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handout05

# handout05 - Math 5020 Handout 5 Basic Concepts of Axiomatic...

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Math 5020 Handout 5 Basic Concepts of Axiomatic Set Theory (revised) Throughout this handout we assume Extensionality, Pairing, Union, Power Set, Separation, Replacement, and the axiom of Existence There exists a set. 1. The empty set exists and is unique. 2. A set x is transitive if x x . 3. A set x is an ordinal if x is transitive and well-ordered by , i.e., y x ( y ̸∈ y ) y x z x u x ( y z z u y u ) y x z x ( y z y = z z y ) w ∈ P ( x ) y w z w ( y = z y z ) 4. (a) is an ordinal. (b) If α is an ordinal and β α , then β is an ordinal. (c) If α, β are ordinals and α β , then α β . (d) If α, β are ordinals then either α β or β α . 5. If α, β are ordinals, we define α < β ⇐⇒ α β. 6. Define Ord to be the class of all ordinals. Ord is a proper class. < is a linear order on Ord. 7. If α is an ordinal, then so is α + 1 := α ∪ { α } . α + 1 is called the successor of α . An ordinal β is a successor ordinal if β = α + 1 for some ordinal α . 8. If C is a class of ordinals, then there is a unique ordinal inf C with the property that inf C C and inf C α for all α C . 9. If X is a nonempty set of ordinals, then X is an ordinal, and is in fact the least ordinal β with α β for all α X (thus it is also denoted as sup X ). 10. If α is an ordinal but not a successor ordinal, then α = sup { β : β < α } = α . In this case, α is called a limit ordinal . 11. 0 := is a limit ordinal. The ordinals 1 := 0 + 1 , 2 := 1 + 1 , , 3 := 2 + 1 , . . .

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handout05 - Math 5020 Handout 5 Basic Concepts of Axiomatic...

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