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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 19th 2014
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   Author: Mike Shulman
   Format: MarkdownItexNew page: [[plump ordinal]]. Is there a general name for a pair of inequalities $\lt$ and $\le$ which satisfy the expected laws? They're kind of weird from a categorical perspective; $\le$ is a poset (or preorder), of course, and $\lt$ is an endo-profunctor of it, but $\lt$ has a multiplication and a *counit*, so it's neither a monad nor a comonad in $Prof$.

   New page: plump ordinal. Is there a general name for a pair of inequalities $\lt$ and $\le$ which satisfy the expected laws? They’re kind of weird from a categorical perspective; $\le$ is a poset (or preorder), of course, and $\lt$ is an endo-profunctor of it, but $\lt$ has a multiplication and a counit, so it’s neither a monad nor a comonad in $Prof$.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 20th 2014
   • (edited Feb 20th 2014)
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   Author: TobyBartels
   Format: MarkdownItexI don\'t know a name, but there should be one! Another place that this pair comes up is in non-located ([[one-sided real number|one-sided]] or [[MacNeille real number|MacNeille]]) real numbers. We can leave transitivity of $\lt$ out of the list of axioms (since it follows in two ways), but we also should add its irreflexivity: If $x \lt x$, then False. An un[[bias]]ed list of axioms is (in addition to $\lt \Rightarrow \leq$) this: * If $x_0 \leq \cdots \leq x_n$, then $x_0 \leq x_n$; * The conclusion can be strengthened to $x_0 \lt x_n$ iff at least one of the premises can be strengthened to $x_i \lt x_{i+1}$. The entire forward direction of the second item here is missing from your list; it may be too strong.

   I don't know a name, but there should be one! Another place that this pair comes up is in non-located (one-sided or MacNeille) real numbers.

   We can leave transitivity of $\lt$ out of the list of axioms (since it follows in two ways), but we also should add its irreflexivity: If $x \lt x$, then False.

   An unbiased list of axioms is (in addition to $\lt \Rightarrow \leq$) this:

   • If $x_0 \leq \cdots \leq x_n$, then $x_0 \leq x_n$;
   • The conclusion can be strengthened to $x_0 \lt x_n$ iff at least one of the premises can be strengthened to $x_i \lt x_{i+1}$.

   The entire forward direction of the second item here is missing from your list; it may be too strong.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 20th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI think it is too strong. Consider the ordinal $x=\{0|P\}$. Then $0\le x$ and $x\le 1$ and $0\lt 1$, but "$0\lt x$ or $x\lt 1$" makes P decidable.

   I think it is too strong. Consider the ordinal $x=\{0|P\}$. Then $0\le x$ and $x\le 1$ and $0\lt 1$, but “$0\lt x$ or $x\lt 1$” makes P decidable.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 20th 2014
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   Author: Mike Shulman
   Format: MarkdownItex(And all those ordinals are plump.)

   (And all those ordinals are plump.)

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 20th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAnother way to say what we're talking about is that the same set is equipped with both a [[partial order]] and a [[quasiorder]] in a "compatible" way.

   Another way to say what we’re talking about is that the same set is equipped with both a partial order and a quasiorder in a “compatible” way.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 20th 2014
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYes, this would restrict attention to some sort of super-plump ordinal (and so not be for the page [[plump ordinal]]). But I don\'t exactly understand Taylor\'s motivation for restricting to plump ordinals in the first place.

   Yes, this would restrict attention to some sort of super-plump ordinal (and so not be for the page plump ordinal).

   But I don't exactly understand Taylor's motivation for restricting to plump ordinals in the first place.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 24th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI don't entirely understand his motivation myself, but it seems to be at least partly that he wants a successor satisfying $x \le y$ iff $x \lt s y$. Joyal-Moerdijk also arrived at an equivalent notion of plump ordinal from algebraic-set-theory considerations. My recent interest in plump ordinals comes from the fact that [they embed constructively in the surreals](http://homotopytypetheory.org/2014/02/22/surreals-plump-ordinals/).

   I don’t entirely understand his motivation myself, but it seems to be at least partly that he wants a successor satisfying $x \le y$ iff $x \lt s y$. Joyal-Moerdijk also arrived at an equivalent notion of plump ordinal from algebraic-set-theory considerations.

   My recent interest in plump ordinals comes from the fact that they embed constructively in the surreals.

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 24th 2014
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   Author: TobyBartels
   Format: MarkdownItex>My recent interest in plump ordinals comes from the fact that [they embed constructively in the surreals](http://homotopytypetheory.org/2014/02/22/surreals-plump-ordinals/). Yes, I noticed that! I also see that you have a higher inductive definition of plump ordinals themselves. This makes them more interesting to me too.

   My recent interest in plump ordinals comes from the fact that they embed constructively in the surreals.

   Yes, I noticed that! I also see that you have a higher inductive definition of plump ordinals themselves. This makes them more interesting to me too.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeSep 21st 2023
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   Author: TobyBartels
   Format: MarkdownItexExamples <a href="https://ncatlab.org/nlab/revision/diff/plump+ordinal/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/plump+ordinal/5">v5</a>, <a href="https://ncatlab.org/nlab/show/plump+ordinal">current</a>

   Examples

   diff, v5, current