This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 wellordered 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 ....
View
Full
Document
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.
 Spring '11
 staff
 Set Theory

Click to edit the document details