An impredicative framework for Freges Basic Law V Giovanni M. Martino - - PowerPoint PPT Presentation

Too Big (has) to Fail A new BLV-Theory Limitations (without) size

An impredicative framework for Frege’s Basic Law V

Giovanni M. Martino1

Vita-Salute San Raffaele University, Milan giovanni.martino3@outlook.it

April 25, 2019 PhDs in Logic XI, Bern

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size

Outline

• 1. Too Big (has) to Fail
• 2. A (new) BLV-Theory
• 3. Limitations (without) size

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size

Outline

• 1. Too Big (has) to Fail
• 2. A (new) BLV-Theory
• 3. Limitations (without) size

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size

Outline

• 1. Too Big (has) to Fail
• 2. A (new) BLV-Theory
• 3. Limitations (without) size

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary I

Let me consider (an axiomatic form of) Frege’s Basic Law V: BLV : ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx)]. wherein, ǫFx is the value range of the concept F. BLV states that for all F and G, the value range of F is the same as the value range of G if and only if F and G are coexstensive. It is easy to show that from BLV it is possible to derive the Russell’s Paradox: ∃X∀x(Xx ↔ ∃X[x = ǫX ∧ ¬Xx]), indeed, is an instance of the impredicative comprehension axiom: CA : ∃X∀x(Xx ← → ϕ(x)) wherein, in a classical impredicative view, ϕ(x) does not contain X free.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary I

Let me consider (an axiomatic form of) Frege’s Basic Law V: BLV : ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx)]. wherein, ǫFx is the value range of the concept F. BLV states that for all F and G, the value range of F is the same as the value range of G if and only if F and G are coexstensive. It is easy to show that from BLV it is possible to derive the Russell’s Paradox: ∃X∀x(Xx ↔ ∃X[x = ǫX ∧ ¬Xx]), indeed, is an instance of the impredicative comprehension axiom: CA : ∃X∀x(Xx ← → ϕ(x)) wherein, in a classical impredicative view, ϕ(x) does not contain X free.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary II

A restriction, (or variation), of BLV due to George Boolos is called New V: ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx) ∨ (Big(F) ∧ Big(G))]. Let V = [x : x = x] be the concept of everything, under which all objects fall: New V states that a concept F is small if V does not go into F. Thus, if ∀x(Fx ↔ Gx), then G is small. According to this, a concept is Big whether is equinumerous with the universal concept V and a concept is Small if it is not big.

• Remark. Of course, V is not small.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary II

A restriction, (or variation), of BLV due to George Boolos is called New V: ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx) ∨ (Big(F) ∧ Big(G))]. Let V = [x : x = x] be the concept of everything, under which all objects fall: New V states that a concept F is small if V does not go into F. Thus, if ∀x(Fx ↔ Gx), then G is small. According to this, a concept is Big whether is equinumerous with the universal concept V and a concept is Small if it is not big.

• Remark. Of course, V is not small.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary II

A restriction, (or variation), of BLV due to George Boolos is called New V: ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx) ∨ (Big(F) ∧ Big(G))]. Let V = [x : x = x] be the concept of everything, under which all objects fall: New V states that a concept F is small if V does not go into F. Thus, if ∀x(Fx ↔ Gx), then G is small. According to this, a concept is Big whether is equinumerous with the universal concept V and a concept is Small if it is not big.

• Remark. Of course, V is not small.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Preliminary II

A restriction, (or variation), of BLV due to George Boolos is called New V: ∀F∀G[ǫFx = ǫGx ← → ∀x(Fx ↔ Gx) ∨ (Big(F) ∧ Big(G))]. Let V = [x : x = x] be the concept of everything, under which all objects fall: New V states that a concept F is small if V does not go into F. Thus, if ∀x(Fx ↔ Gx), then G is small. According to this, a concept is Big whether is equinumerous with the universal concept V and a concept is Small if it is not big.

• Remark. Of course, V is not small.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Stairway to (Set Theory?)

