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

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Unformatted text preview: 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 ( u, v 1 , . . . , v k ) is a formula in the language of set theory and y 1 , . . . , y k are sets, then C = { x : ( x, y 1 , . . . , y k ) } is a class . 3. Two classes are considered equal if they have the same elements (sets): If C = { x : ( x, y 1 , . . . , y k ) } and D = { x : ( x, z 1 , . . . , z l ) } , then C = D iff for all x , ( x, y 1 , . . . , y k ) ( 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. Russells class { x : x x } is a proper class. Likewise, V is a proper class (Separation)....
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handout04 - Math 5020 Handout 4 Basic Concepts of Axiomatic...

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