Under what conditions does an element in a set have an immediate successor ?

Question

Apparently with the axiom of choice, there are orderings for which every set is well ordered...and thus every subset has a least element with respect to that order. If such a subset is bounded above, the set of upper bounds must have a least element right ? Implying that any set under a well ordering has the least upper bound property ?

As for immediate successors to exist, however, what are the conditions ? For example {0} union with {...1/16, 1/4, 1/2, 1} is well ordered under the typical ordering...but 0 has no immediate successor....so what are the conditions for immediate successors to exist ?

Answer 1

If you read your statements carefully, you will find you have basically answered your question.



Under what conditions does an element in a set have an immediate successor ?



You state that "there are orderings for which every set is well-ordered. " So there is an ordering for which the subset of elements "not less than" the specified element is well-ordered. In that ordering, the least element is the successor of the specified element.



Unfortunately, the Well-ordering Theorem is an existence theoem, so there is no way of knowing what the least element is, but that element is the successor.

So the answer to "Under what conditions ..." is whenever the Axiom of Choice is valid in the system.

Answer 2

The Axiom of Choice implies that there exists some well ordering of any set.



If the subset is bounded from above then is set of upper bounds could be empty. If it is nonempty, it need not have a least element, we just know that we can order it in some way in which ever subset has a least element.



For example, consider the set (0,1). Clearly, there is no minimal element. But, by the axiom of choice there is some ordering which there is a minimum. Therefore this ordering, gaurenteed to exist, must not be the natural ordering in this situation.

Answer 3

It sounds like your problem is one of the following:



The well-ordering theorem (which is equivalent to the axiom of choice under ZF-set theory) isn't a statement about the normal ordering (unless it's boring like the normal ordering on the natural numbers).



The well-ordering theorem is an existence theorem... what it says is basically there exists _some way_ to order any set so that every subset has a least element as defined under that order.



Further, (here's my second point) this ordering is non-constructive in general. This means that nobody can tell you what such an ordering "looks" like except for very special cases.



In your example of {... 1/16, 1/4, 1/2, 1}, we could order it

1 < 1/2 < 1/4 < 1/16 < ... . This would be a well-ordering.



If you wanted me to show you what a well-ordering of the reals looks like, I'd have to say no. It's impossible to create such an order, even though it exists by the theorem.