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  • 1. ORDINALSCARDINALS ORDINALS AND CARDINALS SEPErik A. AndrejkoUniversity of Wisconsin - MadisonSummer 2007ERIK A. ANDREJKOORDINALS AND CARDINALS

2. ORDINALSCARDINALS NEUMANN ORDINALS VON FIGURE: John von NeumannERIK A. ANDREJKO ORDINALS AND CARDINALS 3. ORDINALSCARDINALS ORDINALSDEFINITION An ordinal is a set x that is transitive and well ordered by .The class of ordinals is denoted ON. 0 ON zero/ ON = {} ON successor For any set X ,X ON =X ON limit ERIK A. ANDREJKO ORDINALS AND CARDINALS 4. ORDINALSCARDINALS ORDINALS0=0/0 {0} = {0} = 1// /{0} {{0}} = {0, {0}} = 2 //// . . . . . . . . . ERIK A. ANDREJKO ORDINALS AND CARDINALS 5. ORDINALS CARDINALS ORDER TYPES THEOREM If A, R is a well-ordering then there is a unique ordinal such that A, R = i.e. with A, R = , .DEFINITION is the order type of the well ordering A, R also denoted type(A, R) = . ERIK A. ANDREJKO ORDINALS AND CARDINALS 6. ORDINALSCARDINALS ORDINAL SUMS... ... FIGURE: + e.g. 1+ = = +1ERIK A. ANDREJKO ORDINALS AND CARDINALS 7. ORDINALS CARDINALS ORDINAL SUMS FACT For any ordinals , , + ( + ) = ( + ) + , 1 + 0 = , 2 + 1 = S(), 3 + S( ) = S( + ), 4 If is a limit ordinal 5 + = sup( + : < ).ERIK A. ANDREJKO ORDINALS AND CARDINALS 8. ORDINALSCARDINALS ORDINAL PRODUCTS............ ... ... ... ... FIGURE: ERIK A. ANDREJKOORDINALS AND CARDINALS 9. ORDINALS CARDINALS ORDINAL PRODUCTS ...... FIGURE: e.g. 2 = = 2 = +ERIK A. ANDREJKO ORDINALS AND CARDINALS 10. ORDINALSCARDINALS ORDINAL PRODUCTS FACT For ordinals , , ( ) = ( ) , 1 0 = 0 = 0, 2 1 = 1 = , 3 S( ) = + , 4 For limit 5 = sup{ : < } ( + ) = + . 6ERIK A. ANDREJKO ORDINALS AND CARDINALS 11. ORDINALS CARDINALS ORDINAL WARNINGSWARNING The + operation is not commutative: + = + . (except on the natural numbers)WARNING The operation is not commutative except on the natural numbers:2 = = 2 = + .The right distributive law does not hold: (1 + 1) = = 1 + 1 = + .ERIK A. ANDREJKO ORDINALS AND CARDINALS 12. ORDINALS CARDINALS ORDINAL EXPONENTIATIONFor ordinals , dene by 0 = 1, 1 +1 = , 2 For limit 3 = sup{ : < } ERIK A. ANDREJKO ORDINALS AND CARDINALS 13. ORDINALS CARDINALS CANTOR NORMAL FORMTHEOREM (Cantors Normal Form Theorem) Every ordinal > 0 can be written as = 1 k1 + + n kn for ki {0}, 1 > > n .Note that it is possible for = 1 . The least such ordinal is 0 . i.e.0 = 0 ERIK A. ANDREJKO ORDINALS AND CARDINALS 14. ORDINALSCARDINALS SPECIAL ORDINALS DEFINITION1 = .0n+1 = n . 2 0 = sup{n : n < } 3 Then 0 = 0 0 is the least ordinal such that = . DEFINITION 1CK is the least non-computable ordinal. ERIK A. ANDREJKO ORDINALS AND CARDINALS 15. ORDINALS CARDINALS CARDINALSDEFINITION Let A be a set that can be well-ordered. Then |A| is dened to be the least ordinal such that |A| .Under AC every A can be well ordered and so |A| is dened for all sets A. DEFINITION An ordinal is called a cardinal if = ||. ERIK A. ANDREJKO ORDINALS AND CARDINALS 16. ORDINALS CARDINALS CARDINALSDEFINITION Given an ordinal , dene + to be the least cardinal > .DEFINITION The cardinals = are dened as 0 = 0 = , 1 +1 = +1 = ( )+ , 2 For limit , = = sup{ : < }. 3 ERIK A. ANDREJKO ORDINALS AND CARDINALS 17. ORDINALS CARDINALS CARDINALS FACT 1. Each is a cardinal, 2. Every innite cardinal is equal to for some . 3. < implies < ,ERIK A. ANDREJKO ORDINALS AND CARDINALS 18. ORDINALSCARDINALS CARDINAL ARITHMETIC For cardinals and dene the sum = | {0} {1}|and the product = | | FACT and are commutative.FACT For cardinals , k = = max(, ), 1 | < | = . 2ERIK A. ANDREJKO ORDINALS AND CARDINALS 19. ORDINALS CARDINALS CARDINAL ARITHMETIC DEFINITION Using AC, for cardinals and dene = | |.FACT For and 2 then 2 P( ) FACT (AC) For cardinals , , = and ( ) = i.e. the normal rules for exponentiation apply. ERIK A. ANDREJKOORDINALS AND CARDINALS 20. ORDINALSCARDINALS HARTOG FUNCTIONDEFINITION Given a set X dene (X ), Hartogs Aleph Function, (X ) = sup{ : f X f is 1 1}FACT (AC) (X ) = |X |+ERIK A. ANDREJKO ORDINALS AND CARDINALS 21. ORDINALS CARDINALS CARDINAL TYPES DEFINITION1 is a limit cardinal if and only if is a limit ordinal., is a successor cardinal if and only if is a successor 2ordinal. ERIK A. ANDREJKO ORDINALS AND CARDINALS 22. ORDINALSCARDINALS CARDINAL TYPESDEFINITION Let f : . Then f maps conally if ran(f ) is unbounded in .DEFINITION The conality of , denoted cf( ) is the least such that there exists a map from conally into .DEFINITION A cardinal is regular if cf() = , 1 A cardinal is singular if cf() < . 2FACT + is regular for any cardinal . ERIK A. ANDREJKO ORDINALS AND CARDINALS