Boolos’s NewV is a neo-fregean account of the limitation of size conception in set theory. Indeed, a set is defined as: Set(x) ↔ ∃F[x = ǫF ∧ ¬Big(F)], namely, a set is the extension of a small property and the membership relation is defined in terms of ǫ-operator: x ∈ y ↔ ∃G[Gx ∧ y = ǫG].

• Claim. New V is consistent. NewV entails the second-order extensionality,

separation, empty set, pairing, and replacement axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Stairway to (Set Theory?)

Boolos’s NewV is a neo-fregean account of the limitation of size conception in set theory. Indeed, a set is defined as: Set(x) ↔ ∃F[x = ǫF ∧ ¬Big(F)], namely, a set is the extension of a small property and the membership relation is defined in terms of ǫ-operator: x ∈ y ↔ ∃G[Gx ∧ y = ǫG].

• Claim. New V is consistent. NewV entails the second-order extensionality,

separation, empty set, pairing, and replacement axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Stairway to (Set Theory?)

Boolos’s NewV is a neo-fregean account of the limitation of size conception in set theory. Indeed, a set is defined as: Set(x) ↔ ∃F[x = ǫF ∧ ¬Big(F)], namely, a set is the extension of a small property and the membership relation is defined in terms of ǫ-operator: x ∈ y ↔ ∃G[Gx ∧ y = ǫG].

• Claim. New V is consistent. NewV entails the second-order extensionality,

separation, empty set, pairing, and replacement axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Rise and Fall I

A model for NewV is M = κ ∪ {⊕}, I, wherein, κ is an infinite cardinal and ⊕ is an arbitrary set for all Bad extensions. According to Uzquiano and Jané 2004, M satisfies the axiom of infinity iff k is uncountable, the axiom of union iff κ, the axiom of power set iff κ is a strong limit.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Rise and Fall I

A model for NewV is M = κ ∪ {⊕}, I, wherein, κ is an infinite cardinal and ⊕ is an arbitrary set for all Bad extensions. According to Uzquiano and Jané 2004, M satisfies the axiom of infinity iff k is uncountable, the axiom of union iff κ, the axiom of power set iff κ is a strong limit.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Rise and Fall II

Quoting Boolos by the way: «limitation of size (in either version) is not a natural view, for one would come to entertain it only after one’s preconceptions had been sophisticated by knowledge of the set theoretic antinomies, including not just Russell’s paradox, but those of Cantor and Burali–Forti as well». Indeed, Russell’s set is big, Universal Set V is Big and, of course, the set of all ordinals is Big – otherwise the Burali-Forti Paradox can be derived from NewV. Moreover, when the ZF-membership relation ∈ is defined by ǫ operator, ∈ is non well founded: New V has founded and non well-founded model, i.e., in some model κ satisfies Aczel’s anti-foundation axiom.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Rise and Fall II

Quoting Boolos by the way: «limitation of size (in either version) is not a natural view, for one would come to entertain it only after one’s preconceptions had been sophisticated by knowledge of the set theoretic antinomies, including not just Russell’s paradox, but those of Cantor and Burali–Forti as well». Indeed, Russell’s set is big, Universal Set V is Big and, of course, the set of all ordinals is Big – otherwise the Burali-Forti Paradox can be derived from NewV. Moreover, when the ZF-membership relation ∈ is defined by ǫ operator, ∈ is non well founded: New V has founded and non well-founded model, i.e., in some model κ satisfies Aczel’s anti-foundation axiom.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Rise and Fall II

Quoting Boolos by the way: «limitation of size (in either version) is not a natural view, for one would come to entertain it only after one’s preconceptions had been sophisticated by knowledge of the set theoretic antinomies, including not just Russell’s paradox, but those of Cantor and Burali–Forti as well». Indeed, Russell’s set is big, Universal Set V is Big and, of course, the set of all ordinals is Big – otherwise the Burali-Forti Paradox can be derived from NewV. Moreover, when the ZF-membership relation ∈ is defined by ǫ operator, ∈ is non well founded: New V has founded and non well-founded model, i.e., in some model κ satisfies Aczel’s anti-foundation axiom.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Recap towards a new theory

• 1. Limitation of size conception avoids Russell’s Paradox and

Burali-Forti paradox with the same strategy. However, they are different.

