*Please read between your lines included in bold* letters
*John
*
On Thu, Jul 16, 2009 at 4:13 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

>
>  On 15 Jul 2009, at 00:50, John Mikes wrote:
>
>  Bruno,
> I appreciate your grade-school teaching. We (I for one) can use it.
> I still find that whatever you explain is an 'extract' of what can be
> thought of a 'set' (a one representing a many).
> Your 'powerset' is my example.
> All those elements you put into { }s are the same as were the physical
> objects to Aristotle in his 'total' - the SUM of which was always MORE than
> the additives of those objects.
> Relations!
> The set is not an inordinate heap (correct me please, if I am off) of the
> elements, the elements are in SOME relation to each other and the "set"-idea
> of their ensemble, to *form* a SET.
> You stop short at the naked elements *together*, as I see.
>
>
> You get the idea.
> We can add structure to sets, by explicitly endowing them with operations
> and relations.
>
>
> *Furhter below you also expose the contrary (to simplify) - **I am afraid
> your "operations and relations" are restricted to the numbers-based (math?)
> domain, which is not what I mean by 'totality'. *
>
  **
>
>  *They wear cloths and hold hands. Mortar is among them.*
> Maybe your math-idea can tolerate any sequence and hiatus concerning to the
> 'set', and it still stays the same, as far as the *"math-idea you need"*goes,
>
>
> Yes, it is the methodology.
>
>
>
>  but if I go further (and you indicated that ANYTHING can form a set)
>
>
> More precisily, we can form a set of multiple thing we can conceive or
> defined.
>


>   *I would not restrict 'a set' to what WE can conceive, or define now.
> (Not even within the 'math'-related domain).*
>
>  the relations of the set-partners comes into play. Not only those which
> WE choose for 'interesting' to such set, but ALL OF THEM influencing the
> character of that *"ONE".*
> *Just musing.*
>
>
> It is OK. The idea consists in simplifying the things as much as possible,
> and then to realize that despite such simplification we are quickly driven
> to the unprovable, unnameable, un-reductible, far sooner than we could have
> imagine.
>

*I may suggest (or: assume?) that instead of "despite" it would make more
sense to write: "AS A CONSEQUENCE" *
*- think about it.*
**

