Re: proof of uncountable von Neumann ordinals in ZFC

David Bernier wrote:

In ZFC, every set can be well-ordered.
The von Neumann definition of an ordinal
is (according to "Other definitions" in
http://en.wikipedia.org/wiki/Ordinal_number )
"A is an ordinal if A is transitive and
every element of A is transitive."

A set S is transitive if x e S and
y e x implies y e S.

(I'll trust that this is correct ...)


This definition requires the Axiom of Foundation (aka Regularity). I am more familiar with the definition that an ordinal is a transitive set which is well-ordered by set membership (not set containment - as subsets - as it says in the Wikipedia article).



The Wikipedia article on the
Axiom Schema of Replacement states:

``Clearly then, the existence of assignment
of an ordinal to every well-ordered set
requires replacement as well."

After looking at the other axioms, I'm not
surprised. What's not obvious to me is,
for example, how to use Replacement to
show that some von Neumann ordinal is
order-isomorphic to some well-ordering
of the reals obtained by using the Axiom
of Choice. In general, I don't see how to
go from a well-ordered set to a von Neumann
ordinal with the same order type.


If x and y are well-ordered sets, then either x and y are order-isomorphic or one is order- isomorphic to an initial segment of the other. This is a standard beginning set theory result (at least in a graduate-level course); see Kunen if you can't find it anywhere else.

Given a well-ordered set x, let a be the supremum (i.e., union) of all ordinals that are order-isomorphic to either x or an initial segment of x. Then a is order-isomorphic to x. I *think* this works; I may have left out a subtle detail. Again, see Kunen, who explicitly notes where Replacement comes into play. His argument is much more elegant than the one you find in the Wikipedia article.

In other words, even without choice, the ordinals provide the complete class of all possible order types of well-orderings.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan
.

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