On Sun, Aug 2, 2009 at 5:23 PM, Kip Murray <[email protected]> wrote:
> Tracy, I abandon my model of finite set theory to absorb the attractive
> proposal
> of Dan and Fraser. Fraser's set-creating verb is
>
> Set =: (/:~)@~. NB. sorted nub
>
> and I take your suggestion to be, maybe "sorted" can be dispensed with. I
> have
> to punt to the group while I absorb.
>
> I propose that any model of set theory should complete the following
> "catechism":
>
> 1. Question: What is a set?
> Answer:
>
> 2. Question: What is an element of a set?
> Answer:
>
> 3. Question: When are two sets the same set?
> Answer: Set H is set K provided each element of H is an element of K and
> each element of K is an element of H.
>
> 4. Question: What is a subset of a set?
> Answer: To say H is a subset of set K means H is a set, and each element
> of H
> is an element of K.
>
> 5. Question: What is an empty set?
> Answer: The empty set, called phi, is the set which has no element.
>
> 6. Question: Why did you say "The" empty set?
> Answer: If H is a set which has no element, then there is no element of
> H
> that is not an element of phi, and there is no element of phi
> that is
> not an element of H -- because there is no element of H or phi.
> By 3,
> H is phi. Thus phi is the only empty set.
>
> (It is standard to interpret "Each one is" to mean "Not one is not".)
>
> 7. Question: Why is the empty set a subset of every set?
> Answer:
>
> 8. Question: Can the empty set be an element of a set?
> Answer: That depends on 2, but the answer to 2 should make this answer
> "Yes."
>
> 9. Question: Can a set be a subset of itself?
Answer: According to Question 4 the answer is yes.
>
>
> (Add questions)
>
>
> Kip
>
>
> Tracy Harms wrote:
> > On Sun, Aug 2, 2009 at 4:27 AM, Kip Murray<[email protected]> wrote:
> >> ...
> >> Footnote:
> >>
> >> The empty is a subset (not an element) of every set.
> >>
> >
> > This fact suggests to me that the empty box that's been included as an
> > element of the set-representations should be omitted. I think doing so
> > would streamline this model, most obviously (as others have proposed)
> > by allowing non-box arrays to stand as sets.
> >
> > A set could then be taken to be any array where (-: ~.) y.
> >
> > --
> > Tracy
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