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."
...
(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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