• 2. What we talk about, when we talk about Set and Fregean extension?

Are they the same mathematical objects? Maybe no, any Fregean extension seems non well ordered (from ZF point of view).

• 3. New V fails to entails the axiom of infinity, the union and the power
• set. It is possible to develop with BLV a new account in Set Theory?

Maybe, it is possible to recover at least infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Recap towards a new theory

• 1. Limitation of size conception avoids Russell’s Paradox and

Burali-Forti paradox with the same strategy. However, they are different.

• 2. What we talk about, when we talk about Set and Fregean extension?

Are they the same mathematical objects? Maybe no, any Fregean extension seems non well ordered (from ZF point of view).

• 3. New V fails to entails the axiom of infinity, the union and the power
• set. It is possible to develop with BLV a new account in Set Theory?

Maybe, it is possible to recover at least infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Recap towards a new theory

• 1. Limitation of size conception avoids Russell’s Paradox and

Burali-Forti paradox with the same strategy. However, they are different.

• 2. What we talk about, when we talk about Set and Fregean extension?

Are they the same mathematical objects? Maybe no, any Fregean extension seems non well ordered (from ZF point of view).

• 3. New V fails to entails the axiom of infinity, the union and the power
• set. It is possible to develop with BLV a new account in Set Theory?

Maybe, it is possible to recover at least infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Recap towards a new theory

• 1. Limitation of size conception avoids Russell’s Paradox and

Burali-Forti paradox with the same strategy. However, they are different.

• 2. What we talk about, when we talk about Set and Fregean extension?

Are they the same mathematical objects? Maybe no, any Fregean extension seems non well ordered (from ZF point of view).

• 3. New V fails to entails the axiom of infinity, the union and the power
• set. It is possible to develop with BLV a new account in Set Theory?

Maybe, it is possible to recover at least infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The limitation of size view

Recap towards a new theory

• 1. Limitation of size conception avoids Russell’s Paradox and

Burali-Forti paradox with the same strategy. However, they are different.

• 2. What we talk about, when we talk about Set and Fregean extension?

Are they the same mathematical objects? Maybe no, any Fregean extension seems non well ordered (from ZF point of view).

• 3. New V fails to entails the axiom of infinity, the union and the power
• set. It is possible to develop with BLV a new account in Set Theory?

Maybe, it is possible to recover at least infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

The Theory

Let me consider a second order language L2K plus a unary function symbol ǫ that attaching to second-order Fx yields a first-order term, i.e. VR-term, ǫFx. Let me call this theory TK. The theory employs a full impredicative comprehension scheme (CA): ∃X∀x(Xx ← → ϕ(x)), wherein ϕ(x) does not contains X free. TK is augmented by an axiomatic form of Basic Law V (BLV): ∀F∀G[ǫF(x) = ǫG(x) ← → ∀x(F(x) ↔ G(x))], wherein ǫF(x) and ǫG(x) are any L2K variable. According to my semantical approach, also BLV has not syntactical restriction.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Concepts and objects

Let ϑ(x) be a metavariable for any second-order variable, or concept, with at most one free variable, M1 the first-order domain and M2 the second-order domain. I define E(ϑ(x)) ⊆ M1 the extension of the concept, namely, all those

• bject that fall under ϑ. Let V be the universe: I define

A(ϑ(x)) := V − E(ϑ(x)), namely, all those objects that do not fall under ϑ with E ∩ A = ∅ and E ∪ A = ℘(ω). Moreover, A− ⊆ A(ϑ(x)) when E ∩ A = ∅

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Concepts and objects

Let ϑ(x) be a metavariable for any second-order variable, or concept, with at most one free variable, M1 the first-order domain and M2 the second-order domain. I define E(ϑ(x)) ⊆ M1 the extension of the concept, namely, all those

• bject that fall under ϑ. Let V be the universe: I define

A(ϑ(x)) := V − E(ϑ(x)), namely, all those objects that do not fall under ϑ with E ∩ A = ∅ and E ∪ A = ℘(ω). Moreover, A− ⊆ A(ϑ(x)) when E ∩ A = ∅

