Alpha-Ramsey Theory Timothy Trujillo 2016 Joint Meeting of the - PowerPoint PPT Presentation

Alpha-Ramsey Theory Timothy Trujillo 2016 Joint Meeting of the Intermountain and Rocky Mountain Sections - Colorado Mesa University Timothy Trujillo α -Ramsey Theory 1/20

Overview 1 Nonstandard Analysis 2 The Alpha-Theory 3 Alpha-Trees 4 Alpha-Ramsey Theory 5 Applications Timothy Trujillo α -Ramsey Theory 2/20

Nonstandard analysis (1961) Nonstandard analysis was introduced by Robinson to reintroduced infinitesimals and infinite numbers into analysis. (1961) Using model theory, Robinson gave a rigorous development of the calculus of infinitesimals. • Ultrafilters • Ultrapowers • ∗ -transfer principle (2003) Benci and Di Nasso in [2] have introduced a simplified presentation of nonstandard analysis called the Alpha-Theory. Timothy Trujillo α -Ramsey Theory 3/20

The Alpha-Theory Idea Extend the natural numbers N by adjoining a new number α that behaves like a very large natural number. (1958) Laugwitz [3] adjoin a new symbol Ω and assume that a ‘formula’ is true at Ω if it is true for all sufficiently large natural numbers. (2003) Alpha-Theory approach can be seen as a strengthening of the Ω-Theory. (2003) Alpha-Theory conservatively extends ZFC with five new axioms which describe a new symbol α . Timothy Trujillo α -Ramsey Theory 4/20

Sequences Definition A sequence is any function f whose domain is N . Example ( . 9 , . 99 , . 999 , . 9999 , . . . ) (1 , 1 / 2 , 1 / 3 , 1 / 4 , 1 / 5 , . . . ) ( R , R 2 , R 3 , R 4 , R 5 , . . . ) ( α, α, α, α, α, α, . . . ) Timothy Trujillo α -Ramsey Theory 5/20

Axiom 1, 2 & 3 Axiom (Extension) For all sequences f : N → X there a unique element f [ α ], called the “ideal value of f .” Timothy Trujillo α -Ramsey Theory 6/20

Axiom 1, 2 & 3 Axiom (Extension) For all sequences f : N → X there a unique element f [ α ], called the “ideal value of f .” Axiom (Number) 1 For all n ∈ N the constant sequence ( n , n , n , . . . ) has ideal value n . Timothy Trujillo α -Ramsey Theory 6/20

Axiom 1, 2 & 3 Axiom (Extension) For all sequences f : N → X there a unique element f [ α ], called the “ideal value of f .” Axiom (Number) 1 For all n ∈ N the constant sequence ( n , n , n , . . . ) has ideal value n . 2 The identity sequence (1 , 2 , 3 , 4 , . . . ) has ideal value α �∈ N . Timothy Trujillo α -Ramsey Theory 6/20

Axiom 1, 2 & 3 Axiom (Extension) For all sequences f : N → X there a unique element f [ α ], called the “ideal value of f .” Axiom (Number) 1 For all n ∈ N the constant sequence ( n , n , n , . . . ) has ideal value n . 2 The identity sequence (1 , 2 , 3 , 4 , . . . ) has ideal value α �∈ N . Axiom (Pair) The ideal value of ( { x 0 , y 0 } , { x 1 , y 1 } , { x 2 , y 2 } , { x 3 , y 3 } , . . . ) is { x [ α ] , y [ α ] } . Timothy Trujillo α -Ramsey Theory 6/20

Axiom 4 Axiom (Composition) Let ( x 1 , x 2 , x 3 , . . . ) and ( y 1 , y 2 , y 3 , . . . ) be sequences and f be a function such that ( f ( x 1 ) , f ( x 2 ) , f ( x 3 ) , . . . ) and ( f ( y 1 ) , f ( y 2 ) , f ( y 3 ) , . . . ) are well-defined. Timothy Trujillo α -Ramsey Theory 7/20

Axiom 4 Axiom (Composition) Let ( x 1 , x 2 , x 3 , . . . ) and ( y 1 , y 2 , y 3 , . . . ) be sequences and f be a function such that ( f ( x 1 ) , f ( x 2 ) , f ( x 3 ) , . . . ) and ( f ( y 1 ) , f ( y 2 ) , f ( y 3 ) , . . . ) are well-defined. If x [ α ] = y [ α ] then ( f ( x 1 ) , f ( x 2 ) , f ( x 3 ) , . . . ) and ( f ( y 1 ) , f ( y 2 ) , f ( y 3 ) , . . . ) have the same ideal values. Timothy Trujillo α -Ramsey Theory 7/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. Timothy Trujillo α -Ramsey Theory 8/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. The ideal value of the sequence consists of the ideal values of sequences of elements from A . Timothy Trujillo α -Ramsey Theory 8/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. The ideal value of the sequence consists of the ideal values of sequences of elements from A . { y [ α ] : ∀ n ∈ N , y n ∈ A } . Timothy Trujillo α -Ramsey Theory 8/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. The ideal value of the sequence consists of the ideal values of sequences of elements from A . { y [ α ] : ∀ n ∈ N , y n ∈ A } . Definition For all sets A , we let ∗ A denote the ideal value of the constant sequence ( A , A , A , . . . ). Timothy Trujillo α -Ramsey Theory 8/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. The ideal value of the sequence consists of the ideal values of sequences of elements from A . { y [ α ] : ∀ n ∈ N , y n ∈ A } . Definition For all sets A , we let ∗ A denote the ideal value of the constant sequence ( A , A , A , . . . ). We call ∗ A the ∗ -transform of A . Timothy Trujillo α -Ramsey Theory 8/20

