rule "\≡I"; rule CP)
 fix x v
 assume "[\∃!y. \Aφ y (x^P) in v]"
 hence "[\∃α. \Aφ α (x^P) lparr>backd (🚫
 unfolding exists_unique_def by simp
 then obtain α where "[\Aφ α (x^P) & (\∀β. \Aφ β (x^P) \→ β = α) in v]"
 by (rule Instantiate)
 hence "[\A(α^P = x^P & {x^P,H}) in v]"
 unfolding φ_def using "&E" by blast
 hence "[\A({x^P,H}) in v]"
 using Act_Basic_2[equiv_lr] "&E" by blast
 thus "[{x^P,H} in v]"
 _10[equi] by ssimsimp
 next
 fix x v
 assume "[{x^P,H} in v]"
 hence 1: "[) in v]"
 using en_eq_10[equiv_rl] by blast
 have "[x = x in v]"
 using id_eq_1[where 'a=ν] by simp
 hence "[\A(x = x) in v]"
 using id_act_3[equiv_lr] by fast
 hence "[\A(x^P = x^P & {x^blast
java.lang.NullPointerException
java.lang.NullPointerException
 unfolding φ_def by simp
 moreover have "[\∀β. \Aφ β (x^P) \→ β = x in v]"
 proof (rule "\∀I"; rule CP)
 fix β
java.lang.NullPointerException
 hence "[\A(β-paradox}
 unfolding φ_def identity_ν_def
java.lang.NullPointerException
 thus "[β = x in v]" using id_act_3[equiv_rl] by fast
 qed
 ultimately have "[paradox:
 using "&I" by fast
 hence "[phi>. IsProperInX (λ>G\<><
 by (rule existential)
 thus "[\∃!y. \Aφ y (x^P) in v]"
 unfolding exists_unique_def by simp
 qed
 have "[\◻(\∀x. (F,x^P) \≡ {x^P,H}) in v]"
 apply (PLM_subst_goal_method
java.lang.NullPointerException
 "λ x . (\∃
 using 2 F_def by auto
 thus "[\∃ F . backdoor (λ\<exists >" by auto
 by (rule existential)
 qed

 
 lemma
 assumes is_propositional: "(∧G φ. IsProperInX (λx. (G,\ιy. φ y x)))"
 and Abs_x: "[(A!,x^P) in v]"
 and Abs_y: "[(A!,y^P) in v]"
 and noteq: "[x \≠ y in v]"
 shows diffprop: "[\∃ F . \¬((F,x^P) \≡ (F,y^P)) in v]"
 proof -
 have "[\∃ F . \¬({x^] by fast
 using noteq unfolding exists_def
 proof (rule reductio_aa_2)
 assume 1: "[then x where x_prop:
 {
 fix F
 have "[({sup>P)∀P,F\<rbrace  \^bold>= K))in dw"
 using 1[THEN "\∀E"] useful_tautologies_1[deduction] by blast
 }
 hence "[\∀F. {x^P,F} \≡ {y^P,F} in v]" by (rule "\∀I")
 thus "[x = y in v]"
 unfolding identity_ν_def
 using ab_obey_1[deduction, deduction]
 Abs_x Abs_y "&I" by blast
 qed
 then obtain H where H_def: "[\¬({x^P, H} \≡ {y^P, H}) in v]"
 by (rule Instantiate)
java.lang.NullPointerException
 apply - by PLM_solver
 have "[\∃
java.lang.NullPointerException
 then obtain F where "[\◻(\∀x. (F,x^P) \≡ {x^P,H}) in v]"
 by (rule Instantiate)
 hence F_prop: "[\∀x. (F,x^P) \≡ {x^P,H} in v]"
 using qml_2[axiom_instance, deduction] by blast
 hence a: "[(F,x^P) \≡ {x^P,H} in v]"
 using "\∀E" by blast
 have b: "[(F,y^P) \≡ {y^P,H} in v]"
 using F_prop "\∀E" by blast
 {
 assume 1: "[{x^P, H} & \¬{y^P, H} in v]"
 hence "[(F,x^P) in v]"
 using a[equiv_rl] "&E" by blast
 moreover have "[\¬(F,y^P) in v]"
 using b[THEN oth_class_taut_5_d[equiv_lr], equiv_rl] 1[conj2] by auto
java.lang.NullPointerException
 by (rule "&I")
 hence "[((F,x^P) & \¬(F,y^P)) \∨ (\¬(F,x^P) & (F,y^P)) in v]"
java.lang.NullPointerException
 hence "[\¬ her F_def
 using oth_class_taut_5_j[equiv_rl] by blast
 }
 moreover {
 assume 1: "[\¬{x^P, H} & {y^P, H} in v]"
 hence "[(F,y^P) in v]"
 using b[equiv_rl] "&E" by blast
java.lang.NullPointerException
 using a[THEN oth_class_taut_5_d[equiv_lr], equiv_rl] 1[conj1] by auto
 ultimately have "[\¬(F,x^P) K in dw
 using "&I" by blast
java.lang.NullPointerException
 using "\∨I" by blast
 hence "[\¬((F,x^P) >&E" un K_def by bl
 using oth_class_taut_5_j[equiv_rl] by blast
 }
java.lang.NullPointerException
 using 2 intro_elim_4_b reductio_aa_1 by blast
java.lang.NullPointerException
 by (rule existential)
 qed
 
 lemma original_paradox:
 assumes is_propositional: "(\<>G
 shows "False"
 proof -
 fix v
 have "[\∃x y. (A!,x^P) & (A!,y^P) & x \≠ y & (\∀F. (F,x^P) \≡}
 using aclassical2 by auto
 then obtain x where
 "[\∃ y. (A!,x^P) & (A!,y^P) by bl
  (rul Instantia)
 then obtain y where xy_def:
java.lang.NullPointerException
 by (rule Instantiate)
 have "[[equ],
 using diffprop[OF assms, OF xy_def[conj1,conj1,conj1],
 OF xy_def[conj1,conj1,conj2],
 OF xy_def[conj1,conj2]]
 by auto
 then obtain F where "[K_def by blast
 by (rule Instantiate)
 moreover have "[(^bold>\>\<equiv "
 using xy_def[conj2] by (rule "\∀E")
 ultimately have "∧φ.[φ in v]"
 using PLM.raa_cor_2 by blast
 thus "False"
 using Semantics.T4 by auto
 qed

(*<*)
endlbracein"
(*>*)

end