Thread: Immediate successors in Well-ordered sets

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, 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 (**)?

Originally Posted by Mimi89

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 (**)?

Look at the set , and assuming that x isn't the greatest element of X and that X i well-ordered take now a first element in S ...

Tonio

Originally Posted by tonio

Look at the set , and assuming that x isn't the greatest element of X and that X i well-ordered take now a first element in S ...

Tonio

Thank you!!! Though it might seem obvious to you, it didn't occur to me to look at this set and say it must have a minimal element!