Footnote: Verb "set" is forgiving and correctly treats argument <2 as ,<2 . Kip
Kip Murray wrote:
> This has to do with Raul's question,
>
> Raul Miller wrote:
> > . . . how do you distinguish between an element of a
> > set and a set containing only that element?
>
>
> I will discuss three version of sets. Bear with me, we will get to "2 is the
> set whose only element is 2" in the second version.
>
>
> In mathematical set theory,
>
> 2 looks different from {2} -- but how do you know 2 isn't {2}? The
> question
> isn't as fanciful as it looks, for it is common in set theory to say
>
> 0 is defined to be {}, the empty set
> 1 is defined to be {0}, the set whose only element is 0
> 2 is defined to be {0,1}, the set whose only elements are 0 and 1
> ... and so on: the natural numbers are sets, and how do we know set 2 isn't
> the
> set {2}?
>
> W see 0 is {}, 1 is { {} }, and
>
> 2 is { {} , { {} } }, whereas
> {2} is { { {} , { {} } } }, and we are sure 2 is not {2}. (For one thing,
> {2}
> has only one element, and 2 has two elements!)
>
>
> But in the current J model of sets we have
>
> <"_1 ] 2 NB. <"_1 boxes items
> +-+
> |2|
> +-+
>
> which means 2 is the only element of set 2 -- we have defined sets to be open
> or
> boxed arrays and the elements of open arrays to be their items.
>
>
> In my original J model of sets,
>
> 2 NB. looks different from (set ,<2) the set whose only element
> is 2
> 2
> set ,<2 NB. argument is required to be a list of boxes containing
> elements
> +-++
> |2||
> +-++
>
> count set <2 NB. number of elements
> 1
> 2 iselement set <2
> 1
> 2 -: set <2 NB. not only looks different, is different
> 0
>
>
> Reference material:
>
>
> Current implementor catechism
>
> Question 1: What is a set?
> Answer: A set is an array.
>
> Question 2: What is an element of a set?
> Answer: If the set is open, an element is an item of the set. If the set
> is
> boxed, an element is the open of an atom of the set.
>
>
> Earlier implementor catechism (revised)
>
> Question 0: What is a sequence?
> Answer: A "sequence" is a list of boxes. A "term" of a sequence is the
> open
> of one of its boxes.
>
> Question 1: What is a set?
> Answer: A "set" is a sequence whose last term is i.0 0 .
>
> Question 2: What is an element of a set?
> Answer: An "element" of a set is a term of its curtail.
>
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