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

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Unformatted text preview: 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 is an ordinal and , then is an ordinal. (c) If , are ordinals and , then . (d) If , are ordinals then either or . 5. If , are ordinals, we define < . 6. Define Ord to be the class of all ordinals. Ord is a proper class. < is a linear order on Ord. 7. If is an ordinal, then so is + 1 := { } . + 1 is called the successor of . An ordinal is a successor ordinal if = + 1 for some ordinal . 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 for all C . 9. If X is a nonempty set of ordinals, then X is an ordinal, and is in fact the least ordinal with for all X (thus it is also denoted as sup X ). 10. If is an ordinal but not a successor ordinal, then = sup { : < } = . In this case, is called a limit ordinal ....
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This note was uploaded on 04/11/2011 for the course MATH 5020 taught by Professor Staff during the Spring '11 term at North Texas.

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

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