The Axiom of Specification Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, too. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � x ∈ S : p ( x ) too. It is denoted logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � � � x ∈ S : p ( x ) x ∈ S | p ( x ) too. It is denoted or . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � � � x ∈ S : p ( x ) x ∈ S | p ( x ) too. It is denoted or . Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � � � x ∈ S : p ( x ) x ∈ S | p ( x ) too. It is denoted or . Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the objects must come out of an existing set. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � � � x ∈ S : p ( x ) x ∈ S | p ( x ) too. It is denoted or . Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the objects must come out of an existing set. (But that’s crucial.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model The Axiom of Specification If S is a set and p ( · ) is an open sentence for the elements of S , then the collection of all elements x ∈ S that satisfy p ( x ) is a set, � � � � x ∈ S : p ( x ) x ∈ S | p ( x ) too. It is denoted or . Note how similar the Axiom of Specification is to the third assumption in Russell’s Paradox. The only difference is that the objects must come out of an existing set. (But that’s crucial.) So now we have recaptured the parts of the assumptions in Russell’s Paradox that we definitely cannot live without. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . � � p ( x ) = ( x �∈ S ) ∧ ( x ∈ S ) is an open sentence. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . � � p ( x ) = ( x �∈ S ) ∧ ( x ∈ S ) is an open sentence. By the Axiom of � � � � Specification, A : = x ∈ S : p ( x ) = x ∈ S : ( x �∈ S ) ∧ ( x ∈ S ) is a set. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . � � p ( x ) = ( x �∈ S ) ∧ ( x ∈ S ) is an open sentence. By the Axiom of � � � � Specification, A : = x ∈ S : p ( x ) = x ∈ S : ( x �∈ S ) ∧ ( x ∈ S ) is a set. But because p ( x ) is always false, A has no elements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . � � p ( x ) = ( x �∈ S ) ∧ ( x ∈ S ) is an open sentence. By the Axiom of � � � � Specification, A : = x ∈ S : p ( x ) = x ∈ S : ( x �∈ S ) ∧ ( x ∈ S ) is a set. But because p ( x ) is always false, A has no elements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. There is a set that contains no elements. Proof. By our first axiom, there is a set S . � � p ( x ) = ( x �∈ S ) ∧ ( x ∈ S ) is an open sentence. By the Axiom of � � � � Specification, A : = x ∈ S : p ( x ) = x ∈ S : ( x �∈ S ) ∧ ( x ∈ S ) is a set. But because p ( x ) is always false, A has no elements. The set will be called the empty set , denoted / 0. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. Let C be a nonempty family of sets. Then there is a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. Let C be a nonempty family of sets. Then there is a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. Let C be a nonempty family of sets. Then there is a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. Proof. ∀ C ∈ C : x ∈ C is an open sentence. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. Let C be a nonempty family of sets. Then there is a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. Proof. ∀ C ∈ C : x ∈ C is an open sentence. Let C 0 ∈ C . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification

Specifying Subsets Empty Set Intersection Complement Model Proposition. Let C be a nonempty family of sets. Then there is a set I so that an object x is an element of I iff for all C ∈ C we have x ∈ C. Proof. ∀ C ∈ C : x ∈ C is an open sentence. Let C 0 ∈ C . By the � � Axiom of Specification I : = x ∈ C 0 : [ ∀ C ∈ C : x ∈ C ] is a set. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Specification