By Larry M. Hyman

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Zm then _C zn and zm x zm _c z n . Clearly, Also, m u Y X Y ' n,m=O ," zn zm m C - x z 'max(m,n n m=O max (m,n) m C - x U (y x y) Hence, Therefore, 5 'max (m, n +1 y. i s a s u b s e t of a s e t which c a n b e w e l l - x ordered, so t h a t U n,m=O x c a n be w e l l - o r d e r e d a l s o , q . e . d . When A l f r e d T a r s k i was s t u d y i n g t h e n o t i o n of f i n i t e (see T a r s k i [1924a,1938a]), h e d i s c o v e r e d s e v e r a l d e f i n i t i o n s o f f i n i t e which w e r e e q u i v a l e n t t o t h e axiom of c h o i c e .

Choice function on f " x ) (See Theorem 2 . 9 ) . Therefore, it was interesting to observe that the following statement is equivalent to WO 1. - WO 8: Every set on which there is a choice function, can be well ordered. WO 1 because if Clearly WO 1 -+ WO 8. Conversely, WO 8 x is any non-empty set then there is a choice function on Thus, WO 8 implies x can be well ordered. {{u}:u E XI. -+ 52. THE AXIOM OF CHOICE Apparently, the first specific reference to the axiom of choice was given in a paper by G.

Let f be a choice function on the set of all non-empty subsets of x and define f(B) = u where u 4 x. We define a function G as follows: For all ordinal numbers a, G ( a ) = f(x G"a). ) - is 1-1 on g ( G ) f l x. If a < 8 , then G ( a ) # G ( B ) since G ( B ) = f(x (1) G-l G(B) E x, and G(a) E G"8. - and G"f3) G(a), f x - G"B (2) There is an ordinal number a such that G " a = x. For there must be some ordinal number 8 such that G ( B ) 4 x. If this were not so, by (1) G would be a 1-1 mapping of On into x, which contradicts x being a set.

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