Paraconsistent First-Order Logic with infinite hierarchy levels of contradiction LP# .Axiomatical system HST# ,as inconsistent generalization of Hrbacek set theory HST. Jaykov Foukzon Israel Institute of Technology jaykovfoukzon@list.ru Abstract: In this paper paraconsistent first-order logic LP# with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth # is proposed. Axiomatical system HST # ,as inconsistent generalization of Hrbacek set theory HST is considered. Content I.Introduction. II.Paraconsistent Logic with n levels of contradiction LPn#. III. Paraconsistent Logic with infinite hierarchy levels of contradiction LP# . IV. Paraconsistent set theory KSth # . V. Generalized Incompleteness Theorems. VI. Set theory HST # . VI.1.Axiomatical system HST # ,as inconsistent generalization of Hrbacek set theory HST. VI.2 The universe of HST # . VI.3 The axioms of the external inconsistent universe. VI.4. Axioms for standard and internal sets. VII.5. Well-founded inconsistent sets. List of designations VCon consistent universum VInc inconsistent universum U complete universum U  VConVInc    relation of the classical consistent equivalence  s  relation of the strong consistent nonclassical equivalence or s-equivalence  s   VCon      ∈  classical consistent membership relation  ∈s  strong consistent membership relation or s-membership relation  ∈s   VCon   ∈   w  relation of the weak equivalence or w-equivalence  w   VCon      w1  relation of the weak inconsistent equivalence order 1 or w1-equivalence  w1  ↔  w  ∧  ≠w   wn  relation of the weak inconsistent equivalence order n or wn-equivalence  ∈w  weak membership relation or w-membership relation  ∈w1  weak inconsistent membership relation order 1 or w1-membership relation w-weak empty set I.Introduction. The real history of non-Aristotelian logic begins on May 18,1910 when N.A. Vasiliev presented to the Kazan University faculty a lecture "On Partial Judgements, the Triangle of Opposition, the Law of Excluded Fourth" [Vasiliev 1910] to satisfy the requirements for obtaining the title of privat-dozent. In this lecture Vasiliev expounded for the first time the key principles of non-Aristotelian, imaginary, logic. In this work he likewise constructed his "imaginary" logic free of the laws of contradiction and excluded middle in the informal, so-to-speak Aristotelian, manner (although imaginary logic is in essense nonAristotelian).Thus the birthday of new logic was exactly fixed in the annals of history. Vasiliev's reform of logic was radical, and he did his best to determine whether it was possible for the new logic with new laws and new subject to imply a new logical Universe. Vasiliev began the modern non-classical revolution in logic, but he certainly did not complete it. The founder of paraconsistent logic, N.A. Vasiliev, stated as a characteristic feature of his logic, three kinds of sentence, i.e. "S is A", "S is not A", "S is and is not A". Thus Vasiliev logic rejected the law of non-contradiction: A ∧ A and the law of excluded middle: A ∨ A.However Vasiliev's logic preserve the law of excluded fourth: A ∨ A ∨A ∧ A. Possible formalized versions of Vasiliev's logic with one level of contradiction LP1 # was proposed by A.I.Arruda [1]. In this paper we proposed paraconsistent first-order logic LP# with infinite hierarchy levels of contradiction. Corresponding paraconsistent set theory KSth # is discussed. The postulates (or their axioms schemata) of Vasiliev-Amida propositional paraconsistent logic VA1 are the following: The language L1 of paraconsistent logic VA1  VA1V has as primitive symbols (i) countable set of a clalassical propositional variables, (ii) countable set V  P ii∈N of a non clalassical propositional variables,(iii) the connectives , ,∧,∨,→ and (iv) the parentheses (,). I. Logical postulates: 1 A → B → A, 2 A → B → A → B → C → A → C, 3 A → B → A ∧ B, 4 A ∧ B → A, 5 A ∧ B → B, 6 A → A ∨ B, 7 B → A ∨ B, 6 A → A ∨ B, 7 B → A ∨ B, 8 A → C → B → C → A ∨ B → C, 9 A ∨ A, 10 B → B → A if B ∉ V. II.Rules of a conclusion: Anrestricted Modus Ponens rule MP : A,A → B  B. Theorem 1.1.[1]. (1) If B ∉ V, then B,B  A; (2) A ↔ A iff A ∉ V; (3) A → A. The postulates (or their axioms schemata) of Vasiliev-Amida propositional paraconsistent logic VA2 are the following: The language L2 of paraconsistent logic VA2  VA2V has as primitive symbols (i) countable set of a clalassical propositional variables, (ii) countable set V  P ii∈N of a non clalassical propositional variables,(iii) the connectives , ,∧,∨,→ and (iv) the parentheses (,). I. Logical postulates: 1 A → B → A, 2 A → B → A → B → C → A → C, 3 A → B → A ∧ B, 4 A ∧ B → A, 5 A ∧ B → B, 6 A → A ∨ B, 7 B → A ∨ B, 6 A → A ∨ B, 7 B → A ∨ B, 8 A → C → B → C → A ∨ B → C, 9 A ∨ A, 10 B → B → A if B ∉ V, 11 Pi ∧ P i iff Pi∈ V, i  1,2, . . . . II.Rules of a conclusion: Anrestricted Modus Ponens rule MP : A,A → B  B. II.Paraconsistent Logic with n levels of contradiction LPn#. Modern formalized versions of Vasiliev's logic with one level of contradiction may be found in Amida [1980], [Puga and Da Costa 1988], Smimov [Smirnov 1987], and [Smimov 1987a, 161-169]. There is also the presentation Smimov given at the International Congress of Logic, Methodology and Philosophy of Science in Uppsala in 1991. Paraconsistent Logic with one levels of contradiction LP1 #. Let us consider now Vasiliev-Amida type paraconsistent logic LP1 #  LP1#V,Δ with one level of contradiction. The postulates (or their axioms schemata) of propositional paraconsistent logic LP1 # are the following: The language L1# of paraconsistent logic LP1#  LP1#V,Δ has as primitive symbols (i) countable set of a clalassical propositional variables, (ii) countable set V  P ii∈N of a non clalassical propositional variables, (iii) the connectives w,s,∧,∨,→ and (iv) the parentheses (,). Remark.2.1.We distinguish a weak negation w and a strong negation s. The definition of formula is the usual. We denote the set of the all formulae of LP1 #V1,Δ by F1#, where V1  V0  V1 and Δ is a given subset of F1#. Here we used the following definitions: V0  V,V1  1 | ∈ V,1   ∧ w. A,B,C, ... will be used as metalanguage variables which indicate formulas of LP1 #V,Δ. We assume through that V1⊂ Δ  F1#. I. Logical postulates: 1 A → B → A, 2 A → B → A → B → C → A → C, 3 A → B → A ∧ B, 4 A ∧ B → A, 5 A ∧ B → B, 6 A → A ∨ B, 7 B → A ∨ B, 8 A → C → B → C → A ∨ B → C, 9 P i ∧ wP i iff Pi∈ V, i  1,2, . . . 10 A ∨ wA iff A ∉ V, 11 B → wB → A if B ∉ V1, 12 A ∨ wA ∨A ∧wA iff A ∈ V1, 13 A ∨ sA if A ∈ F1#, 14 B → sB → A if A,B ∈ F1 #. II.Rules of a conclusion: Restricted Modus Ponens rule MPR : A,A → B  B iff A ∉ Δ. Unrestricted Modus Tollens rules: P → Q,wQ  wP;P → Q,sQ  sP. The rule of a strong contradiction: A ∧ sA  B. III.Quantification Corresponding to the propositional paraconsistent relevant logic LP1 #V,Δ we construct the corresponding paraconsistent relevant first-order predicate calculus LP1 #  LP1 # Ṽ, Δ . The language of the paraconsistent predicate calculus LP1 #, denoted by L1 # , is an extension of the language L1# introduced above, by adding, as usually,for every m, denumerable families of m-ary predicate symbols R1 m,R2 m, . . . ,Rnm, . . . ,and m-ary function symbols f1 m, f2 m, . . . , fnm, . . . , and the universal ∀ and existential ∃ quantifiers. We assume throughout that: the language L1 # contains also (i) the classical numerals 0,1,... ; (ii) countable set Γ of the classical consistent set variables Γ  x,y, z, . . . ; (iii) countable set Γ of the non classical inconsistent set variables Γ  x, y, z, . . . ; (iv) countable set Θ of the classical non-logical constants Θ  a,b,c. . . ; (iv) countable set Θ of the non classical non-logical constants Θ  ă, b, c. . . ; Definition 2.1. An LP1 # wff  (well-formed formula ) is a LP1 #sentence iff it hasn't free variables; a wff Ψ is open if it has free variables. We'll use the slang 'k – place open wff' to mean a wff with k distinct free variables. Definition 2.2. An LP1 # wff  is a classical iff it hasn't non classical variables and non classical constants. Definition 2.3. An LP1 # wff  is a non classical iff it has an non classical variables or non classical constants.We denote the set of the all formulae of LP1 # Ṽ, Δ by F1 # , where Ṽ ⊃ V1 and Δ ⊃ Δ is a given subsets of F1 # .We assume through that Ṽ ⊂ Δ  F1 # . The postulates of LP1 # Ṽ, Δ are those of LP1# Ṽ, Δ (suitably adapted) plus the following: (I)  → x  → ∀xx , (II) ∀xx → y, (III) x → ∃xx, (IV) x →  ∃xx →  , (V) ∀xx1 → ∀xx1, (VI) ∀xx1 → ∃xx1, (VII) ∀xx1 → ∀xx ∧ ∀xwx, (VIII) ∀xx1 → ∃xx ∧ ∃xwx, where we used the following definitions: 0  ,1  w ∧ w and 0  ,1   ∧ w and where the variables x and y and the formulas  and  satisfy the usual definition. From the calculi LP1 # Ṽ, Δ ,one can construct the following predicate calculus with equality. This is done by adding to their languages the binary predicates symbol of strong equality    or  s  and weak equality w  with suitable modifications in the concept of formula, and by adding the following postulates: (IX) ∀xx s x, (X) ∀x∀y x s y1  B , (XI) ∀x∀yx s y → x ↔ y, (XII) ∀x∀y∀zx s y ∧ y s z → x s z, (XIII) ∀y∃xy w x, (XIV) ∀y∃xy w x1, (XV) ∀x∀yx w y → x ↔ y (XVI) ∀x∀y x w y1 ↔ 1x ↔ 1y , (XVII) ∀x∀y∀zx w y ∧ y w z → x w z, (XVIII) ∀x∀y∀z x w y1 ∧ y w z1 → x w z1 , (XIX) ∀x∀y∀zx w y ∧ y s z → x w z, (XX) ∀x∀y∀z x w y1 ∧ y s z → x w z1 , (XXI) ∀x∀y∀zx s y ∧ y w z → x w z, (XXII) ∀x∀y∀z x s y ∧ y w z1 → x w z1 . II.Rules of a conclusion: Restricted Modus Ponens rule MPR : A,A → B  B iff A ∉ Δ. Unrestricted Modus Tollens rules: P → Q,wQ  wP;P → Q,sQ  sP. The rule of a strong contradiction: A ∧ sA  B. Definition 2.4. Classical V-object ICl  ICl Ṽ, Δ is the object such that from any classical formula of the form PICl ∧ wPICl,where PINCl ∉ Δ by using principles as in paraconsistent logical calculas LP1 # Ṽ, Δ using Restricted Modus Ponens rule, one can deduce any formula i.e., classical object ICl is the object which hasn't any inconsistent property with respect to a weak negation w. Definition 2.5. Non classical V-object INCl  INCl Ṽ, Δ of the 1-degree of inconsistency is the object INCl such that: from any non classical formula of the form PINCl ∧ wPINCl,where PINCl ∉ Δ by using principles as in paraconsistent logical calculas LP1 # Ṽ, Δ using Restricted Modus Ponens rule one can't deduce any formula whatsoever i.e., non classical object of the 1-degree of inconsistency is the object INCl which has at least one inconsistent property of the 1-degree with respect to a weak negation w. The simplest example of non classical objects 1-degree inconsistency is inconsistent numbers ă such that ă w 1 ∧ wă w 1, 2.1 or b w 1 ∧ b w 2 . 