Ordinal Numbers and the Axiom of Substitution Bernd Schr oder - PowerPoint PPT Presentation

Counting Ordinal Numbers Axiom of Substitution Ordinal Numbers and the Axiom of Substitution Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. 2. But we also would like to extend the idea of counting past infinity. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. 2. But we also would like to extend the idea of counting past infinity. 3. “She can’t do that.” logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. 2. But we also would like to extend the idea of counting past infinity. 3. “She can’t do that.” 4. Well-ordered sets provide such a mechanism. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. 2. But we also would like to extend the idea of counting past infinity. 3. “She can’t do that.” 4. Well-ordered sets provide such a mechanism. 5. But we also want to have standardized numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 1. We spent significant effort to extend N from the counting system that the Peano Axioms provide to a framework that accommodates algebra and analysis. 2. But we also would like to extend the idea of counting past infinity. 3. “She can’t do that.” 4. Well-ordered sets provide such a mechanism. 5. But we also want to have standardized numbers. 6. This is where ordinal numbers come in. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / = { / 0 } = { 0 } 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / = { / 0 } = { 0 } 1 � � = 0 , { / 0 } = { 0 , 1 } 2 / logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / = { / 0 } = { 0 } 1 � � = 0 , { / 0 } = { 0 , 1 } 2 / � � = 0 , { / 0 } , { / 0 , { / 0 }} = { 0 , 1 , 2 } 3 / logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / = { / 0 } = { 0 } 1 � � = 0 , { / 0 } = { 0 , 1 } 2 / � � = 0 , { / 0 } , { / 0 , { / 0 }} = { 0 , 1 , 2 } 3 / . . . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Introduction 0 = 0 / = { / 0 } = { 0 } 1 � � = 0 , { / 0 } = { 0 , 1 } 2 / � � = 0 , { / 0 } , { / 0 , { / 0 }} = { 0 , 1 , 2 } 3 / . . . Every natural number contains all the natural numbers before it as elements and as strict subsets. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . The base step n = 0 is trivial, because 0 = / 0 has no elements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . The base step n = 0 is trivial, because 0 = / 0 has no elements. Induction step n → n ′ = n ∪{ n } : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . The base step n = 0 is trivial, because 0 = / 0 has no elements. Induction step n → n ′ = n ∪{ n } : Let m ∈ n ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . The base step n = 0 is trivial, because 0 = / 0 has no elements. Induction step n → n ′ = n ∪{ n } : Let m ∈ n ′ . If m ∈ n , then m = { k ∈ n : k ⊂ m } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

Counting Ordinal Numbers Axiom of Substitution Proposition. Consider N 0 , where N is constructed as for the Peano Axioms and 0 : = / 0 . Then every n ∈ N 0 is so that for all m ∈ n we have m = { k ∈ n : k ⊂ m } . Proof. Induction on n . The base step n = 0 is trivial, because 0 = / 0 has no elements. Induction step n → n ′ = n ∪{ n } : Let m ∈ n ′ . If m ∈ n , then m = { k ∈ n : k ⊂ m } . Because m ⊆ n , we have n �⊂ m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution