What's wrong with this argument in ZF?

I seem to have "proved" something in ZF that (I think) is not provable
in ZF (without AC). What's wrong? The "result" is

(1) For any ordinal alpha, the set S of bijections of aleph_alpha onto
itself is well-orderable.

Here is the relation that seems ZF-provably to wellorder P_alpha:

There's a beta < aleph_alpha such that for all gamma < beta, F(gamma)
= G(gamma) and F(beta) < G(beta).

(apparent proof)

Irreflexivity is obvious.

Transitivity follows by considering separately the cases beta_0 <
beta_1, beta_0 = beta_1, beta_0 > beta_1 (where these beta_i witness
the existential quantifier in each hypothesis).

Now let X be a subset of S. Then the least element F of X is given by:

For all beta < aleph_alpha, F(beta) =
the least gamma < aleph_alpha such that
there's a G in X such that
[forall delta < beta G(delta) = F(delta)
and G(delta) = gamma].

So S is wellordered.

This pseudo-result can be extended to:

(2) For any aleph_alpha, for any beta \greq aleph_alpha such that
|beta| = aleph_alpha, the set of bijections of beta onto aleph_alpha
can be wellordered.



The problem is,

(3) for any ordinal alpha, P(alpha) can be wellordered

entails AC.

And (2) seems to entail (3). For take arbitrary ordinal alpha. Then
there's a bijection F from P(alpha) to the set T of bijections from
alpha onto |alpha| (I guess |alpha| is an aleph). But (2) says T
wellordered. So the wellordering of T induces a wellordering of
P(alpha) via the bijection F of P(alpha) onto T.


THANKS!

.

Relevant Pages

  • Re: Whats wrong with this argument in ZF?
    ... Let X be a set of bijections of w on w. ... [forall delta < beta G= F ... entails AC. ... So the wellordering of T induces a wellordering of ...
    (sci.logic)
  • Re: Whats wrong with this argument in ZF?
    ... Let X be a set of bijections of w on w. ... [forall delta < beta G= F ... So the wellordering of T induces a wellordering of ... David C. Ullrich ...
    (sci.logic)
  • Re: Whats wrong with this argument in ZF?
    ... That doesn't give a well-ordering. ... write bijections from w to itself as sequences; ... [forall delta < beta G= F ... So the wellordering of T induces a wellordering of ...
    (sci.logic)