2.2 Remark.2.2. Note that: (i) formula (2.1) meant that ă w 1 ∈ Ṽ and (ii) formula (2.2) meant that b w 1 ∈ Δ and b w 2 ∈ Δ. Paraconsistent Logic with n levels of contradiction LPn#. Let us consider now paraconsistent logic LPn#  LPn#V,Δ with n levels of contradiction. The postulates (or their axioms schemata) of propositional paraconsistent logic LPn#  LPn# V, Δ are the following: The language Ln# of paraconsistent logic LPn# has as primitive symbols (i) countable set of a clalassical propositional variables, (ii) countable set V  P ii∈N of a non clalassical propositional variables, (iii) the connectives w,s,∧,∨,→ and (iv) the parentheses (,). Remark 2.3.We distinguish a weak negation w and a strong negation s. The definition of formula is the usual. We denote the set of the all formulae of LPn# V, Δ by Fn# where V and Δ is a given subsets of Fn#. We assume through that V ⊂ Δ  Fn#. A,B,C, ... will be used as metalanguage variables which indicate formulas of LPn# V, Δ . Definition 2.6. (i) k stands for k−1 ∧ k−11, where 0  , 1  w ∧ w, 0 ≤ k ≤ n. (ii) the (finite) k-order of the level of a weak consistency (w-consistency) is: k, 0 ≤ k ≤ n. Definition 2.7. (i) k stands for k−1 ∧ k−11, where 0  , 1   ∧ w, 0 ≤ k ≤ n. (ii) the (finite) k-order of the level of a weak inconsistency (w-inconsistency) is: n, 1 ≤ k ≤ n. I. Logical postulates: 1 A → B → A, 2 A → B → A → B → C → A → C, 3 A → B → A ∧ B, 4 A ∧ B → A, 5 A ∧ B → B, 6 A → A ∨ B, 7 B → A ∨ B, 8 A → C → B → C → A ∨ B → C, 9 P ∧ wP iff P ∈ V, 10 Pk iff P ∈ V, 11 A ∨ wA if A ∉ V  k0 n Vk. , 12 A ∨ sA if A ∈ Fn#, 13 B → sB → A if A,B ∈ Fn#, 14 A ∨ wA∨A ∧ wA ∨ A2 n−1 ∨. . .∨Ak ∨. . .∨ An if A ∈ Fn#, 15 B → wB → A if B ∉ V  k0 n Vk. II.Rules of a conclusion: Restricted Modus Ponens rule MPR : A,A → B  B iff A ∉ V. Unrestricted Modus Tollens rule: P → Q,wQ  wP;P → Q,sQ  sP. The rule of a strong contradiction: A ∧ sA  B. III.Quantification Corresponding to the propositional paraconsistent relevant logic LPn# V we construct the corresponding paraconsistent relevant first-order predicate calculus. These new calculus will be denoted by LPn # V . The postulates of LPn # V are those of LPn# V (suitably adapted) plus the following: (I)  → x  → ∀xx , (II) ∀xx → y, (III) x → ∃xx, (IV) x →  ∃xx →  , (V) ∀xxk → ∀xxk,k  1,2, . . . ,n, (VI) ∀xxk → ∃xxk,k  1,2. . . ,n, (VII) ∀xxk → ∀xxk,k  1,2. . . ,n. From the calculus LPn # V ,we can construct the following predicate calculus with equality.This is done by adding to their languages the binary predicates symbol of strong equality    or  s  and weak equality w  with suitable modifications in the concept of formula, and by adding the following postulates: (IX) ∀xx s x, (X) ∀x x s x1  B , (XI) ∀x∀yx s y → x ↔ y, (XII) ∀x∀y∀zx s y ∧ y s z → x s z, (XIII) ∀y∃xx w xk,k  0,1,2, . . . ,n, (XIV) ∀x∀y x w yk ↔ ∀∘kx ↔ ky ,k  1,2, . . . ,n, (XV) ∀x∀y∀z x w yk ∧ y w zk → x w zk ,k  0,1,2, . . . ,n, (XVI ∀x∀y∀z x w yk ∧ y s z → x w zk ,k  0,1,2, . . . ,n, (XVII) ∀x∀y∀z x s y ∧ y w zk → x w zk ,k  0,1,2, . . . ,n, (XVIII) ∀y∃xy w xk,k  0,1,2, . . . ,n. III. Paraconsistent Logic with infinite hierarchy levels of contradiction LP# . The postulates (or their axioms schemata) of propositional paraconsistent logic LP#  LP# V, Δ are the following: The language L# of paraconsistent logic LP# has as primitive symbols (i) countable set of a clalassical propositional variables, (ii) countable set V  P ii∈N of a non clalassical propositional variables, (iii) the connectives w,s,∧,∨,→ and (iv) the parentheses (,). Remark.3.1.We distinguish a weak negation w and a strong negation s. The definition of formula is the usual. We denote the set of the all formulae of LP# V, Δ by F# where V and Δ is a given subsets of F# . We assume through that V ⊂ Δ  F# . A,B,C, ... will be used as metalanguage variables which indicate formulas of LP# V, Δ . Definition 3.1. (i) n stands for n−1 ∧ n−11, where 0  ,1  w ∧ w, 1 ≤ n  . (ii)  stands for ∀nn . (iii) the finite n-order of the level of a weak consistency (w-consistency) is: 0  ,n, 1 ≤ n  . (iv) the ifinite -order of level of a weak consistency (w-consistency) is : . Definition 3.2. (i) n stands for n−1 ∧ n−10, where 0   ∧ w, 1 ≤ n  . (ii)  stands for ∀nn . (iii) the finite n-order of the level of a weak inconsistency (w-inconsistency) is: n, 1 ≤ n  . (iv) the ifinite -order of the level of a weak inconsistency (w-inconsistency) is: . I. Logical postulates: 1 A → B → A, 2 A → B → A → B → C → A → C, 3 A → B → A ∧ B, 4 A ∧ B → A, 5 A ∧ B → B, 6 A → A ∨ B, 7 B → A ∨ B, 8 A → C → B → C → A ∨ B → C, 9 P i ∧ wP i iff Pi∈ V, i  1,2, . . . , 10 Pi n iff Pi∈ V, i  1,2, . . . ; 1 ≤ n  , 11 A ∨ wA if A ∉ V  k∈N Vk, 12 A ∨ sA if A ∈F# , 14 B → sB → A if A,B ∈ F# , 15 A ∨ wA ∨ A1 ∨ A2 n ∨. . .∨ An if A ∈ F# , 1 ≤ n  , 16 B → wB → A if B ∉ V   k∈N Vk. II.Rules of a conclusion: Restricted Modus Ponens rule MPR : A,A → B  B iff A ∉ V. Unrestricted Modus Tollens rule: P → Q,wQ  wP;P → Q,sQ  sP. The rule of a strong contradiction: A ∧ sA  B. III.Quantification Corresponding to the propositional paraconsistent relevant logic LP# V we construct the corresponding paraconsistent relevant first-order predicate calculus. These new calculus will be denoted by LP # V . The postulates of LP # V are those of LP# V (suitably adapted) plus the following: (I)  → x  → ∀xx , (II) ∀xx → y, (III) x → ∃xx, (IV) x →  ∃xx →  , (V) ∀xxn → ∀xxn, 1 ≤ n  , (VI) ∀xxn → ∃xxn, 1 ≤ n  , (VII) ∀xxn → ∀xxn, 1 ≤ n  , . . From the calculus LP # V ,we can construct the following predicate calculus with equality.This is done by adding to their languages the binary predicates symbol of strong equality    or  s  and weak equality w  with suitable modifications in the concept of formula, and by adding the following postulates: (IX) ∀xx s x, (X) ∀x x s x1  B , (XI) ∀x∀yx s y → x ↔ y, (XII) ∀x∀y∀zx s y ∧ y s z → x s z, (XIII) ∀y∃xx w xn, 0 ≤ n  , (XIV) ∀x∀y x w yn ↔ ∀∘nx ↔ ny , 1 ≤ n  , (XV) ∀x∀y∀z x w yn ∧ y w zn → x w zn , 0 ≤ n  , (XVI) ∀x∀y∀z x w yn ∧ y s z → x w zn , 0 ≤ n  , (XVII) ∀x∀y∀z x s y ∧ y w zn → x w zn , 0 ≤ n  , (XVIII) ∀y∃xy w xn, 0 ≤ n  . IV. Paraconsistent set theory KSth # Cantor's "naive" set theory KSth was based mainly on two fundamental principles: the postulate of extensionality (if the sets x and y have the same elements, then they are equal), and the postulate of comprehension or separation (every property determines a set, composed of the objects that have this property). The latter postulate, in the standard (first-order) language of set theory, becomes the following schema of formulas: ∃y∀xx ∈ y ↔ Fx,y. 4.1 Now, it is enough to replaces the formula Fx,y in (4.1) by x ∉ x to derive Russell's paradox. That is, the principle of comprehension (4.1) entails an inconsistency. Thus, if one adds (4.1) to classical first-order logic, conceived as the logic of a set-theoretic language, a trivial theory is obtained. Remark.4.1.We distinguish a weakly inconsisten membership relation ∘ ∈w ∘ and a strongly consisten membership relation∘ ∈s ∘. Definition 4.1. (i) the minimal order of the level of a weak consistency (w-consistency) is: 1  0 ∧ w0 ∧ w0,0    x ∈w y; (ii) the minimal order of the level of a weak inconsistency (w -inconsistency) is: 1  0 ∧ w0,0    x ∈w y. Definition 4.2.(i) x ∈w,n y is to stands for x ∈w yn and is to mean "x is a weakly consistent member of y of the n-order (of the n-level) of w-consistency ". (ii) x ∈w,n y is to stands for x ∈w yn and is to mean "x is a weakly inconsistent member of y of the n-order (of the n-level) of w-inconsistency ". Definition 4.1. An LP1 # wff  is a w-wff iff it does not contain the connective: w. We now replace the formula (4.1) by formulae ∃y∀xx ∈w,n y ↔ Fx,y , n  0,1,2, . . . 4.2 and ∃y∀xx ∈w,n y ↔ Fx,y, n  0,1,2, . . . . 4.3 Theorem 4.1. (1) The collections n  ∀xx ∈w,n n↔wx ∈w,n x is contradictory of the n  1-order of w-inconsistency. (2) The collections n  ∀xx ∈w,n n↔wx ∈w,n x is contradictory of the n  1-order of w-inconsistency. Theorem 4.2. (1) The collection   ∀x∀nx ∈w,n ↔wx ∈w,n x is contradictory of the   1-order of w-inconsistency. (2) The collection   ∀x∀nx ∈w,n ↔wx ∈w,n x is contradictory of the   1-order of w-inconsistency. The standard non-classical response to these paradoxes is to find fault with the logical and deduction principles involved in the deduction. Most standard approaches to the paradoxes take them to be important lessons in the behaviour of a Boolean negation. However if you wish to define negation non-classically, there are many options available.You can define negation inferentially, taking A to mean that if A, then something absurd follows,or it can be defined by way of the equivalence between the truth of ~A and the falsity of A, and allowing truth and falsity to have rather more independence from one another than is usually taken to be the case: say, allowing statements to be neither true nor false, or both true and false. The former account takes truth as primary, and defines negation in terms of a rejected proposition and implication. For example, one can to define a strong negation ~sA non-classically [16]: ~sA  A → ∀x∀yx ∈w y ∧ x s y. 4.4 Theorem 4.3. The collection ~s such that x ∈w ~s↔~sx ∈w x i.e., ~s  x~sx ∈w x is contradictory. Proof. Replace Fx,y in the axiom schema of abstraction (4.2) in the definition of collection by ~sx ∈w x, so that the implicit definition of ~s becomes x ∈w ~s↔~sx ∈w x. 4.5 Instantiating in (4.5) x by ~s then by unrestricted modus pones MP, we obtain: (1)  ~s∈w~s↔ ~s~s∈w~s . By unrestricted modus pones MP one obtain the contradiction (2)  ~s∈w~s∧~s~s∈w~s . Thus, if we adds (4.2)-(4.3) to first-order logic LP # V, Δ , conceived as the logic of a set-theoretic language with suitable adapted V and Δ a nontrivial paraconsistent set theory KSth # is obtained. V. Generalized Incompleteness Theorems. 5.1. Let Th be some fixed, but unspecified, paraconsistent, i.e. inconsistent but nontrivial formal theory and in these case we wrote PTh or PconPTh instead Th. For later convenience, we assume that the encoding is done in some fixed consistent formal theory S and that PTh contains S. We do not specify S - it is usually taken to be a formal system of arithmetic, although a weak set theory is often more convenient. The sense in which S is contained in PTh is better exemplified than explained: If S is a formal system of arithmetic and PTh is, say, ZFn, 1  n   or ZFC# then PTh contains S in the sense that there is a well-known embedding, or interpretation, of S in PTh.Since encoding is to take place in S, it will have to have a large supply of constants and closed terms to be used as codes. (E.g. in formal arithmetic, one has 0,1,... .) S will also have certain function symbols to be described shortly. To each formula , of the language of PTh is assigned a closed term, c, called the code of . [N.B. If x is a formula with free variable x, then xc is a closed term encoding the formula x with x viewed as a syntactic object and not as