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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJun 19th 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI\'ve seen two meanings for this term, and they are both at [[limit point]], along with a family of other terms for various arity classes.

   I've seen two meanings for this term, and they are both at limit point, along with a family of other terms for various arity classes.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJun 19th 2013
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   Author: Mike Shulman
   Format: MarkdownItexNice! Unfortunately, though, $\{0\}$ is not an arity class -- arity classes have to contain 1. Maybe this is somewhere where you really do want a regular cardinal rather than an arity class? Why 'indexed subset' rather than 'family'? Finally, for the HoTT book we settled on 'disequality' as a name for negated equality, on the grounds that 'inequality' is commonly used to refer instead to $\lt$ and $\leq$. Apparently 'disequality' is not uncommon among type theorists. We might want to consider this issue for the nlab.

   Nice! Unfortunately, though, $\{0\}$ is not an arity class – arity classes have to contain 1. Maybe this is somewhere where you really do want a regular cardinal rather than an arity class?

   Why ’indexed subset’ rather than ’family’?

   Finally, for the HoTT book we settled on ’disequality’ as a name for negated equality, on the grounds that ’inequality’ is commonly used to refer instead to $\lt$ and $\leq$. Apparently ’disequality’ is not uncommon among type theorists. We might want to consider this issue for the nlab.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 19th 2013
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   Author: Mike Shulman
   Format: MarkdownItexActually, it's not obvious to me why you even want to exclude singular cardinals. Not that I have any idea what use the notion would be for cardinals larger than $\aleph_1$, but at least it doesn't seem to collapse into triviality or anything at a singular cardinal. Does it?

   Actually, it’s not obvious to me why you even want to exclude singular cardinals. Not that I have any idea what use the notion would be for cardinals larger than $\aleph_1$, but at least it doesn’t seem to collapse into triviality or anything at a singular cardinal. Does it?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJun 19th 2013
   • (edited Jun 19th 2013)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>Unfortunately, though, $\{0\}$ is not an arity class -- arity classes have to contain 1. Maybe this is somewhere where you really do want a regular cardinal rather than an arity class? H\'m, yeah, that *should* be the arity class $\{1\}$, only that doesn\'t give us the correct result. (It gives us that $x$ is an accumulation point of $A$ if it is an adherent point, I think, which is not an interesting condition.) >Why 'indexed subset' rather than 'family'? More likely to be immediately understood, and implying that repetition is irrelevant. >for the HoTT book we settled on 'disequality' as a name for negated equality In constructive mathematics, I see ‘inequality’ a lot. But this is a generic term, *not* necessary negated equality (which is called ‘the denial inequality’). In general, an inequality could be any irreflexive symmetric relation, although one often restricts to tight inequalities. >it's not obvious to me why you even want to exclude singular cardinals I was thinking that you turn a $\kappa$-ary family of $\kappa$-ary families into a single $\kappa$-ary family, so may as well assume that $\kappa$ is regular, but maybe not.

   Unfortunately, though, $\{0\}$ is not an arity class – arity classes have to contain 1. Maybe this is somewhere where you really do want a regular cardinal rather than an arity class?

   H'm, yeah, that should be the arity class $\{1\}$, only that doesn't give us the correct result. (It gives us that $x$ is an accumulation point of $A$ if it is an adherent point, I think, which is not an interesting condition.)

   Why ’indexed subset’ rather than ’family’?

   More likely to be immediately understood, and implying that repetition is irrelevant.

   for the HoTT book we settled on ’disequality’ as a name for negated equality

   In constructive mathematics, I see ‘inequality’ a lot. But this is a generic term, not necessary negated equality (which is called ‘the denial inequality’). In general, an inequality could be any irreflexive symmetric relation, although one often restricts to tight inequalities.

   it’s not obvious to me why you even want to exclude singular cardinals

   I was thinking that you turn a $\kappa$-ary family of $\kappa$-ary families into a single $\kappa$-ary family, so may as well assume that $\kappa$ is regular, but maybe not.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 19th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI've never heard of an "indexed subset" before, but I know what a "family" is.

   I’ve never heard of an “indexed subset” before, but I know what a “family” is.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeJun 20th 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexBut ‘family’ is ambiguous; what if I say ‘indexed family’?

   But ‘family’ is ambiguous; what if I say ‘indexed family’?

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJun 20th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs there a "non-indexed" kind of family? I've never heard of one.

   Is there a “non-indexed” kind of family? I’ve never heard of one.

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeJun 20th 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYeah, I often see ‘family’ used to just mean a set (or class, or subset, whatever). It\'s an informal usage, along with ‘collection’ (and ‘class’, sometimes, when that\'s not being used for size distinctions). More careful people use it in the indexed sense.

   Yeah, I often see ‘family’ used to just mean a set (or class, or subset, whatever). It's an informal usage, along with ‘collection’ (and ‘class’, sometimes, when that's not being used for size distinctions). More careful people use it in the indexed sense.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 20th 2013
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI think Toby means 'indexed family' in the context of a collection of families indexed by another set.

   I think Toby means ’indexed family’ in the context of a collection of families indexed by another set.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 20th 2013
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHuh. (-:

    Huh. (-:

11. • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeJun 21st 2013
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>I was thinking that you turn a $\kappa$-ary family of $\kappa$-ary families into a single $\kappa$-ary family, so may as well assume that $\kappa$ is regular, This is indeed wrong. Given a $\kappa$-ary family of $\kappa$-ary families, each family has an element out of it, giving a $\kappa$-ary family of these exceptions (using choice!), which then has its own exception, but that need not be out of the union of the original families, so regularity is not relevant. (I already changed the text to deemphasize arity classes.)

    I was thinking that you turn a $\kappa$-ary family of $\kappa$-ary families into a single $\kappa$-ary family, so may as well assume that $\kappa$ is regular,

    This is indeed wrong. Given a $\kappa$-ary family of $\kappa$-ary families, each family has an element out of it, giving a $\kappa$-ary family of these exceptions (using choice!), which then has its own exception, but that need not be out of the union of the original families, so regularity is not relevant. (I already changed the text to deemphasize arity classes.)

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeJun 21st 2013
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexThe definition of accumulation point was wrong, so I had to redesign things slightly. It should be working now.

    The definition of accumulation point was wrong, so I had to redesign things slightly. It should be working now.