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

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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

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logo1 Counting Ordinal Numbers Axiom of Substitution

Ordinal Numbers and the Axiom of Substitution

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 4

logo1 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 5

logo1 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.”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 6

logo1 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 7

logo1 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 8

logo1 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= /

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= / 1 = {/ 0} = {0}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= / 1 = {/ 0} = {0} 2 =

  • /

0,{/ 0}

  • = {0,1}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= / 1 = {/ 0} = {0} 2 =

  • /

0,{/ 0}

  • = {0,1}

3 =

  • /

0,{/ 0},{/ 0,{/ 0}}

  • = {0,1,2}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= / 1 = {/ 0} = {0} 2 =

  • /

0,{/ 0}

  • = {0,1}

3 =

  • /

0,{/ 0},{/ 0,{/ 0}}

  • = {0,1,2}

. . .

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Introduction

= / 1 = {/ 0} = {0} 2 =

  • /

0,{/ 0}

  • = {0,1}

3 =

  • /

0,{/ 0},{/ 0,{/ 0}}

  • = {0,1,2}

. . . Every natural number contains all the natural numbers before it as elements and as strict subsets.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Proposition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := / 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 is so that for all

m ∈ n we have m = {k ∈ n : k ⊂ m}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 is so that for all

m ∈ n we have m = {k ∈ n : k ⊂ m}. Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 is so that for all

m ∈ n we have m = {k ∈ n : k ⊂ m}.

  • Proof. Induction on n.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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}:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 23

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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′.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 24

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 26

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 27

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 28

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 29

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n} (proved in derivation of the Peano Axioms, and it trivially holds for 0).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 30

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n} (proved in derivation of the Peano Axioms, and it trivially holds for 0). Moreover, (good exercise) the containment can be sharpened to strict containment

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-31
SLIDE 31

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n} (proved in derivation of the Peano Axioms, and it trivially holds for 0). Moreover, (good exercise) the containment can be sharpened to strict containment, so m = n = {k ∈ n : k ⊂ n}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-32
SLIDE 32

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n} (proved in derivation of the Peano Axioms, and it trivially holds for 0). Moreover, (good exercise) the containment can be sharpened to strict containment, so m = n = {k ∈ n : k ⊂ n}. Finally, because n′ \n = {n} and because n is not a strict subset of itself, m = n = {k ∈ n′ : k ⊂ n}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 33

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Proposition. Consider N0, where N is constructed as for the

Peano Axioms and 0 := /

  • 0. Then every n ∈ N0 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 and so, because n′ \n = {n}, the latter set is equal to {k ∈ n′ : k ⊂ m}. If m ∈ n, then m = n = {k ∈ n : k ⊆ n} (proved in derivation of the Peano Axioms, and it trivially holds for 0). Moreover, (good exercise) the containment can be sharpened to strict containment, so m = n = {k ∈ n : k ⊂ n}. Finally, because n′ \n = {n} and because n is not a strict subset of itself, m = n = {k ∈ n′ : k ⊂ n}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Definition.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}. Proposition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

  • Proposition. Let α be an ordinal number and define the

successor of α to be α′ := α ∪{α}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 38

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

  • Proposition. Let α be an ordinal number and define the

successor of α to be α′ := α ∪{α}. Then α′ is an ordinal number, too.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-39
SLIDE 39

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

  • Proposition. Let α be an ordinal number and define the

successor of α to be α′ := α ∪{α}. Then α′ is an ordinal number, too. Proof.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-40
SLIDE 40

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

  • Proposition. Let α be an ordinal number and define the

successor of α to be α′ := α ∪{α}. Then α′ is an ordinal number, too.

  • Proof. Good exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 41

logo1 Counting Ordinal Numbers Axiom of Substitution

  • Definition. An ordinal number is a set α of sets that is

well-ordered by set inclusion so that for each β ∈ α we have that β = {γ ∈ α : γ ⊂ β}.