a parameter.] Corresponding to the logical connectives and quantifiers are function symbols, neg, imp, etc., such that, for all formulae , : S  negn c  n c, S  impc, c   → c, etc. Of particular importance is the substitution operator sub, represented by the function symbol sub, . For formulae x, terms t with codes tc : S  subxc, tc  tc. 5.1 Iteration of the substitution operator sub allows one to define function symbols sub3, sub4, . . . , such that S  subnx1,x2, . . . ,xnc, t1 c, t2 c, . . . , tn c  t1, t2, . . . , tnc. 5.2 It well known [17] that one can also encode derivations and have a binary relation ProvThx,y (read "x proves y " or "x is a proof of y") such that for closed t1, t2 : S  ProvTht1, t2 iff t1 is the code of a derivation in PTh of the formula with code t2.It follows that PTh   iff S  ProvPTht, c 5.3 for some closed term t. Definition 5.1.Thus one can define PrPThy ↔ ∃xProvPThx,y, 5.4 and therefore one obtain a predicate asserting provability. Remark 5.1. We note that it is not always the case that : PTh   iff S  PrPThc. 5.5 It well known [17] that the above encoding can be carried out in such a way that the following important conditions D1,D2 and D3 are met for all sentences [17]: D1.PTh   implies S  PrPThc, D2.S  PrPThc → PrPThPrPThcc, D3.S  PrPThc ∧ PrPTh → c → PrPThc. 5.6 Generalized Incompleteness Theorems depend on the following. Theorem 5.1. (Diagonalization Lemma). Let x in the language of PTh have only the free variable indicated. Then there is a sentence  such that S   ↔ c. 5.7 Proof. Given x, let x ↔ subx,x be the diagonalization of x. Let m  xc and   m. Then we claim that S   ↔ c.For x in S, we see that  ↔ m ↔ subm,m ↔ subxc,m ↔ mc ↔ c. 5.8 We apply now (5.7) to nPrThx. Theorem 5.2. (Generalized First Incompleteness Theorem).Let (1) PconnTh and (2) Th  ↔ nPrThc. Then (i) Th  , 5.9 (ii) under an additional assumption Th  n. 5.10 Proof. (i) Observe Th  implies Th  PrThc by D1, which implies Th  n, contradicting the paraconsistency of Th. (ii) The additional assumption is a strengthening of the converse to D1, namely Th  PrThc implies Th .We have Th  n,hence Th  nnPrThc so that Th  PrThc and, by the additional assumption,Th  , again contradicting the paraconsistency of Th. Theorem 5.3. (Generalized Second Incompleteness Theorem). Let PconnTh be nPrThn c,where n  A ∧ nA is any convenient n-contradictory statement. Then Th  PconnTh. 5.11 Proof. Let  be as in the statement of Theorem 5.2.. We show: S   ↔ PconnTh. Observe that S   → nPrThc implies S   → nPrThn c, since S   → n implies S  PrTh → n c,by D1, which implies S  PrThn c → PrThc,by D3. But  → nPrThn c is just  → PconnTh and we have proven half of the equivalence. Conversely, by D2,S  PrThc → PrThPrThcc , which implies S  PrThc → → PrThn c,by D1,D3, since  → nPrThc.This yields S  PrTh ∧ n c, by D1,D3, and logic, which implies S  PrThc → PrThn c by D1,D3, and logic. By contraposition, S  nPrThn c → nPrThc,which is S PconnTh → , by definitions. Theorem 5.4. S  PconnTh → Pconn Th  nPconnTh . Proof.By the proof of Theorem 5.3, (i) S  PconnTh → nPrThc, (ii) S  PconnTh ↔ .Using now D2,D3, it follows that S  PconnTh → nPrThPconnTh c,so that S PconnTh → nPrThnPconnTh → n  c 5.12 which gives S  PconnTh → Pconn Th  nPconnTh . Definition 5.2.Define: (i) ProvTh  x,y ↔ ProvThx,y ∧ ∧∀zw ≤ xProvThz,w → y ≠ negnw ∧ w ≠ negny 5.13 (ii) PrTh  y ↔ ∃xProvTh x,y 5.14 and (iii) Pconn  Th ↔ PrTh  n c. 5.15 Theorem 5.5. (Generalized Rossers Theorem).Let (1) PconnTh and (2) Th  ↔ nPrTh  c. Then (i) Th  , 5.16 (ii) Th  n. 5.17 (iii) Th  Pconn Th. 5.18 Proof.(i) By the paraconsistency of Th, ProvTh and ProvTh  binumerate the same relation. Hence D1 holds: Th   Th  PrTh c.Thus, the proof of the first part of the First Incompleteness Theorem yields the result. (ii) This follows from (iii). (iii) Follows immediately from the remarks that Th is paraconsistent and Th  nn. Theorem 5.6. (Generalized Löb's Theorem). Let be (1) PconnTh and (2)  be closed. Then Th  PrThc →  iff Th  . 5.19 Proof. The one direction is obvious. For the other, assume that Th  . Then Th  n is consistent and we may appeal to the Generalized Second Incompleteness Theorem to conclude that Th  n does not yield PconnTh  n, hence not nPrTh → n c. Thus Th  n  nPrThc.Contraposition yields Th  PrThc → . Let be PconnTh.Now we focuses our attention on the following schemata: (I) Generalized Local Reflection Principle RfnTh : PrThc → ,  closed. 5.20 (II) Generalized First Uniform Reflection Principle RFNTh : ∀xPrThxc → ∀xx, x has only x free. 5.21 (III) Generalized Second Uniform Reflection Principle RFN′Th : ∀xPrThxc → xx, x has only x free. 5.22 Theorem 5.7. (Generalized First Incompleteness Theorem).Let be PconnTh.Then for some true, unprovable  Th  PrThc →  5.23 Theorem 5.8. (Generalized Second Incompleteness Theorem).Let be PconnTh.Then for any refutable  Th  PrThc →  5.24 Theorem 5.6 simply yields Th  PrThc →  iff Th , 5.25 VI. Set theory HST# . VI.1.Axiomatical system HST# ,as inconsistent generalization of Hrbacek set theory HST. In this chapter we introduces HST # , inconsistent generalization of Hrbacek set theory HST and describes the basic structure of the HST # set universe. Syntactically, HST # is a theory in the sts-∈s -stw-∈w -language, which contains: (1) a binary consistent predicate of strong or consistent membership ∈s and consistent unary predicate of strong or consistent standardness sts (and