>
> Bruno
>   *John*
>
>  On Tue, Jul 14, 2009 at 4:40 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>> Hi Kim, Marty, Johnathan, John, Mirek, and all...
>>
>> We were studying a bit of elementary set theory to prepare ourself to
>> Cantor's theorem, and then Kleene's theorem, which are keys to a good
>> understanding of the universal numbers, and to Church thesis, which are the
>> keys of the seven steps.
>> I intend to bring you to the comp enlightenment :)
>>
>> But first some revision. Read the following with attention!
>>
>> A set is a collection of things, which in general can themselves be
>> anything. Its use consists in making a many into a one.
>> If something, say x,  belongs to a set S, it is usually called "element"
>> of S. We abbreviate this by (x \belongs-to S).
>>
>> Example:
>>
>> A = {1, 2, 56}. A is a set with three elements which are the numbers 1, 2
>> and 56.
>>
>> We write:
>>
>> (1 \belongs-to {1, 2, 56}), or (1 \belongs-to A), or simply 1 \belongs-to
>> A, when no confusions exist. The parentheses "(" and ")" are just delimiters
>> for easing the reading. I write \belongs-to the relation "belongs to" to
>> remind it is a mathematical symbol.
>>
>> B = {Kim, Marty, Russell, Bruno, George, Jurgen} is  a set with 5 elements
>> which are supposed to be humans.
>>
>> C = {34, 54, Paul, {3, 4}}
>>
>> For this one, you may be in need of spectacles. In case of doubt, you can
>> expand it a little bit:
>>
>> C = {    34,     54,    Paul,   {3, 4}     }
>>
>> You see that C is a sort of hybrid set which has 4 elements:
>>
>>    - the number 34
>>    - the number 54
>>    - the human person Paul
>>    - the set {3, 4}
>>
>> Two key remarks:
>> 1) the number 3 is NOT an element of C. Nor is the number 4 an element of
>> C. 3 and 4 are elements of {3, 4}, which is an element of C. But, generally,
>> elements of elements are not elements! It could happen that element of
>> element are element, like in D = {3, 4, {3, 4}}, the number 3 is both an
>> element of D and element of an element of D ({3, 4}), but this is a special
>> circumstance due to the way D is defined.
>> 2) How do I know that "Paul" is a human, and not a dog. How do I know that
>> "Paul" does not refer just to the string "paul". Obvioulsy the expression
>> "paul" is ambiguous, and will usually be understood only in some context.
>> This will not been a problem because the context will be clear. Actually we
>> will consider only set of numbers, or set of mathematical objects which have
>> already been defined. Here I have use the person Paul just to remind that
>> typically set can have as elements any object you can conceive.
>>
>> What is the set of even prime number strictly bigger than 2. Well, to
>> solve this just recall that ALL prime numbers are odd, except 2. So this set
>> is empty. The empty set { } is the set which has no elements. It plays the
>> role of 0 in the world of sets.
>>
>>
>> We have seen some *operations* defined on sets.
>>
>> We have seen INTERSECTION, and UNION.
>>
>> The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7, 8} will
>> be written (S1 \inter S2), and is equal to the set of elements which belongs
>> to both S1 and S2. We have
>>
>> (S1 \inter S2) = {2, 3}
>>
>> We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and (x
>> belongs-to S2))}
>>
>> 2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2 belongs-to
>> S2))
>> 8 does not belongs to (S1 \inter S2) because it is false that ((2
>> belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.
>>
>> Of course some sets can be disjoint, that is, can have an empty
>> intersection:
>>
>> {1, 2, 3} \inter {4, 5, 6} = { }.
>>
>> Similarly we can define (S1 \union S2) by the set of the elements
>> belonging to S1 or belonging to S2:
>>
>> (S1 \union S2) = {x such-that ((x belongs-to S1) or (x belongs-to S2))}
>>
>> We have, with S1 and S2 the same as above (S1 = {1, 2, 3} and S2 = {2, 3,
>> 7, 8}):
>>
>> (S1 \union S2) = {1, 2, 3, 7, 8}.
>>
>> OK. I suggest you reread the preceding post, and let me know in case you
>> have a problem.
>>
>>
>> We have seen also a key *relation* defined on sets: the relation of
>> inclusion.
>>
>> We say that (A \included-in B) is true when all elements of A are also
>> elements of B.
>>
>> Example:
>> The set of ferocious dogs is included in the set of ferocious animals.
>> The set of even numbers is included in the set of natural numbers.
>> The set {2, 6, 8} is included in the set {2, 3, 4, 5, 6, 7, 8}
>> The set {2, 6, 8} is NOT included in the set  {2, 3, 4, 5, 7, 8}.
>>
>> When a set A is included in a set B, A is called a subset of B.
>>
>> We were interested in looking to all subsets of a some set.
>>
>> What are the subsets of {a, b} ?
>>
>> They are { }, {a}, {b}, {a, b}. Why?
>>
>> {a, b} is included in {a, b}. This is obvious. All elements of {a,b} are
>> elements of {a, b}.
>> {a} is included in {a, b}, because all elements of {a} are elements of {a,
>> b}
>> The same for {b}.
>> You see that to verify that a set with n elements is a subset of some set,
>> you have to make n verifications.
>> So, to see that the empty set is a subset of some set, you have to verify
>> 0 things. So the empty set is a subset of any set.
>>
>> proposition: { } is included-in any set.
>>
>> So the subsets of {a, b} are { }, {a}, {b}, {a, b}.
>>
>> But set have been invented to make a ONE from a MANY, and it is natural to
>> consider THE set of all subsets of a set. It is called the powerset of that
>> set.
>>
>> So the powerset of {a, b} is THE set {{ }, {a}, {b}, {a, b}}. OK?
>>
>> Train yourself on the following exercises:
>>
>> What is the powerset of { }
>> What is the powerset of {a}
>> What is the powerset of {a, b, c}
>>
>> Any question?
>>
>> This was a bit of revision, to let Kim catch up.
>>
>> The sequel will appear asap. Be sure everything is OK, and please, ask
>> question if it is not.
>> You can also ask any question on the first sixth steps of UDA ('course).
>>
>> Bruno
>>
>>   http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
>>
>>
>>
>
>
>
>
>  http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>

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