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Too Big (has) to Fail A new BLV-Theory Limitations (without) size An impredicative framework for Frege’s Basic Law V Giovanni M. Martino 1 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 The limitation of size view Limitations (without) size Preliminary I Let me consider (an axiomatic form of) Frege’s Basic Law V: BLV : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ x ( Fx ↔ Gx )] . wherein, ǫ Fx 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 = ǫ X ∧ ¬ Xx ]) , indeed, is an instance of the impredicative comprehension axiom: ∃ X ∀ x ( Xx ← → ϕ ( x )) CA : 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 The limitation of size view Limitations (without) size Preliminary I Let me consider (an axiomatic form of) Frege’s Basic Law V: BLV : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ x ( Fx ↔ Gx )] . wherein, ǫ Fx 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 = ǫ X ∧ ¬ Xx ]) , indeed, is an instance of the impredicative comprehension axiom: ∃ X ∀ x ( Xx ← → ϕ ( x )) CA : 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 The limitation of size view Limitations (without) size Preliminary II A restriction, (or variation), of BLV due to George Boolos is called New V : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ 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 The limitation of size view Limitations (without) size Preliminary II A restriction, (or variation), of BLV due to George Boolos is called New V : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ 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 The limitation of size view Limitations (without) size Preliminary II A restriction, (or variation), of BLV due to George Boolos is called New V : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ 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 The limitation of size view Limitations (without) size Preliminary II A restriction, (or variation), of BLV due to George Boolos is called New V : ∀ F ∀ G [ ǫ Fx = ǫ Gx ← → ∀ 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 The limitation of size view Limitations (without) size 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 = ǫ F ∧ ¬ 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 = ǫ G ] . 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 The limitation of size view Limitations (without) size 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 = ǫ F ∧ ¬ 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 = ǫ G ] . 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 The limitation of size view Limitations (without) size 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 = ǫ F ∧ ¬ 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 = ǫ G ] . 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 The limitation of size view Limitations (without) size 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 The limitation of size view Limitations (without) size 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 The limitation of size view Limitations (without) size 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