strong or consistent equality s of course) as the consistent primary notions and (2) a binary inconsistent predicate of weak or inconsistent membership ∈w and inconsistent unary predicate of weak or inconsistent standardness stw (and weak or inconsistent equality w of course) as the inconsistent primary notions. Formula x ∈w y reads: x weakly belongs to y, or x is an weak element of y, with the usual set theoretic understanding of inconsistent membership. The formula stwx reads: x is a weakly standard, its meaning will be explained below. A sts-∈s -stw-∈w -formula is a formula of the sts-∈s -stw-∈w -language. An ∈w -formula is a formula of the ∈w -language having ∈w as the only atomic predicate. Thus an ∈w -formula is a stw-∈w -formula in which the standardness predicate does not occur. ∈w -formulas are also called weak internal formulas, in opposition to weak external formulas, i.e., those stw-∈w -formulas containing stw. VI.2. The universe of HST# Inconsistent set theory HST # deals with eight major types of sets: (i) strongly external or s-external,(ii) strongly internal or s-internal, (iii) strongly standard or s-standard, (iv) strongly well-founded or s-well-founded,(v) weakly external or w-external,(vi) weakly internal or w-internal, (vii) weakly well-founded or w-well-founded. First of all, strongly standard sets are those consistent sets x which satisfy stsx and weakly standard sets are those inconsistent sets x which satisfy stwx.Strongly internal sets are those consistent sets y which satisfy intsy, where intsy is the formula ∃stsxy ∈s x ≡ ∃xstsx ∧ y ∈s x (saying: y strongly belongs to a strongly standard set), weakly internal sets are those inconsistent sets y which satisfy intwy, where intwy is the formula ∃stwxy ∈w x (saying: y weakly belongs to a weakly standard set). Thus, i Ss  x : stsxs is the class of all consistent standard sets, ii Is  y : intsys  y : ∃stsxy ∈s xs is the class of all consistent internal sets, iii Sw  x : stwxw is the class of all inconsistent standard sets, iv Iw  y : intwyw  y : ∃stwxy ∈w xw is the class of all inconsistent internal sets, v S#  Ss s Sw  x : stsxs s x : stwxw is the class of all consistent and inconsistent standard sets, vi I#  Is s Iw  y : intsys s y : intwyw is the class of all consistent and inconsistent internal sets. The class Is is the source of some typical objects of consistent "nonstandard" mathematics like consistent hyperintegers and consistent hyperreals, the class Iw is the source of some typical objects of inconsistent "nonstandard" mathematics like inconsistent hyperintegers and inconsistent hyperreals [], Blanket agreement 1.1. Thus, internal sets are precisely all sets which are elements of consistent or inconsistent standard sets. This understanding of the notion of internality and the associated notions like I#,∃sts ,∃stw ,∃st# ≡ ∃sts ∨ ∃stw ,∀sts ,∀st# ≡ ∀sts ∧ ∀stw is default throughout this paper. All exceptions (e.g., when IST # is considered) will be explicitly indicated. External sets consistent and inconsistent, are simply all sets in the nonstandard universe of HST # . We shall use H# to denote the class of all consistent and inconsistent external sets. Thus, H# is the "universe of discourse", the universe of all sets considered by the theory, including the class WF# of all well-founded sets. WF# will satisfy all axioms of ZFC# . The class S # of all standard sets {determined by the predicate st, as above) will be shown to be ∈s -∈w -isomorphic to WF# . In a sense, S# is an "isomorphic expansion" of WF# into H# . Given that S # is not transitive, I# arises naturally as the class of all elements of sets in S#. It is viewed as an elementary extension of S# {in ∈s -∈w language), and thereby also of WF# . Finally, H# is a comprehensive universe in which all these classes coexist in a reasonable common set theoretic structure, with ∈s -∈w having the natural meaning in all mentioned universes. VI.3. The axioms of the external inconsistent universe. This group includes the ZFC# Extensionality, Pair, Union, Infinity axioms and the schemata of Separation and Collection (therefore also Replacement, which is a consequence of Collection, as usual) for all sts-∈s stw-∈w -formulas or for all st#-∈# formulas for short. VI.4. Axioms for standard and internal sets Notation 4.1. (1).Let quantifiers ∃sts ,∀sts ,∃stw and ∀stw be shortcuts meaning: there exists a strongly standard..., for all strongly standard,there exists a weakly standard..., for all weakly standard, ..., formally: (i) ∃stsxx means ∃xstsx ∧ x, (ii) ∀stsxx means ∀xstsx  x, (iii) ∃stwxx means ∃xstwx ∧ x, (iv) ∀stwxx means ∀xstwx  x. Quantifiers ∃int and ∀int (meaning there exists an internal ... , for all internal ...) are introduced similarly. If g, is an E-formula then g,st, the relativization of g to S, is the formula obtained by restriction of all quantifiers in g to the class S, so that all occurrences of 3 x ... are changed to 3stx ... while all occurrences of V x ... are changed to ystx .... In other words, g,st says that g is true in S. Rela-tivization g,int, which displays the truth of an e-formula g in the universe 0, is defined similarly: the quantifiers 3, V change to 3int, yint. The following axioms specify the behaviour of standard and internal sets. Notation 4.2.For all sts-∈s stw-∈w -formulas or for all st#-∈# formulas for short. ZFCst# : The collection of all formulas of the form g,st, where g is an estatement which is an axiom of ZFC# . In other words, it is postulated that the universe S # is a ZFC# universe. (Note that the ZFC# axioms are assumed to be formulated as certain closed ∈# formulas in this definition.) This is enough to prove the following statement: Lemma 4.1. (1) Ss ⊆ Is, (2) Sw ⊆ Iw. Proof.