On the Failure of BD- N Robert S. Lubarsky Florida Atlantic - PowerPoint PPT Presentation

Introduction Topological Example Application Realizability Models Questions References On the Failure of BD- N Robert S. Lubarsky Florida Atlantic University Constructive Mathematics: Proofs and Computation Fraueninsel, Chiemsee June 7-11, 2010 Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Introduction Definition A subset A of N is pseudo-bounded if every sequence ( a n ) of members of A is eventually bounded by the identity function: ∃ N ∀ n > N a n < n (equivalently, a n ≤ n ). Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Introduction Definition A subset A of N is pseudo-bounded if every sequence ( a n ) of members of A is eventually bounded by the identity function: ∃ N ∀ n > N a n < n (equivalently, a n ≤ n ). Example Any bounded set. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Introduction Definition A subset A of N is pseudo-bounded if every sequence ( a n ) of members of A is eventually bounded by the identity function: ∃ N ∀ n > N a n < n (equivalently, a n ≤ n ). Example Any bounded set. BD- N : Every countable pseudo-bounded set is bounded. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Introduction Definition A subset A of N is pseudo-bounded if every sequence ( a n ) of members of A is eventually bounded by the identity function: ∃ N ∀ n > N a n < n (equivalently, a n ≤ n ). Example Any bounded set. BD- N : Every countable pseudo-bounded set is bounded. BD- N is true classically, intuitionistically, computably. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Introduction Definition A subset A of N is pseudo-bounded if every sequence ( a n ) of members of A is eventually bounded by the identity function: ∃ N ∀ n > N a n < n (equivalently, a n ≤ n ). Example Any bounded set. BD- N : Every countable pseudo-bounded set is bounded. BD- N is true classically, intuitionistically, computably. Question: How could it fail? Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References A topological counter-example Let T be { f : ω → ω | range( f ) is finite } . Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References A topological counter-example Let T be { f : ω → ω | range( f ) is finite } . A basic open set p is given by an unbounded sequence g p of integers, with a designated integer stem ( p ), beyond which g p is non-decreasing. f ∈ p if f ( n ) = g p ( n ) for n < stem ( p ) and f ( n ) ≤ g p ( n ) otherwise. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References A topological counter-example Let T be { f : ω → ω | range( f ) is finite } . A basic open set p is given by an unbounded sequence g p of integers, with a designated integer stem ( p ), beyond which g p is non-decreasing. f ∈ p if f ( n ) = g p ( n ) for n < stem ( p ) and f ( n ) ≤ g p ( n ) otherwise. Without loss of generality, g p ( stemp ) ≥ max { g p ( i ) | i < stem ( p ) } . Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References A topological counter-example Let T be { f : ω → ω | range( f ) is finite } . A basic open set p is given by an unbounded sequence g p of integers, with a designated integer stem ( p ), beyond which g p is non-decreasing. f ∈ p if f ( n ) = g p ( n ) for n < stem ( p ) and f ( n ) ≤ g p ( n ) otherwise. Without loss of generality, g p ( stemp ) ≥ max { g p ( i ) | i < stem ( p ) } . Let G be the canonical generic: p � G ( n ) = x iff n < stem ( p ) and g p ( n ) = x . Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References A topological counter-example Let T be { f : ω → ω | range( f ) is finite } . A basic open set p is given by an unbounded sequence g p of integers, with a designated integer stem ( p ), beyond which g p is non-decreasing. f ∈ p if f ( n ) = g p ( n ) for n < stem ( p ) and f ( n ) ≤ g p ( n ) otherwise. Without loss of generality, g p ( stemp ) ≥ max { g p ( i ) | i < stem ( p ) } . Let G be the canonical generic: p � G ( n ) = x iff n < stem ( p ) and g p ( n ) = x . Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N A is unbounded: ∀ N ∃ i ∈ A i > N Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N A is unbounded: ∀ N ∃ i ∈ A i > N Notice that if A ⊆ N is countable and pseudo-bounded then it is not unbounded. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N A is unbounded: ∀ N ∃ i ∈ A i > N Notice that if A ⊆ N is countable and pseudo-bounded then it is not unbounded. What if A is not assume to be countable? Then DC (even CC) + A pseudo-bounded implies A is not unbounded. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N A is unbounded: ∀ N ∃ i ∈ A i > N Notice that if A ⊆ N is countable and pseudo-bounded then it is not unbounded. What if A is not assume to be countable? Then DC (even CC) + A pseudo-bounded implies A is not unbounded. Question: Is there an example of A pseudo-bounded and yet unbounded? Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References DC and boundedness Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. A is bounded: ∃ N ∀ i ∈ A i < N A is not bounded: ¬∃ N ∀ i ∈ A i < N A is unbounded: ∀ N ∃ i ∈ A i > N Notice that if A ⊆ N is countable and pseudo-bounded then it is not unbounded. What if A is not assume to be countable? Then DC (even CC) + A pseudo-bounded implies A is not unbounded. Question: Is there an example of A pseudo-bounded and yet unbounded? Conjecture: In the topological model over the space of unbounded sets of naturals, the generic is pseudo-bounded and unbounded. Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N

Introduction Topological Example Application Realizability Models Questions References Proof Theorem T � rng ( G ) is countable, pseudo-bounded, but not bounded. Also, T � DC. The proof that rng(G) is pseudo-bounded depends crucially on the following Lemma Let p be an open set forcing “ t ∈ rng ( G )” , and I an integer such that max n < stem ( p ) g p ( n ) ≤ I ≤ g p ( stem ( p )) . Then there is a q extending p with the same stem and g q ( stem ( q )) ≥ I forcing “ t ≤ I ” . Robert S. Lubarsky Florida Atlantic University On the Failure of BD- N