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handout04 - Math 5020 Handout 4 Basic Concepts of Axiomatic...

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Math 5020 Handout 4 Basic Concepts of Axiomatic Set Theory 1. The language of set theory contains only one relation symbol whose intended interpretation is the membership relation between sets. 2. If φ ( u, v 1 , . . . , v k ) is a formula in the language of set theory and y 1 , . . . , y k are sets, then C = { x : φ ( x, y 1 , . . . , y k ) } is a class . 3. Two classes are considered equal if they have the same elements (sets): If C = { x : φ ( x, y 1 , . . . , y k ) } and D = { x : ψ ( x, z 1 , . . . , z l ) } , then C = D iff for all x , φ ( x, y 1 , . . . , y k ) ψ ( x, z 1 , . . . , z l ) . 4. The set theoretic universe is the class V = { x : x = x } . 5. Every set can be considered a class: If y is a set then y can be considered as { x : x y } . 6. A class that is not a set is a proper class . 7. Russell’s class { x : x ̸∈ x } is a proper class. Likewise, V is a proper class (Separation). 8. The following relations and operations can be defined for classes: C D x ( x C x D ) C D = { x : x C x D } C D = { x : x C x D } C D = { x : x C x ̸∈ D } C = { x : y C ( x y ) } = { y : y C } 9. (Kuratowski) The ordered pair ( x, y ) is defined as {{ x } , { x, y }} . (Note that it is an element of P ( P ( { x, y } )); also
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