(1) See [18] Lemma 1.1.3. (2) Let x ∈w Sw. The formula ∃yx ∈w y is a theorem of ZFC# , therefore ∃yx ∈w ystw that is the formula ∃stwyx ∈w y,is true. In other words, x is an element of a standard set, which means x ∈w Iw. 1.Strong or Consistent Transfer (s-Transfer): ints  sts , where  is an arbitrary closed ∈s -formula containing only consistent standard sets as parameters. To be more exact, Consistent Transfer is the collection of all statements of the form ∀stsx1. . .∀stsxnintsx1, . . . ,xn  stsx1, . . . ,xn 2.Strong Consistent Transitivity of Is : ∀intsx∀yy ∈s x  intsy. 3.Consistent Regularity over Is : For any non empty consistent set X there exists x ∈s X such that x ∩s X ⊆s Is. (The full Regularity of ZFC requires x ∩s X  s.) 4.Consistent Standardization: ∀X∃stwyX ∩s Ss  ∩s Ss). (Such consistent standard set Y, unique by Consistent Transfer and Consistent Extensionality, is sometimes denoted by SsX. ) 5.Weak Transfer (w-Transfer): intw  stw , where  is an arbitrary closed ∈s -∈w -formula containing only consistent and inconsistent standard sets as parameters. To be more exact, Weak Transfer is the collection of all statements of the form ∀stsx1. . .∀stsxn∀stwy1. . .∀stwymintsx1, . . . ,xn;y1, . . . ,ym  stwx1, . . . ,xn;y1, . . . ,ym 6.Weak Transitivity of Iw : ∀intsx∀yy ∈w x  intwy. 7.Weak Regularity over Iw : For any non empty consistent set X there exists x ∈w X such that x ∩w X ⊆w Iw. (The full Regularity of ZFC requires x ∩w X  w.) 8.Strictly Weak Regularity (Strictly w-Regularity): For any non empty inconsistent set X there exists x ∈w X such that x ∩w X w w ∧ x ∩w X w w. 9.Weak Standardization (w-Standardization): ∀X∃stwYX ∩w Sw w Y ∩w Sw. 9.Weak Standardization: ∀X∃stwyX ∩w Sw  ∩w Sw). (Such consistent standard set Y, unique by Consistent Transfer and Consistent Extensionality, is sometimes denoted by SwX. ) Such inconsistent standard set Y, w-unique by w-Transfer and weak Extensionality, is sometimes denoted by SwX. Remark 4.1. (i) w-Transfer can be considered as saying that: Iw, the universe of all inconsistent internal sets, is an elementary extension of Sw in the ∈s -∈w -language. It fo llows, by ZFC#  stw , that the class Iw of all inconsistent internal sets satifies ZFC# (in the ∈s -∈w -language ), in fact, we can replace ZFC#  stw by ZFC#  intw , with relativization to Iw, in the list of HST # axioms. See also Theorem 1.3.9 below. (ii) w-Transitivity of Iw postulates that: inconsistent internal sets to form the basement of the ∈s -∈w -structure of the universe H# . This axiom is very important since it implies that some set operations in Iw retain their sense in the whole universe H# . (iii) w-Regularity over Iw organizes the HST # set universe H# in general case as a sort of hierarchy over the internal universe Iw, in the same way as the w-Regularity axiom organizes the universe in the von Neumann w-hierarchy over the w-empty set w in ZFC# . (iv) Strictly w-Regularity organizes the HST # set universe H# in the von Neumann w-hierarchy over the w-empty set w,but in a strictly inconsistent sense only. (v) w-Standardization postulates that H# does not contain collections of standard sets other than those of the form S ∩w Sw for inconsistent standard set S. Remark 4.2. It well known that the ZFC Regularity fails in H  Hs : the set of all nonstandard Is-natural numbers does not contain an ∈s -minimal element, (see for example [18], Exercise 1.2. 15(3)). In contrast with a classical case, ZFC# w-Regularity valid in H# , but in a strictly inconsistent sense only. For example the set of all nonstandard Iw-natural contain an inconsistent ∈w -minimal element, see [22]-[23]. VII.5. Well-founded inconsistent sets. Now we can introduce the last principal class: well-founded inconsistent sets. Recall the following notions from general inconsistent set theory. Definition 5.1. (i) A binary weak relation w on inconsistent set or inconsistent class X is a strictly well-founded if any nonempty set Y ⊆w X contains consistent w -minimal w-element x∗ ∈w Y, that is there exists x ∈w Y such that no y ∈w Y satisfies y w x. (ii) A binary weak relation w on inconsistent set or inconsistent class X is weakly wellfounded (or w-well-founded) if: (1) w is not a strictly well-founded and (2) any nonempty set Y ⊆w X contains a w -minimal w-element x∗ ∈w Y, that is there exists y ∈w Y satisfies: y w x ∧ x w y, i.e. y w x ∧ y w x. (iii) Inconsistent set or inconsistent class X is w-transitive if any x ∈w X satisfies x ⊆w X, i.e., weak elements of weak elements of X are weak elements of X. (iv) Inconsistent set or inconsistent class X is w-complete if we have y ∈w X whenever y ⊆w x ∈w X, that is a weak subsets of weak elements of X are weak elements of X. (v) Inconsistent set x is a strictly well-founded if there is a w-transitive set X such that x ⊆w X and the restriction ∈w  X is a strictly well-founded weak relation. (vi) Inconsistent set x is w-well-founded if there is a w-transitive set X such that x ⊆w X and the restriction ∈w  X is a w-well-founded weak relation. Remark 5.1. It is known that all sets are well-founded in ZFC by the Regularity axiom. This is not the case in HST : the set ∗N of all Is-natural numbers is ill-founded [18]. Remark 5.2. In contrast with a classical case, all inconsistent sets are w-well-founded in HST # by the Strictly w-Regularity axiom. For example, the set #N  ∗N inc of all Iw-natural numbers is w-well-founded by the Strictly Weak Regularity axiom. Definition 5.2.