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Concepts and objects

Let ϑ(x) be a metavariable for any second-order variable, or concept, with at most one free variable, M1 the first-order domain and M2 the second-order domain. I define E(ϑ(x)) ⊆ M1 the extension of the concept, namely, all those

• bject that fall under ϑ. Let V be the universe: I define

A(ϑ(x)) := V − E(ϑ(x)), namely, all those objects that do not fall under ϑ with E ∩ A = ∅ and E ∪ A = ℘(ω). Moreover, A− ⊆ A(ϑ(x)) when E ∩ A = ∅

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Concepts and objects

Let ϑ(x) be a metavariable for any second-order variable, or concept, with at most one free variable, M1 the first-order domain and M2 the second-order domain. I define E(ϑ(x)) ⊆ M1 the extension of the concept, namely, all those

• bject that fall under ϑ. Let V be the universe: I define

A(ϑ(x)) := V − E(ϑ(x)), namely, all those objects that do not fall under ϑ with E ∩ A = ∅ and E ∪ A = ℘(ω). Moreover, A− ⊆ A(ϑ(x)) when E ∩ A = ∅

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Extensions

The abstraction operator ǫ is interpreted by the function π : M2 → M1: however, BLV does not delivers always admissible extensions. χ(ϑ(x)) =

• 1

if x ∈ E(ϑ) if x / ∈ E(ϑ) i.e. x ∈ A(ϑ) In agreement with the full-impredicative view, the quantification in CA is unrestricted: the quantifiers range over the full M2 domain: U | = ∀F n(Fx) if U | = F nx for all F n ∈ M2 and U | = ∃X(Xx) if U | = Xnx for some Xn ∈ M2.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Extensions

The abstraction operator ǫ is interpreted by the function π : M2 → M1: however, BLV does not delivers always admissible extensions. χ(ϑ(x)) =

• 1

if x ∈ E(ϑ) if x / ∈ E(ϑ) i.e. x ∈ A(ϑ) In agreement with the full-impredicative view, the quantification in CA is unrestricted: the quantifiers range over the full M2 domain: U | = ∀F n(Fx) if U | = F nx for all F n ∈ M2 and U | = ∃X(Xx) if U | = Xnx for some Xn ∈ M2.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Extensions

The abstraction operator ǫ is interpreted by the function π : M2 → M1: however, BLV does not delivers always admissible extensions. χ(ϑ(x)) =

• 1

if x ∈ E(ϑ) if x / ∈ E(ϑ) i.e. x ∈ A(ϑ) In agreement with the full-impredicative view, the quantification in CA is unrestricted: the quantifiers range over the full M2 domain: U | = ∀F n(Fx) if U | = F nx for all F n ∈ M2 and U | = ∃X(Xx) if U | = Xnx for some Xn ∈ M2.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Extensions

The abstraction operator ǫ is interpreted by the function π : M2 → M1: however, BLV does not delivers always admissible extensions. χ(ϑ(x)) =

• 1

if x ∈ E(ϑ) if x / ∈ E(ϑ) i.e. x ∈ A(ϑ) In agreement with the full-impredicative view, the quantification in CA is unrestricted: the quantifiers range over the full M2 domain: U | = ∀F n(Fx) if U | = F nx for all F n ∈ M2 and U | = ∃X(Xx) if U | = Xnx for some Xn ∈ M2.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Hierarchy of interpretations

Let me sketch a hierarchy S of temporary interpretation of ϑ(x) based on (E). Let ∆ be a metavariable for any TK-formulas: S0: with M1 = ∅, E is empty because there is no object that falls under any concept. Sn+1: U | = ∆, ∀x ∈ {E(ϑ)} Sσ:

λ<σ E.

