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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 24th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIf I have a well-ordered set $(J,\lt)$, a countably infinite subset $I \subset J$ with the induced order and a given $a\in I$, what can we say about the subset $L(a) := \{b\in I| b \lt a\}$? 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 $\omega+1$ and $a$ could be the top element, and then $L(a) = \omega$. 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.

   If I have a well-ordered set $(J,\lt)$, a countably infinite subset $I \subset J$ with the induced order and a given $a\in I$, what can we say about the subset $L(a) := \{b\in I| b \lt a\}$? 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 $\omega+1$ and $a$ could be the top element, and then $L(a) = \omega$. 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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 24th 2010
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   Author: Mike Shulman
   Format: MarkdownItexI can't think of anything you could say that would ensure it other than the tautological "there are only finitely many elements preceeding $a$".

   I can’t think of anything you could say that would ensure it other than the tautological “there are only finitely many elements preceeding $a$”.

3. • CommentRowNumber3.
   • CommentAuthorIan_Durham
   • CommentTimeMay 24th 2010
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   Author: Ian_Durham
   Format: MarkdownItexCould 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 25th 2010
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   Author: DavidRoberts
   Format: MarkdownItex@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!

   @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!