On Sun, 02 Aug 2009, Kip Murray 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:
iirc this should be undefined.
>
> 2. Question: What is an element of a set?
> Answer:
iirc this should be undefined, but you can/should define a test for
membership, eg. `e.' in J
>
> 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:
using negation of Q4: no element of empty set which is not an element of
any set. Or there is no counter example that there exits one element of
phi that not belonged to other sets.
Last night I googled and there is quotation that elements inside a set
need not be distinct, but 2 sets are consider equal if they only
differ in repeated elements eg.
{ 1 2 3 } = { 1 1 2 3} = { 1 1 2 2 3 3 3}
the equality follows from Q3.
--
regards,
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