Remark 1. At level S1 I may introduce any pure second-order concept, namely, any instance of CA without ǫ. Such instance are interpreted by (E); at level S2 I may introduce ǫ-term, i.e. VR-terms, of the former instances of CA. And so on. Remark 2. The hierarchy is, of course, order-preserving.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Fixed Point

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. Moreover, by poset properties is possibile to define a function φ over M such that: let φ an unary-function and D a domain, if ∀x, y, such that x ≤ y then φ(x) ≤ φ(y), where φ is ordered preserving, φ is called monotone. Let me apply the monotone condition φ over (E): At level 0: E = ∅; At level n + 1: En+1 extends the interpretation of En: if En ≤ En+1, by monotonicity, En ≤ φ(En+1). At the limit stage σ, I have Eσ = Eσ+1; by monotonicity, Eσ = φ(Eσ+1), i.e.

λ<σ E = Eσ+1. According to Moschovakis, φ

has least fixed point property: φ(Eσ+1) = φ(Eσ).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

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Fixed Point

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. Moreover, by poset properties is possibile to define a function φ over M such that: let φ an unary-function and D a domain, if ∀x, y, such that x ≤ y then φ(x) ≤ φ(y), where φ is ordered preserving, φ is called monotone. Let me apply the monotone condition φ over (E): At level 0: E = ∅; At level n + 1: En+1 extends the interpretation of En: if En ≤ En+1, by monotonicity, En ≤ φ(En+1). At the limit stage σ, I have Eσ = Eσ+1; by monotonicity, Eσ = φ(Eσ+1), i.e.

λ<σ E = Eσ+1. According to Moschovakis, φ

has least fixed point property: φ(Eσ+1) = φ(Eσ).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Fixed Point

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. Moreover, by poset properties is possibile to define a function φ over M such that: let φ an unary-function and D a domain, if ∀x, y, such that x ≤ y then φ(x) ≤ φ(y), where φ is ordered preserving, φ is called monotone. Let me apply the monotone condition φ over (E): At level 0: E = ∅; At level n + 1: En+1 extends the interpretation of En: if En ≤ En+1, by monotonicity, En ≤ φ(En+1). At the limit stage σ, I have Eσ = Eσ+1; by monotonicity, Eσ = φ(Eσ+1), i.e.

λ<σ E = Eσ+1. According to Moschovakis, φ

has least fixed point property: φ(Eσ+1) = φ(Eσ).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Fixed Point

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. Moreover, by poset properties is possibile to define a function φ over M such that: let φ an unary-function and D a domain, if ∀x, y, such that x ≤ y then φ(x) ≤ φ(y), where φ is ordered preserving, φ is called monotone. Let me apply the monotone condition φ over (E): At level 0: E = ∅; At level n + 1: En+1 extends the interpretation of En: if En ≤ En+1, by monotonicity, En ≤ φ(En+1). At the limit stage σ, I have Eσ = Eσ+1; by monotonicity, Eσ = φ(Eσ+1), i.e.

λ<σ E = Eσ+1. According to Moschovakis, φ

has least fixed point property: φ(Eσ+1) = φ(Eσ).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Fixed Point

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. Moreover, by poset properties is possibile to define a function φ over M such that: let φ an unary-function and D a domain, if ∀x, y, such that x ≤ y then φ(x) ≤ φ(y), where φ is ordered preserving, φ is called monotone. Let me apply the monotone condition φ over (E): At level 0: E = ∅; At level n + 1: En+1 extends the interpretation of En: if En ≤ En+1, by monotonicity, En ≤ φ(En+1). At the limit stage σ, I have Eσ = Eσ+1; by monotonicity, Eσ = φ(Eσ+1), i.e.

