The following a most Fundamental Fixed Point Theorem is proved in   ZF   set theory

without the Axiom of Choice.  It is " a most fundamental" in the sense that it gives a

necessary and sufficient condition for the existence of a fixed point of a mapping of a set   S

into itself where absolutely no algebraic or analytic or order theoretic or any other special

structures are imposed on   S.

  THEOREM (Abian). Let   f   be a mapping from a set S into itself. Then   f   has a fixed point if

and only if:
  There exists an element   a   of   P   such that   f^k(a)   is an element of   P   for every ordinal   k,   and
for every ordinal   v

(1)   if   f^v(a)   is not a fixed point of   f   then  the   f^u(a) 's are all distinct for every ordinal  u  < v.

  PROOF.  First we show that (1) implies that   f   has a fixed point.  Assume on the contrary that   f

  has no fixed point and let   p   and   q   be any two distinct ordinal numbers.  Clearly, there always

exists an ordinal   v   such that     p < v   and   q < v.    But then since by our assumption   f^v(a)

cannot be a fixed point of   f   hence (1) implies that   f^p(a)   and   f^q(a)   are two distinct elements

of   S.  Thus, our assumption implies that for every two distinct ordinals   p   and   q   there

correspond two distinct elements   f^p(a)   and   f^q(a)   of the set   S.  Consequently, every ordinal can

be assigned in a one-to-one way to every element of a subset of   S.  But then the Axiom Scheme

of Replacement of   ZF   would imply that the set of all ordinals exists, which is a contradiction.

Thus, our assumption is false and   f   has a fixed point.

  Next, assume that   f   has a fixed point   a.  Then  (1 ) is obviously satisfied by setting

a = f(a) = f^k(a)   for every ordinal   k.

  Thus,  the Theorem is proved.

  REMARK.  Note that   f^k(a)   in the above does not necessarily indicate the k-th or any other

iterates of   f.  Thus, it is not even required that  f(f^k(a))   be equal to   f^k+1(a).   Obviously, by

"  f^k(a)  is a fixed point of   f  "   is meant that   f(f^k(a))   =   f^k(a),  and as mentioned above,

without necessitating that   f(f^k(a))   be equal to   f^k+1(a).

  The fundamental significance of the Theorem lies in the fact that a great many fixed point

theorems can be reduced to the special cases of the Theorem, in as much as, no special structures

are required by the set   S   to have, and, no iterative rules are imposed on   f.

Alexander Abian
Dept .of Math. Iowa state Univ.
Ames, IA, 50011, USA.
e-mail: abian@iastate.edu