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From: John Mikes <jamikes.domain.name.hidden>

Date: Thu, 16 Jul 2009 09:17:23 -0400

*Please read between your lines included in bold* letters

*John

*

On Thu, Jul 16, 2009 at 4:13 AM, Bruno Marchal <marchal.domain.name.hidden> 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
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*> WE choose for 'interesting' to such set, but ALL OF THEM influencing the
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*> 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 <marchal.domain.name.hidden> 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"
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*>> of S. We abbreviate this by (x \belongs-to S).
*

*>>
*

*>> Example:
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*>>
*

*>> A = {1, 2, 56}. A is a set with three elements which are the numbers 1, 2
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*>> and 56.
*

*>>
*

*>> We write:
*

*>>
*

*>> (1 \belongs-to {1, 2, 56}), or (1 \belongs-to A), or simply 01 \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 05 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 04 elements:
*

*>>
*

*>> - the number 34
*

*>> - the number 54
*

*>> - the human person Paul
*

*>> - the set {3, 4}
*

*>>
*

*>> Two key remarks:
*

*>> 1) the number 03 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
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*>> 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
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*>> solve this just recall that ALL prime numbers are odd, except 2. So this set
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*>> is empty. The empty set { } is the set which has no elements. It plays the
*

*>> role of 00 in the world of sets.
*

*>>
*

*>>
*

*>> We have seen some *operations* defined on sets.
*

*>>
*

*>> We have seen INTERSECTION, and UNION.
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*>>
*

*>> The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7, 8} will
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*>> be written (S1 \inter S2), and is equal to the set of elements which belongs
*

*>> to both S1 and S2. We have
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*>>
*

*>> (S1 \inter S2) = {2, 3}
*

*>>
*

*>> We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and (x
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*>> belongs-to S2))}
*

*>>
*

*>> 02 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2 belongs-to
*

*>> S2))
*

*>> 08 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
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*>> 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:
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*>>
*

*>> (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,
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*>> 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:
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*>>
*

*>> 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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Received on Thu Jul 16 2009 - 09:17:23 PDT

Date: Thu, 16 Jul 2009 09:17:23 -0400

*Please read between your lines included in bold* letters

*John

*

On Thu, Jul 16, 2009 at 4:13 AM, Bruno Marchal <marchal.domain.name.hidden> wrote:

**

*I may suggest (or: assume?) that instead of "despite" it would make more

sense to write: "AS A CONSEQUENCE" *

*- think about it.*

**

--~--~---------~--~----~------------~-------~--~----~

You received this message because you are subscribed to the Google Groups "Everything List" group.

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Received on Thu Jul 16 2009 - 09:17:23 PDT

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