  • Proposition. Let α be an ordinal number and define the

successor of α to be α′ := α ∪{α}. Then α′ is an ordinal number, too.

  • Proof. Good exercise.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 44

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 45

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set). Axiom.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

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SLIDE 47

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-48
SLIDE 48

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution. Let A be a set.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-49
SLIDE 49

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution. Let A be a set. If S(a,b) is a

sentence such that for each a ∈ A the set

  • b : S(a,b)
  • can be

formed

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-50
SLIDE 50

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution. Let A be a set. If S(a,b) is a

sentence such that for each a ∈ A the set

  • b : S(a,b)
  • can be

formed (“so that a sensible substitute for a can be constructed”)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-51
SLIDE 51

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution. Let A be a set. If S(a,b) is a

sentence such that for each a ∈ A the set

  • b : S(a,b)
  • can be

formed (“so that a sensible substitute for a can be constructed”), then there exists a function F with domain A such that F(a) =

  • b : S(a,b)
  • for all a ∈ A.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-52
SLIDE 52

logo1 Counting Ordinal Numbers Axiom of Substitution

Define ω +n to be the n-fold successor of ω (no problem). Want to define 2ω to be the union of the ω +n (problem: no guarantee for a surrounding set).

  • Axiom. Axiom of Substitution. Let A be a set. If S(a,b) is a

sentence such that for each a ∈ A the set

  • b : S(a,b)
  • can be

formed (“so that a sensible substitute for a can be constructed”), then there exists a function F with domain A such that F(a) =

  • b : S(a,b)
  • for all a ∈ A. (And because F

has a range R that is a set,

  • F(a) ∈ R : a ∈ A
  • is a set, obtained

by substituting F(a) for each a ∈ A.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-53
SLIDE 53

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-54
SLIDE 54

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-55
SLIDE 55

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-56
SLIDE 56

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)].

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-57
SLIDE 57

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-58
SLIDE 58

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-59
SLIDE 59

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-60
SLIDE 60

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0}

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-61
SLIDE 61

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-62
SLIDE 62

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω), ω +1 = ω′ = C1(ω)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-63
SLIDE 63

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω), ω +1 = ω′ = C1(ω), ...

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-64
SLIDE 64

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω), ω +1 = ω′ = C1(ω), ..., ω +n = Cn(ω)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-65
SLIDE 65

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω), ω +1 = ω′ = C1(ω), ..., ω +n = Cn(ω), etc.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-66
SLIDE 66

logo1 Counting Ordinal Numbers Axiom of Substitution

For 2ω, let C(x) := x∪{x}, A := ω = N0 and S(n,b) := [b = Cn(ω)]. Then F(n) =

  • b : S(n,b)
  • = {Cn(ω)}

and

  • F(n) : n ∈ N0
  • = {Cn(ω) : n ∈ N0} contains

ω +0 = ω = C0(ω), ω +1 = ω′ = C1(ω), ..., ω +n = Cn(ω),

  • etc. Now 2ω := ω ∪{Cn(ω) : n ∈ N0} is an ordinal number

(good exercise).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-67
SLIDE 67

logo1 Counting Ordinal Numbers Axiom of Substitution

Final Remarks

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-68
SLIDE 68

logo1 Counting Ordinal Numbers Axiom of Substitution

Final Remarks

  • 1. There is no set of all ordinal numbers (Burali-Forti

paradox).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-69
SLIDE 69

logo1 Counting Ordinal Numbers Axiom of Substitution

Final Remarks

  • 1. There is no set of all ordinal numbers (Burali-Forti

paradox).

  • 2. The ordinal numbers are “totally ordered” by inclusion.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution

slide-70
SLIDE 70

logo1 Counting Ordinal Numbers Axiom of Substitution

Final Remarks

  • 1. There is no set of all ordinal numbers (Burali-Forti

paradox).

  • 2. The ordinal numbers are “totally ordered” by inclusion.
  • 3. We won’t go into set theory with classes, etc.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Ordinal Numbers and the Axiom of Substitution