errata : p. 382, last line but one, delete "in the theory of selections (*83’92) and."
382 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
- 54-56. J-: a"-’€O u 1 u 2. = . (x,y,z). x,y,z€a .x=f=y. x=f=z. y=fZ
Dem. J- . *54-55 . *11’52 . :> J-:. at’..J€O u 1 u 2. = : (x,y). x,y€a. x=f=y. a=f= t’x u t’y: [*51-2.*22-59] = : (x, y). ,,’x u t’y Ca. x=f y. a=f= t’x u "’y: [ *24’6] = : (x, y) . ,,’x u "’y C Co( . x =f= y . ! a - (t ’x u "’y) : [*51 0 232.Transp] = : (x, y): t’x u t’y Ca. x =f= y: (z). z €a. z=f=x. z=f=y: [ *51’2.*22’ 59] = : (x, y, z) . x, y, Z € a . x =f= y . x =f= z . y =f= z :. ::> . Prop In virtue of this proposition, a class which is neither null nor a unit class nor a couple contains at least three distinct members_ Hence it will follow that any cardinal number other than 0 or 1 or 2 is equal to or greater than 3. The above proposition is used in *104-43, which is an existence-theorem of considerable importance in cardinal arithmetic_
- 54-6. :. a n {3 = 1. . x, x’ € a . y, y’ € {3 . :> :
t ’x u t ’y = t’ x’ u t ’y' . = . x = x’ . y = y’ Dem, J- . *51-2 . ::> J- :. H p . ::> : t’ X Ca. ,,’x' Ca. t ’y C {3 . t ’y' C {3 . a n {3 = A : [*24-48] ::>: t’x u t’y = ,,’x' U t’y' . = . t’x = t’x' . t’y = t’y' . [*51’23] = . x = x’ . y = y’ :. ::> J- . Prop The above proposition is useful in dealing with sets of couples formed of one member of a class a and one member of a class {3, where a and f3 have no members in common_ It is used in the theory of selections (*83-92) and in the theory of cardinal multiplication (*113-148)_
