# Thread: Question about upper bounds for sets of ordinals.

1. ## Question about upper bounds for sets of ordinals.

Hi,

I need to show that there is a set A of ordinals (von Neumann) with a largest element a such that UA (union of A) is strictly less than a. But if a is an element of A and UA is in a, then UA is an element of itself.

I'm probably missing something very simple here, but any help would be much appreciated.

Regards
Sam

For reference, the problem is in Discovering Modern Set Theory: The ... - Google Books

page 159.

2. Sorry, I just realized that the page 159 isn't on google books, so here's the problem in full:

"Exercise12: (a) show that if A is a set of ordinals that contains a largest element alpha then UA is less than or equal to alpha.
(b) give examples of A and alpha as in point (a) such that:
(b1) UA = alpha
(b2) UA is strictly less than alpha"

And it's b2 that I'm having trouble with.

Again, any help would be great.

Regards
Sam

3. I agree, this is puzzling. If $\alpha\in A$, then $\alpha\subseteq\bigcup A$ by the axiom of union. Therefore, if $\bigcup A<\alpha$, i.e., if $\bigcup A\in\alpha$, then $\bigcup A\in\bigcup A$, which cannot be.

4. Thanks for the reply!

I'm glad you agree. I might just have to assume that they made a mistake. If they meant something else, I can't figure it out.

Regards
Sam

5. I looked at the book. As far as I can tell (maybe I'm mistaken?), that exercise is incorrectly written.

The exercise should be just this:

Exercise12: show that if A is a set of ordinals that contains a largest element alpha then UA EQUALS [this is what I changed] alpha.

So (b1) is subsumed already; there is no need for examples, since it holds for ALL sets as described above. And the incorrect (b2) is deleted.

Or are the authors perhaps asking you to say to (b2) "There are no such examples."?

6. Hi MoeBlee,

I don't have an instructor, sorry. I might just e-mail one of the authors and ask (I also can't find a corrections page for the book), but will post back here if I get a reply.

Regards
Sam

7. Yes, I would be very interested in what the authors say. Thanks.

8. Hi MoeBlee,

Winfried Just said:

"Sam,

this morning I did look at the (fortunately quite small, but nevertheless
nonempty) list of known mistakes that I keep in case this book ever will
have another edition. It says:

"page 159, Exercise 12(b): This is nonsense."