Axiom 5 Axiom (Internal Set) Let ( A , A , A , . . . ) be a constant sequence of non-empty sets. The ideal value of the sequence consists of the ideal values of sequences of elements from A . { y [ α ] : ∀ n ∈ N , y n ∈ A } . Definition For all sets A , we let ∗ A denote the ideal value of the constant sequence ( A , A , A , . . . ). We call ∗ A the ∗ -transform of A . Timothy Trujillo α -Ramsey Theory 8/20

The Hypernatural Numbers Definition The elements of ∗ N are called hypernatural numbers . Timothy Trujillo α -Ramsey Theory 9/20

The Hypernatural Numbers Definition The elements of ∗ N are called hypernatural numbers . Remark The Internal Axiom, the set of hypernatural numbers is exactly the collection of ideal values of sequences of natural numbers. Timothy Trujillo α -Ramsey Theory 9/20

The Hypernatural Numbers Definition The elements of ∗ N are called hypernatural numbers . Remark The Internal Axiom, the set of hypernatural numbers is exactly the collection of ideal values of sequences of natural numbers. By the Number Axiom, α ∈ ∗ N N � ∗ N Timothy Trujillo α -Ramsey Theory 9/20

The Hypernatural Numbers Definition The elements of ∗ N are called hypernatural numbers . Remark The Internal Axiom, the set of hypernatural numbers is exactly the collection of ideal values of sequences of natural numbers. By the Number Axiom, α ∈ ∗ N N � ∗ N Definition The elements of ∗ N \ N are called nonstandard hypernatural numbers . Timothy Trujillo α -Ramsey Theory 9/20

The ∗ -Transform Theorem (Proposition 2.2, [2]) For all sets A and B the following hold: 1 A = B ⇐ ⇒ ∗ A = ∗ B ∗ ( A ∪ B ) = ∗ A ∪ ∗ B 6 2 A ∈ B ⇐ ⇒ ∗ A ∈ ∗ B ∗ ( A ∩ B ) = ∗ A ∩ ∗ B 7 3 A ⊆ B ⇐ ⇒ ∗ A ⊆ ∗ B ∗ ( A \ B ) = ∗ A \ ∗ B 8 ∗ { A , B } = { ∗ A , ∗ B } 4 ∗ ( A , B ) = ( ∗ A , ∗ B ) ∗ ( A × B ) = ∗ A × ∗ B 5 9 Timothy Trujillo α -Ramsey Theory 10/20

The ∗ -Transform Remark Recall that in ZFC a binary relation R between two sets A and B is identified with the set { ( x , y ) ∈ A × B : xRy } ⊆ A × B . Timothy Trujillo α -Ramsey Theory 11/20

The ∗ -Transform Remark Recall that in ZFC a binary relation R between two sets A and B is identified with the set { ( x , y ) ∈ A × B : xRy } ⊆ A × B . So ∗ { ( x , y ) ∈ A × B : xRy } ⊆ ∗ A × ∗ B . Timothy Trujillo α -Ramsey Theory 11/20

The ∗ -Transform Remark Recall that in ZFC a binary relation R between two sets A and B is identified with the set { ( x , y ) ∈ A × B : xRy } ⊆ A × B . So ∗ { ( x , y ) ∈ A × B : xRy } ⊆ ∗ A × ∗ B . Hence, ∗ R is a binary relation between ∗ A and ∗ B . Timothy Trujillo α -Ramsey Theory 11/20

The ∗ -Transform Remark Recall that in ZFC a binary relation R between two sets A and B is identified with the set { ( x , y ) ∈ A × B : xRy } ⊆ A × B . So ∗ { ( x , y ) ∈ A × B : xRy } ⊆ ∗ A × ∗ B . Hence, ∗ R is a binary relation between ∗ A and ∗ B . Theorem (Proposition 2.3, [2]) If f : A → B is a function then ∗ f : ∗ A → ∗ B is also a function. Timothy Trujillo α -Ramsey Theory 11/20

Initial segment relation Definition (Initial segment relation) Let s , X ⊆ N with | s | finite. Timothy Trujillo α -Ramsey Theory 12/20

Initial segment relation Definition (Initial segment relation) Let s , X ⊆ N with | s | finite. The relation s ⊑ X , Timothy Trujillo α -Ramsey Theory 12/20

Initial segment relation Definition (Initial segment relation) Let s , X ⊆ N with | s | finite. The relation s ⊑ X , means that the first | s | -elements of X is exactly the set s . Timothy Trujillo α -Ramsey Theory 12/20

Initial segment relation Definition (Initial segment relation) Let s , X ⊆ N with | s | finite. The relation s ⊑ X , means that the first | s | -elements of X is exactly the set s . Example { 0 , 3 , 5 } ⊑ { 0 , 3 , 5 , 8 , 9 , 12 } { 1 , 15 } ⊑ { 1 , 15 , 25 , . . . } { 1 , 15 } �⊑ { 0 , 3 , 5 , 8 , 9 , 12 } { 1 , 15 } �⊑ N Timothy Trujillo α -Ramsey Theory 12/20

Trees Definition A collection T of finite subsets of N is called a tree on N Timothy Trujillo α -Ramsey Theory 13/20

Trees Definition A collection T of finite subsets of N is called a tree on N if 1 T � = ∅ Timothy Trujillo α -Ramsey Theory 13/20