WEEK 4
 
M: Proof of Proposition 1.33. Discussion of consequences. 
      Consequences: If (A,R) is a set well-ordering and (B,S) is a proper class set-like well-ordering then 
                                  (A,R) is isomorphis to an initial segment of (B,S)
                                  If both (B,S) and (B',S') are  proper class set-like well-orderings then they are isomorphic.
W: Extensional relations. Mostowski Collapsing theorem. Ordinals. 
       Consequence: If (A,R) is a set-like  well-ordering then (A,R) is not isomorphic to any of its proper initial segments.
      1.34. Definition. Extensionality of a relation R on a class A.
      1.35. Theorem. Mostowski Collapsing Theorem. If R is a well-founded set-like relation that is extensional on A 
                               then there is a unique transitive class U and a unique isomorophism s : (A,R) --> (U,\in).  
      1.36. Corollary. If  U, U' are transitive classes such that \in is well-founded on either of them  and (U,\in) is  
                                 isomorphic to (U',\in) then U = U' and the isomorphism is the identity map. 
      1.37. Proposition. If (R,A) is a set-like well-ordering then R is extensional on A. 
      1.38. Proposition. If (R,A) is a set-like well-ordering then there is a unique transitive class U and a unique
                                   isomorphism s : (R,A) --> (U,\in). In this case U is a transitive class well-ordered under \in.
      1.39. Defintion. A transitive set that is well ordered under \in is called an ordinal.  
F: Basic properties of ordinals. 
     1.40. Proposition. Let A be a  transitive class well-ordered under \in. 
                                   (a) every element of A is an ordinal. In particular every element of an ordinal is an ordinal. 
                                   (b) any proper initial segment of A is an element of A.  
     1.41. Proposition. (a) If x,y are ordinals then  x \in y or  x = y  or  y \in x
                                   (b) Let
                                        On = the class of all ordinals
                                        Then On is transitive and well-ordered under \in
                                        Also, On is unique, i.e. if A is any transitive proper class well-ordered under \in then A = On.
                                   (c) On is a proper class.