λ<σ E = Eσ+1. According to Moschovakis, φ

has least fixed point property: φ(Eσ+1) = φ(Eσ).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Russell’s paradox

Let x / ∈ x be the Russellian condition R express by the concept: ∃F[y = ˆ x(Fx) ∧ ¬Fx]. If ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A: thus ǫR is an unadmissible VR-term, χ(R) = 0. Let me assume that the function π : R → ǫR, delivers to R his extension ǫR: thus, I may consider the russellian instance: R ∈ ǫR ↔ ¬R ∈ ǫR. By model construction, the paradox is blocked from right to left because χ(R ∈ ǫR) = 0 → χ(¬R ∈ ǫR) = 1. In agreement with χ(ϑ(x)) the Russellian set is not an admissible VR-term (extension).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Russell’s paradox

Let x / ∈ x be the Russellian condition R express by the concept: ∃F[y = ˆ x(Fx) ∧ ¬Fx]. If ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A: thus ǫR is an unadmissible VR-term, χ(R) = 0. Let me assume that the function π : R → ǫR, delivers to R his extension ǫR: thus, I may consider the russellian instance: R ∈ ǫR ↔ ¬R ∈ ǫR. By model construction, the paradox is blocked from right to left because χ(R ∈ ǫR) = 0 → χ(¬R ∈ ǫR) = 1. In agreement with χ(ϑ(x)) the Russellian set is not an admissible VR-term (extension).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Russell’s paradox

Let x / ∈ x be the Russellian condition R express by the concept: ∃F[y = ˆ x(Fx) ∧ ¬Fx]. If ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A: thus ǫR is an unadmissible VR-term, χ(R) = 0. Let me assume that the function π : R → ǫR, delivers to R his extension ǫR: thus, I may consider the russellian instance: R ∈ ǫR ↔ ¬R ∈ ǫR. By model construction, the paradox is blocked from right to left because χ(R ∈ ǫR) = 0 → χ(¬R ∈ ǫR) = 1. In agreement with χ(ϑ(x)) the Russellian set is not an admissible VR-term (extension).

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

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A Model for TK

A model U for TK is the pair ω, I, wherein ω is the set of all natural numbers and I is an interpretation of admissible VR-terms. In order to provide denotations for all these VR-terms, I may arrange all VR-terms and formulas on the basis of their complexity, respectively concerning occurrences of the ǫ−operator and second-order quantifier employing a pair function to fix denotation.

• Remark. ω is the smallest model of TK because TK employs second-order

Peano-Dedekind Axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

A Model for TK

A model U for TK is the pair ω, I, wherein ω is the set of all natural numbers and I is an interpretation of admissible VR-terms. In order to provide denotations for all these VR-terms, I may arrange all VR-terms and formulas on the basis of their complexity, respectively concerning occurrences of the ǫ−operator and second-order quantifier employing a pair function to fix denotation.

• Remark. ω is the smallest model of TK because TK employs second-order

Peano-Dedekind Axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

A Model for TK

A model U for TK is the pair ω, I, wherein ω is the set of all natural numbers and I is an interpretation of admissible VR-terms. In order to provide denotations for all these VR-terms, I may arrange all VR-terms and formulas on the basis of their complexity, respectively concerning occurrences of the ǫ−operator and second-order quantifier employing a pair function to fix denotation.

• Remark. ω is the smallest model of TK because TK employs second-order

Peano-Dedekind Axioms.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Benefits

I may form the concept N(x) =def Pred+(0, x) because only with a predicative fragment I have at least Dedekind-infinitely many M1 individuals that fall under it. If Pred+(y, x) = ∃F∃u(Fu ∧ y = #F ∧ x = #[λz.Fz ∧ z = u]), it easy to show that applying φ to F, F is in the least fixed point of φ. Moreover, TK proofs both U | = ∀x(Hx ↔ x = x), namely, the concept

• f everything is in the fixed point and U |

= ∀x(Hx ↔ x = x). According to this, it is possibile to show a representation of the Fregean universe employing partial orders properties of the model.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Benefits

I may form the concept N(x) =def Pred+(0, x) because only with a predicative fragment I have at least Dedekind-infinitely many M1 individuals that fall under it. If Pred+(y, x) = ∃F∃u(Fu ∧ y = #F ∧ x = #[λz.Fz ∧ z = u]), it easy to show that applying φ to F, F is in the least fixed point of φ. Moreover, TK proofs both U | = ∀x(Hx ↔ x = x), namely, the concept

• f everything is in the fixed point and U |

= ∀x(Hx ↔ x = x). According to this, it is possibile to show a representation of the Fregean universe employing partial orders properties of the model.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Benefits

