On 07 Jul 2009, at 04:03, m.a. wrote:

> Questions and comments interspersed below (in bold)
>>
>> {3, 5} belongs-to {3, 5} True
>
> Not OK. The elements of {3, 5} are 3 and 5. {3, 5} is not an  
> *element* of {3, 5}.Why not? They look like elements to me. Please  
> define "elements" as applies to this example..

Think about a set as it is a sort of box. For example the set {3, 5}  
can be seen as the empty box { } in which you place the object 3, and  
then the object 5. In this case the set {3, 5} contains two elements,  
3 and 5, which appears to be natural numbers. In particular the set  
{3, 5} contains only numbers.

So if I ask you if 3 and 5 are elements of {3, 5}, the answer is TRUE,  
and I guess that this is how you have interpret the question.

But the question was not "is 3 and 5 elements of {3, 5}?". The  
question was "is {3, 5} an element of {3, 5}"? This is really the  
question: "is the mathematical object {3, 5}, which is a set,  an  
element of {3, 5}?"; But just above we have seen that {3, 5} contains  
only numbers, and the object {3, 5} is not a number (indeed it is a  
set), and there is not set in {3,5}, only numbers.

Look: {3, 5} is a box which contains two numbers, 3 and 5, and nothing  
else.
A set in which {3, 5} would be itself an element would be, for example  
{7, 8, {3, 5}}, which can be seen as a box which contains three  
things, the number 3, the number 5, and the box {3, 5}.   {7, 8, {3,  
5}} is an hybrid set which contains two numbers and a set.

Do you see the difference between { }, the empty box, and {{ }}, which  
is a box which contains the empty box. If you put an empty box in a  
box, that box is no more empty: it contains an empty box. OK? All the  
interest of the notion of set, is that it makes a "many" into a "one".  
{3, 5} is the mathematical unique object, a set, which has 3 and 5 as  
element. And it can itself be an element of another set, like {{3,  
5}}, or {{3, 5}, 7}.

You were confusing the question:

- Are the numbers 3, 5 elements of {3, 5}?  (answer: yes)
- Is the set  {3, 5} an element of {3, 5}?  (answer: no).

I give you more examples:

3 belongs-to {0, 1, 2, 3, 4}   TRUE.
{3} belongs to {0, 1, 2, 3, 4}  FALSE
{3} belongs-to {{0}, {1}, {2}, {3}, {4}}  TRUE
{3} belongs-to {0, 1, 2, {3}, 4} TRUE
{3} belongs-to {0, 1, 2, {3, 4}} FALSE
{3,4} belongs-to {0, 1, 2, {3, 4}} TRUE
{3, 4} belongs-to {{0, 1, 2} {3, 4}} TRUE

Tell me if you are OK with those examples. Keep in mind typical  
situation, like:

  {2, 3} is a set with two elements: the number 2, and 3.
{{2, 3}} is a set with one element: the set {2, 3}.


>
> I give you a plan, and some more motivation. To get the seventh step  
> in some proper way, there is a need to understand the mathematical  
> notion of "universal machine".
>
> I've read about Turing machines if that's what you're referring to.
Not exactly.  Turing machines are indeed "mathematical machine", and  
"universal Turing machine" do exist. But most Turing machine are not  
universal. And not all "universal machine" are Turing machine.

So the set of universal machines has a non empty intersection with the  
set of Turing machines, but that is the most we can say. Some Turing  
machine are not universal, and some universal machine are not Turing  
machine. But here we are anticipating.

Bruno


http://iridia.ulb.ac.be/~marchal/




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