A little help with this question?
Let
be a well-ordered set. Show that for each
, either
is the greatest element of
or
has an immediate successor (that is, there exists
such that there does not exist
with
)
Now, here's how I started:
Let X be well-ordered by
. Let
be arbitrary but fixed.
If x is the greatest element of X, then
.
If x is not the greatest element of X, then
(*) and
.
Then I claim that
is the immediate successor of x,
This isn't usually true
i.e. there doesn't exist a
such that
(**)
Questions:
1. Does the proof depend on the choice of well-ordering?
2. How do I show (*)?
3. How do I show (**)?