I may form the concept N(x) =def Pred+(0, x) because only with a predicative fragment I have at least Dedekind-infinitely many M1 individuals that fall under it. If Pred+(y, x) = ∃F∃u(Fu ∧ y = #F ∧ x = #[λz.Fz ∧ z = u]), it easy to show that applying φ to F, F is in the least fixed point of φ. Moreover, TK proofs both U | = ∀x(Hx ↔ x = x), namely, the concept

• f everything is in the fixed point and U |

= ∀x(Hx ↔ x = x). According to this, it is possibile to show a representation of the Fregean universe employing partial orders properties of the model.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

{x = x}

• {x = x}

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size The Strenght of Semantical BLV

Extensions and Poset

Let M = D, ⊆ be a poset where D = ℘(ω) and ⊆ is a relation, reflexive, antisymmetric, and transitive over D. In the corresponding poset, by duality property, I have {x = x}/U is the maximum and the unique maximal element; {x = x}/∅ is the minimum and unique minimal element.

℘(ω) is the set over ranges second-order variables or, better, concepts.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

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Paradoxes I

TK manages to distinguish between Russell’s Paradox and Burali-Forti

• paradox. Indeed, the particular instance of the Russellian condition x ∈ x

leads BLV to inconsistency. For any instance of the type ω ∈ ω or {∅} ∈ {∅}, no problem. In this context, the Russell’s Paradox is a particular instantiation of x ∈ x because it is an self-application from a concept to his extension. Burali-Forti Paradox is different.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes I

TK manages to distinguish between Russell’s Paradox and Burali-Forti

• paradox. Indeed, the particular instance of the Russellian condition x ∈ x

leads BLV to inconsistency. For any instance of the type ω ∈ ω or {∅} ∈ {∅}, no problem. In this context, the Russell’s Paradox is a particular instantiation of x ∈ x because it is an self-application from a concept to his extension. Burali-Forti Paradox is different.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes I

TK manages to distinguish between Russell’s Paradox and Burali-Forti

• paradox. Indeed, the particular instance of the Russellian condition x ∈ x

leads BLV to inconsistency. For any instance of the type ω ∈ ω or {∅} ∈ {∅}, no problem. In this context, the Russell’s Paradox is a particular instantiation of x ∈ x because it is an self-application from a concept to his extension. Burali-Forti Paradox is different.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

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Paradoxes II

The Burali-Forti Paradox states that: 1. Every well ordered set has a unique

• rdinal number; 2. Every segment of ordinals has an ordinal number

which is greater than any ordinal in the segment, and 3. The set α of all

• rdinals in natural order is well ordered.

By (3) and (1), α has an ordinal α. Since α is in α, it follows that α < α by (2), which is a contradiction. In my framework, Burali-Forti arises from the fixed point condition: E(αn) = E(αn+1). Differently, the Russell’s Paradox fails because it fails to belong in his extension: by ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes II

The Burali-Forti Paradox states that: 1. Every well ordered set has a unique

• rdinal number; 2. Every segment of ordinals has an ordinal number

which is greater than any ordinal in the segment, and 3. The set α of all

• rdinals in natural order is well ordered.

By (3) and (1), α has an ordinal α. Since α is in α, it follows that α < α by (2), which is a contradiction. In my framework, Burali-Forti arises from the fixed point condition: E(αn) = E(αn+1). Differently, the Russell’s Paradox fails because it fails to belong in his extension: by ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes II

The Burali-Forti Paradox states that: 1. Every well ordered set has a unique

• rdinal number; 2. Every segment of ordinals has an ordinal number

which is greater than any ordinal in the segment, and 3. The set α of all

• rdinals in natural order is well ordered.

By (3) and (1), α has an ordinal α. Since α is in α, it follows that α < α by (2), which is a contradiction. In my framework, Burali-Forti arises from the fixed point condition: E(αn) = E(αn+1). Differently, the Russell’s Paradox fails because it fails to belong in his extension: by ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes II

The Burali-Forti Paradox states that: 1. Every well ordered set has a unique

• rdinal number; 2. Every segment of ordinals has an ordinal number

which is greater than any ordinal in the segment, and 3. The set α of all

• rdinals in natural order is well ordered.

