Set-theoretical aspects of proof theory via Turing progressions - PowerPoint PPT Presentation

A personal note Proof Theory Turing progressions and ordinal analysis Set-theoretical aspects of proof theory via Turing progressions Joost J. Joosten Universitat de Barcelona Saturday 17-11-2018 Reflections on Set-Theoretic Reflection, Montseny A conference in celebration of Joan Bagaria’s 60th birthday Joost J. Joosten Set theory & proof theory

A personal note Proof Theory Turing progressions and ordinal analysis Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Hilbert: can we safeguard real mathematics using finitistic methods only? Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Hilbert: can we safeguard real mathematics using finitistic methods only? ◮ F ⊢ Con( R )? Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Hilbert: can we safeguard real mathematics using finitistic methods only? ◮ F ⊢ Con( R )? ◮ Gentzen reduces G¨ odel’s negative to an example: Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Hilbert: can we safeguard real mathematics using finitistic methods only? ◮ F ⊢ Con( R )? ◮ Gentzen reduces G¨ odel’s negative to an example: ◮ PRA + TI( ε 0 , Π 0 1 ) ⊢ Con( PA ) Here ε 0 := sup { ω, ω ω , ω ω ω , . . . } ; TI( ε 0 , Π 0 1 ) is the axiom scheme � � ∀ α ∀ β ≺ αϕ ( β ) → ϕ ( α ) → ∀ γϕ ( γ ) with ≺ some natural predicate on the natural numbers that defines a well-order of order-type ε 0 on N . Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } ◮ What is a natural well-order on the natural numbers? Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } ◮ What is a natural well-order on the natural numbers? ◮ Kreisel’s pathological ordering � n < m if ∀ i < max < ( m , n ) ¬ Proof ZFC ( i , � 0 = 1 � ), n ≺ ZFC m = if ∃ i < max < ( m , n ) Proof ZFC ( i , � 0 = 1 � ). m < n Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } ◮ What is a natural well-order on the natural numbers? ◮ Kreisel’s pathological ordering � n < m if ∀ i < max < ( m , n ) ¬ Proof ZFC ( i , � 0 = 1 � ), n ≺ ZFC m = if ∃ i < max < ( m , n ) Proof ZFC ( i , � 0 = 1 � ). m < n ◮ By induction along ≺ ZFC prove ∀ y < x ¬ Proof ZFC ( y , � 0 = 1 � ) Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } ◮ What is a natural well-order on the natural numbers? ◮ Kreisel’s pathological ordering � n < m if ∀ i < max < ( m , n ) ¬ Proof ZFC ( i , � 0 = 1 � ), n ≺ ZFC m = if ∃ i < max < ( m , n ) Proof ZFC ( i , � 0 = 1 � ). m < n ◮ By induction along ≺ ZFC prove ∀ y < x ¬ Proof ZFC ( y , � 0 = 1 � ) ◮ PRA + TI( ≺ ZFC , PRIM) ⊢ Con(ZFC) Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Tentative: | U | Con := min { ot( ≺ ) | PRA + TI( ≺ , PRIM) ⊢ Con( U ) } ◮ What is a natural well-order on the natural numbers? ◮ Kreisel’s pathological ordering � n < m if ∀ i < max < ( m , n ) ¬ Proof ZFC ( i , � 0 = 1 � ), n ≺ ZFC m = if ∃ i < max < ( m , n ) Proof ZFC ( i , � 0 = 1 � ). m < n ◮ By induction along ≺ ZFC prove ∀ y < x ¬ Proof ZFC ( y , � 0 = 1 � ) ◮ PRA + TI( ≺ ZFC , PRIM) ⊢ Con(ZFC) ◮ Other proof theoretical notions | U | sup , | U | Π 0 2 , | U | TI , . . . Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) ◮ Ordinal notation requires small Veblen functions: Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) ◮ Ordinal notation requires small Veblen functions: ◮ ϕ 0 ( α ) := ω α , Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) ◮ Ordinal notation requires small Veblen functions: ◮ ϕ 0 ( α ) := ω α , ◮ ϕ ξ ( α ) := α th simultaneous fixpoint of all the { ϕ ζ } ζ<ξ . Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) ◮ Ordinal notation requires small Veblen functions: ◮ ϕ 0 ( α ) := ω α , ◮ ϕ ξ ( α ) := α th simultaneous fixpoint of all the { ϕ ζ } ζ<ξ . ◮ First Veblen inaccessible is Γ 0 : ∀ α, β ( α, β< Γ 0 → ϕ α ( β ) < Γ 0 ) Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Ramified Analysis (second order arithtmetic) ◮ ATR 0 � �� ◮ ∀ ≺ � wo( ≺ ) → ∃ X ∀ α ∈ field( ≺ ) ∀ n n ∈ X α ↔ ϕ ( n , X <α ) for ϕ arithmetical (or Σ 0 1 ) ◮ Ordinal notation requires small Veblen functions: ◮ ϕ 0 ( α ) := ω α , ◮ ϕ ξ ( α ) := α th simultaneous fixpoint of all the { ϕ ζ } ζ<ξ . ◮ First Veblen inaccessible is Γ 0 : ∀ α, β ( α, β< Γ 0 → ϕ α ( β ) < Γ 0 ) ◮ Essentially, Sch¨ utte, Feferman: | ATR 0 | = Γ 0 Joost J. Joosten Set theory & proof theory

A personal note Foundations and gauging strength Proof Theory Ordinal notation systems Turing progressions and ordinal analysis Fragments of Set Theory ◮ Impredicative notation systems are needed to go substantially beyond Γ 0 Joost J. Joosten Set theory & proof theory