Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
If I have a well-ordered set (J,<), a countably infinite subset I⊂J with the induced order and a given a∈I, what can we say about the subset L(a):={b∈I|b<a}? What conditions are needed on J,I or a to say it is finite? Are there such conditions?
Edit: Hmm, I suppose J=I could be ω+1 and a could be the top element, and then L(a)=ω. I’m still interested in some formal conditions, and I feel my ordinal-fu is insufficient. This seems too much like a homework question, else I’d put it on MathOverflow.
I can’t think of anything you could say that would ensure it other than the tautological “there are only finitely many elements preceeding a”.
Could you say something like a is the least element of the subset of all elements greater than b, or is that equivalent to Mike’s tautology?
@Mike
I think you’re right. In the end I didn’t need this, and I’m sure glad I didn’t ask on MO!
1 to 4 of 4