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Models of set theory in Lukasiewicz logic Zuzana Hanikov a Institute of Computer Science Academy of Sciences of the Czech Republic Prague seminar on non-classical mathematics 11 13 June 2015 (joint work with Petr H ajek) Zuzana

  1. Models of set theory in � Lukasiewicz logic Zuzana Hanikov´ a Institute of Computer Science Academy of Sciences of the Czech Republic Prague seminar on non-classical mathematics 11 – 13 June 2015 (joint work with Petr H´ ajek) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  2. Why fuzzy set theory? try to capture a mathematical world: develop fuzzy mathematics (indicate a direction) study the notion of a set, and rudimentary notions of set theory (some properties may be available on a limited scale; classically equivalent notions need not be available in a weak setting) wider set-theoretic universe: recast the classical universe of sets as a subuniverse of the universe of fuzzy sets Explore the limits of (relative) consistency. (Which logics allow for an interpretation of classical ZF? Which logics give a consistent system?) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  3. Why fuzzy set theory? try to capture a mathematical world: develop fuzzy mathematics (indicate a direction) study the notion of a set, and rudimentary notions of set theory (some properties may be available on a limited scale; classically equivalent notions need not be available in a weak setting) wider set-theoretic universe: recast the classical universe of sets as a subuniverse of the universe of fuzzy sets Explore the limits of (relative) consistency. (Which logics allow for an interpretation of classical ZF? Which logics give a consistent system?) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  4. Why fuzzy set theory? try to capture a mathematical world: develop fuzzy mathematics (indicate a direction) study the notion of a set, and rudimentary notions of set theory (some properties may be available on a limited scale; classically equivalent notions need not be available in a weak setting) wider set-theoretic universe: recast the classical universe of sets as a subuniverse of the universe of fuzzy sets Explore the limits of (relative) consistency. (Which logics allow for an interpretation of classical ZF? Which logics give a consistent system?) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  5. Why fuzzy set theory? try to capture a mathematical world: develop fuzzy mathematics (indicate a direction) study the notion of a set, and rudimentary notions of set theory (some properties may be available on a limited scale; classically equivalent notions need not be available in a weak setting) wider set-theoretic universe: recast the classical universe of sets as a subuniverse of the universe of fuzzy sets Explore the limits of (relative) consistency. (Which logics allow for an interpretation of classical ZF? Which logics give a consistent system?) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  6. Why fuzzy set theory? try to capture a mathematical world: develop fuzzy mathematics (indicate a direction) study the notion of a set, and rudimentary notions of set theory (some properties may be available on a limited scale; classically equivalent notions need not be available in a weak setting) wider set-theoretic universe: recast the classical universe of sets as a subuniverse of the universe of fuzzy sets Explore the limits of (relative) consistency. (Which logics allow for an interpretation of classical ZF? Which logics give a consistent system?) Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  7. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  8. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  9. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  10. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  11. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  12. Programme Work with classical metamathematics. Consider a logic L, magenta weaker than classical logic. (Also, with well-developed algebraic semantics.) Consider an axiomatic set theory T , governed by L. The theory T should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong be consistent (relative to ZF) Between classical and non-classical: classical set-theoretic universe is a sub-universe of the non-classical one Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  13. Plan for talk Logics without the contraction rule 1 � Lukasiewicz logic 2 A set theory can strengthen its logic 3 A -valued universes 4 the theory FST (over � L) 5 generalizations 6 Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  14. A family of substructural logics: FL ew and extensions Consider propositional language F . (FL ew -language: {· , → , ∧ , ∨ , 0 , 1 } .) A logic in a language F is a set of formulas closed under substitution and deduction. “Substructural” — absence of some structural rules (of the Gentzen calculus for INT). In particular, FL ew is contraction free. Structural rules: Γ , ϕ, ψ, ∆ ⇒ χ (e) Γ , ∆ ⇒ χ Γ , ϕ, ϕ, ∆ ⇒ χ (c) (w) Γ , ψ, ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Removal of these rules calls for some changes: splitting of connectives changes to interpretation of a sequent NB: FL ew is equivalent to H¨ ohle’s monoidal logic (ML). Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  15. A family of substructural logics: FL ew and extensions Consider propositional language F . (FL ew -language: {· , → , ∧ , ∨ , 0 , 1 } .) A logic in a language F is a set of formulas closed under substitution and deduction. “Substructural” — absence of some structural rules (of the Gentzen calculus for INT). In particular, FL ew is contraction free. Structural rules: Γ , ϕ, ψ, ∆ ⇒ χ (e) Γ , ∆ ⇒ χ Γ , ϕ, ϕ, ∆ ⇒ χ (c) (w) Γ , ψ, ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Removal of these rules calls for some changes: splitting of connectives changes to interpretation of a sequent NB: FL ew is equivalent to H¨ ohle’s monoidal logic (ML). Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

  16. A family of substructural logics: FL ew and extensions Consider propositional language F . (FL ew -language: {· , → , ∧ , ∨ , 0 , 1 } .) A logic in a language F is a set of formulas closed under substitution and deduction. “Substructural” — absence of some structural rules (of the Gentzen calculus for INT). In particular, FL ew is contraction free. Structural rules: Γ , ϕ, ψ, ∆ ⇒ χ (e) Γ , ∆ ⇒ χ Γ , ϕ, ϕ, ∆ ⇒ χ (c) (w) Γ , ψ, ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Γ , ϕ, ∆ ⇒ χ Removal of these rules calls for some changes: splitting of connectives changes to interpretation of a sequent NB: FL ew is equivalent to H¨ ohle’s monoidal logic (ML). Zuzana Hanikov´ a Models of set theory in � Lukasiewicz logic

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