(HST # ). (i) Let s-wfwx mean that x is a strictly well-founded. We put s-WFw w x : s-wfwxw, the class of all strictly well-founded inconsistent sets and (ii) let w-wfwx mean that x is a w-well-founded. We put w-WFw w x : w-wfwxw, the class of all w-well-founded inconsistent sets. Notation 5.1.We introduce quantifiers ∃s-wfw ,∀s-wfw ,∃w-wfw and ∀w-wfw (meaning: there is a well-founded ... , for any well-founded ... ) and the relativization (1) s-wfw to s-WFw, (2) w-wfw to w-WFw similarly to ∃sts ,∃sts ,sts ,∃stw ,∃stw ,stw in §VII.1.3. In other words, s-wfw says that gj is true in WIF. The main property of the classes s-WFw and w-WFw in HST # is that it admits a definable ∈w -isomorphism w  #w onto the class S of all standard sets. REFERENCES [1] A.I. ARRUDA, 1980.A survey of paraconsistent logic, in A.I. Arruda, R.Chuaqui and N.C.A. Da Costa, (editors) Mathematical logic in Latin America (Amsterdam, North-Holland), 1-41. [2] V.A. BAZHANOV,1988. Nicolai Alexandrovich Vasiliev (1880 1940), Moscow,Nauka. (In Russian). [3] V.A.BAZHANOV, 1990. The fate of one forgotten idea: N.A. Vasiliev and his imaginary logic, Studies in Soviet Thought 39, 333-344. [4] V.A. BAZHANOV ,A.P.YUSHKEVICH, 1992. A.V.Vasiliev as a scientist and a public figure, in A.V. Vasiliev, Nikolai Ivanovich Lobachevskii (1792 1856) (Moscow, Nauka), 221-228. (In Russian). [5] PUGA L.,DA COSTA, N.C.A.1988. On imaginary logic of N.A Vasiliev,Zeitschrift für mathematischen Logik und Grundlagen der Mathematik 34,205-211. Russian translation by V.A.Bazhanov in M.I.Panov (editor),Methodological analysis of the foundations of mathematics (Moscow, Nauka, 1988), 135-142. [6] V.A.SMIRNOV, 1987. Axiomatization of the logical systems of N.A.Vasiliev,in Contemporary logic and methodology of science (Moscow, Nauka),143-151.(In Russian). [7] V.A.SMIRNOV, 1987a.The logical method of analysis in scientific knowledge, Moscow, Nauka. (In Russian). [8] N.A.VASILIEV, 1910. On partial judgements, the triangle of opposition,the law of excluded forth,Record of Studies of Kazan University (October),1-47. (in Russian). Reprinted in (1989), 12-53. [9] N.A.VASILIEV, 1911. Report on academic activities in 1911 -1912. Manuscript.(In Russian). Printed: (1989),149-169. [10] N.A.VASILIEV, 1912(a). Imaginary (non-Aristotelian) Logic, Journal of the Ministry of People's Education (August), 207-246. (In Russian). Reprinted in (1989), 53-94. [11] N.A.VASILIEV, 1912(b). Imaginary logic. Abstract of lectures. (In Russian).Reprinted in (1989), 126-130. [12] N.A.VASILIEV,1912(c). Review of: Encyclopedie der philosophischen Wissenschafien in Verbindung mit W.Windelband herausgeben von A Ruge.I Band: Logic. Verlag von l.C.Mohr. Tübingen, Logos, Book 1/2 (1912-13), 387-389. (In Russian). Reprinted in (1989), 131-134. [13] N.A.VASILIEV,1913. Logic and metalogic, Logos, Book 1/2 (1912-13),53-81. (In Russian). Reprinted in (1989), 94-123. [14] N.A.VASILIEV,1924. Imaginary (non-Aristotelian) logic, G. della Volle (editor),Atti V Congresso Internazionale di Filosofia, Napoli, 5-9 maggio 1924 (Naples,1925; reprinted: Nede ln Iiechtenstein, Kraus Reprint, Ltd.,1968), 107-109. (In English). Russian translation in (1989), 124-126. [15] N.A.VASILIEV,1989. Imaginary logic: Selected works (V.A. Smimov, editor),Moscow, Nauka. (In Russian). [16] A.I. Arruda,Remarks In Da Costa's Paraconsistent Set Theories,Revista colombiana de matematicas.Volume 19 / 1985 / Article. [17] C.Smorynski,HANDBOOK OF MATHEMATICAL LOGIC, Edited by J. Barwise.North-Holland Publishing Company, 1977 http://www.compstat2004.cuni.cz/~krajicek/smorynski.pdf [18] V. Kanovey,M, Reeken, Nonstandard Analysis, Axiomatically, Springer, Aug 11, 2004 Mathematics 408 pp. [19] Da Costa,N.C.A. Paraconsistent mathematics. In: IWorld Congress on Paraconsis tency, 1998, Ghent, Belgium. Frontiers in paraconsistent logic:proceedings.Edited by Batens,D.,Mortensen,C.,Priest,G. Bendegen,J. P. London:King's College Publications,2000.p.165-179. [20] Carvalho,T.F. Sobre o calculo diferencial paraconsistente de da Costa (On da Costa's paraconsistent differential calculus). 2004.200 p. Thesis (Doctorate) Philosophy and Human Sciences Institute, State University of Campinas, 2004. D'Ottaviano,I.M.L.,Carvalho,T.F.Da Costa's Paraconsistent [21] Differential Calculus and a Transference Theorem.2004 ftp://logica.cle.unicamp.br/pub/e-prints/DOttaviano-Carvalho.pdf [22] J.Foukzon, Foundation of paralogical nonstandard analysis and its application to some famous problems of trigonometrical and orthogonal series. Part I,II Volume 2004, Issue 3, March 2004,Pages 343-356.Mathematics Preprint Archive, http://www.sciencedirect.com/preprintarchive http://aux.planetmath.org/files/papers/305/PART.I. PNSA.pdf http://aux.planetmath.org/files/papers/305/PART.II.PNSA.pdf [23] J.Foukzon, Foundation of paralogical nonstandard analysis and its application to some famous problems of trigonometrical and orthogonal series 4ECM Stockholm 2004 Contributed papers. http://www.math.kth.se/4ecm/poster.list.html