Bill, taking your points in order:
Question 1. In abstract set theory, "set" and "is an element of" are
undefined,
but an implementor of a concrete model for set theory must define which objects
he is using to represent sets. That is, an answer for Question 1 is required
of
an implementor of a concrete model.
Question 2. You are correct, it is the relation "is an element of" that must
be
defined. I misstated Question 2.
Question 7. That's exactly right.
Multiple occurrences of an element. I had a teacher who said one day, "You are
not going to like this, but we are going to allow a number to be a member more
than once, and we will take into account 'multiple membership' in finding the
sum of the numbers in a finite set. We will then use the idea of sum for a
finite set to define what is meant by a sum for the numbers in an infinite set.
Not every infinite set of numbers has a sum." So, multiple occurrences of
the
same element are sometimes allowed, but are frowned on! Outside of that one
course, they were never allowed in my training, and I learned later it is more
acceptable to talk about a sum of an infinite sequence (in sequences
repetitions
_are_ allowed), and that the teacher was using "sum for a set" to finesse
absolute convergence. ...more than you wanted to know In brief, I think of
the
names you wrote as different names for the same set of only three numbers.
By the way, what does iirc mean? "if I recall"?
Kip
bill lam wrote:
> 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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