Thanks, Bruno. I think I've got it now. Sorry to be such a slow learner.
marty
----- Original Message -----
From: Bruno Marchal
To: [email protected]
Sent: Tuesday, July 07, 2009 3:30 AM
Subject: Re: The seven step series
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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