By (3) and (1), α has an ordinal α. Since α is in α, it follows that α < α by (2), which is a contradiction. In my framework, Burali-Forti arises from the fixed point condition: E(αn) = E(αn+1). Differently, the Russell’s Paradox fails because it fails to belong in his extension: by ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Paradoxes II

The Burali-Forti Paradox states that: 1. Every well ordered set has a unique

• rdinal number; 2. Every segment of ordinals has an ordinal number

which is greater than any ordinal in the segment, and 3. The set α of all

• rdinals in natural order is well ordered.

By (3) and (1), α has an ordinal α. Since α is in α, it follows that α < α by (2), which is a contradiction. In my framework, Burali-Forti arises from the fixed point condition: E(αn) = E(αn+1). Differently, the Russell’s Paradox fails because it fails to belong in his extension: by ǫR ∈ {E ∩ A}, then ǫR ∈ A− ⊆ A.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

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Founded and non well-founded

Let me considerer again the Russellian instance blocked by means of the characteristic function: χ(R ∈ ǫR) = 0 → χ(¬R ∈ ǫR) = 1. According to this, also in this framework, if I define the set-membership ∈ by ǫ-operator, then ∈ is non well ordered. This results is in agreement with Uzquiano and Jané 2004.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Founded and non well-founded

Let me considerer again the Russellian instance blocked by means of the characteristic function: χ(R ∈ ǫR) = 0 → χ(¬R ∈ ǫR) = 1. According to this, also in this framework, if I define the set-membership ∈ by ǫ-operator, then ∈ is non well ordered. This results is in agreement with Uzquiano and Jané 2004.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Stairway to (Set Theory?) – the Revenge

Differently from Boolos’s limitation of size conception, without any syntactical restriction to BLV, BLV entails infinitely many extension. From a ZF point of view, since BLV entails that ∈ is not well founded, BLV manages to recover the axiom of infinity. According to Tarski’s definiton of finiteness: «every non-empty family of subsets of S has a minimal element with respect to inclusion», BLV with ∈ fails this condition. Thus, BLV entails infinity.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Conclusion

Benefits and problems with a semantical approach: Semantical BLV vs New V or, limitation of size view. Paradoxes: it is possible, formally, to distinguish between Russell’s Paradox and Burali-Forti Paradox. Extension and Set Theory: in order to recover Set Theory within this is framework, Burali-Forti Paradox has to be treat.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Thank You! Buon 25 Aprile (Anniversary of the Resistance – liberation from nazifscism)

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Thank You! Buon 25 Aprile (Anniversary of the Resistance – liberation from nazifscism)

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V

Too Big (has) to Fail A new BLV-Theory Limitations (without) size Some remarks on Set and Extensions References

Burgess J. P., Fixing Frege, Princeton: Princeton University Press, 2005. Ferreira F., and Wehmeier K. F., On the consistency of the ∆1

1–CA fragment of

Frege’s Grundgesetze, Journal of Philosophical Logic, 31 (2002) 4, pp. 301-311. Ferreira, Zig Zag and Frege Arithmetic, http://webpages.fc.ul.pt/~fjferreira/Zigzag.pdf Frege, G., Grundgesetze der Arithmetik. Begriffschriftlich abgeleitet, vol. I-II, Jena:

• H. Pohle, 1893-1903 (trans. by P. A. Ebert and M. Rossberg, The Basic Laws of

Arithmetic, Oxford: Oxford University Press, 2013). Heck, R. K., The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik, History and Philosophical Logic, 17 (1996) 4, pp. 209-220 (originally published under the name "Richard G. Heck, Jr"). Moschovakis, Y., Notes on Set Theory, New York: Springer, 2006 (2nd edition). Uzquiano, G., Jané, I., W ell and Non-W ell-Founded Extesnsions, Journal of Philosophical Logic, 33 (2004), pp. 437-465.

Giovanni M. Martino An impredicative framework for Frege’s Basic Law V