The seven step series

[Johnathan Corgan]

Bruno,

I think you were off to a good start with your planned series of posts
about the seven step argument.  I believe your first installment was a
discussion of set theory as one of the mathematical preliminaries to the
actual argument.

I am looking forward to your next installment.

Regards,

Johnathan Corgan



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[Bruno Marchal]

Hi Johnathan,


On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:

>
> Bruno,
>
> I think you were off to a good start with your planned series of posts
> about the seven step argument.  I believe your first installment was a
> discussion of set theory as one of the mathematical preliminaries to  
> the
> actual argument.
>
> I am looking forward to your next installment.

Well, thanks. I am not sure Kim and Marty are there, but I can provide  
a summary, and recall the motivation.

Marty, did you come back from holiday? Kim? still interested in  
electronical summer's school on mathematics.

The goal of the seven step thread is to make clear the seventh step of  
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to  
make clear that the mind-body problem (or the consciousness/reality  
problem, or the first person/third person) problem is reduced, when we  
do the computationalist assumption, to a pure body appearance or  
discourse problem. UDA shows that if we assume the comp. hyp. then we  
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of  
physics have to be extracted from self-reference.

Later, much later, I could explain, if everyone is OK with UDA, how we  
can already extract from self-reference the general shape of physics,  
so that we can already refute empirically, or confirm, the comp. hyp.  
And it appears that the empirical quantum mechanics,  currently,  
confirms the comp. hyp. Quantum mechanics confirms the partial  
indetermination of the outcomes of our possible experiences, and the  
"high non booleanity" of the propositions describing those outcomes".

The object of the "seventh step thread' consists in making the seventh  
step accessible to non mathematicians. So we have to start from zero.  
I have decided to start from elementary "naive" set theory, without  
which we cannot do anything in math. I will avoid all special  
mathematical symbols, and use instead words with capital letters.

We have not yet done a lot. So I can sum up, with the new "notations".

Definition. A set is just a "many" considered, when clear enough, as a  
"one". So a set is just a collection of objects, and those objects are  
called the element, or the member, of the set. If some x is an element  
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"  
means that we give all elements of the set, enclosed in accolades.  
When the set is not to complex (meaning big or infinite), we can use  
the "...". We can give name to a set, to ease or talk about that set,  
like we do all the times in mathematics. Most of the set we will  
consider are set of mathematical object, mainly numbers in the  
beginning, and then set of ... sets.

Example-exercise:

1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for  
the set {0, 1, 2, 3}. And local means that such a name is used in a  
local context. One paragraph later "A" could designed another, so be  
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".

OK, so with A = {0, 1, 2, 3}. Which of the following propositions are  
true

1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT  
x is a positive integer little than 4}

2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

This makes 14 exercises, which should be easy. I intent to keep it  
that way. I continue after I get either answers (correct or wrong), or  
questions.

Everyone is welcome to participate. Yet, I ask those who are quick to  
respect those who are slow. To be slow in the beginning usually help  
for being deep in the sequel.

Best,

Bruno

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




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[John Mikes]

Hi, Bruno

you know that I am in a different mindset, yet happy to read your train of thoughts. I consider a set a limited model of elements (and conclusions thereof are not applicable to wider domains) - when I read your

"A set can be described in extension or in intension. "in extension" means that we give all elements of the set, enclosed in accolades."

I was really happy with the next sentence:

"When the set is not to complex (meaning big or infinite), we can use the "...".  - "

(I missed here the exemption of the 'infinite' "set", really a contradiction, to which the 'set' considerations cannot apply - OR can they? if you have something on that...) "Many" cannot be infinite (by MY definition).

I loved your words on QM, the (linear) extension of the figment physical world as described in reductionist physical sciences.

I also cannot wait for something more about your approach on

the "self reference" - the basis of physics? - especially as to

'self' of what (who)? I hope the answer will not be "machine" or comp, because then I have to continue "and what is that?"

(in more than a utilitarian explanation of what it does). ('it?')

What boils down to my ignorance as to the originating and maintaining to ANY action we speak about. The 'theos' of a non-assumed and non-supernatural factor (system?) yet involved in conducting all we just find natural and proceeding.

You may substitute 'numbers' for such, but so far did not reply (to my satisfaction at least) WHAT those 'numbers' may be.

Sorry, I am not of the religious kind.

*

Maybe my error is in 'believeing' that a REALITY may exist and 'we' have only access to part of it. Inventing for our comfort (the D. Bohmian idea) 'numbers' at the human level of pre-Platonian thinking. If 'reality' exists only by 'comp' or 'consequences' then I may be in a reversed error, due to brainwashing by in college imprinted  natural sciences - what I try to exceed yet it still sits there.

Our 'perceived reality' (ColinH) may also provide the numbers.

Now that sounds heretical enough in this thread. Forgive me.

*

Waiting for the self-reference, (who's?) - with thanks so far

 

John Mikes

On Tue, Jun 30, 2009 at 6:45 AM, Bruno Marchal <marchal@...> wrote:

Hi Johnathan,

On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:

>
> Bruno,
>
> I think you were off to a good start with your planned series of posts
> about the seven step argument.  I believe your first installment was a
> discussion of set theory as one of the mathematical preliminaries to
> the
> actual argument.
>
> I am looking forward to your next installment.

Well, thanks. I am not sure Kim and Marty are there, but I can provide
a summary, and recall the motivation.

Marty, did you come back from holiday? Kim? still interested in
electronical summer's school on mathematics.

The goal of the seven step thread is to make clear the seventh step of
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
make clear that the mind-body problem (or the consciousness/reality
problem, or the first person/third person) problem is reduced, when we
do the computationalist assumption, to a pure body appearance or
discourse problem. UDA shows that if we assume the comp. hyp. then we
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of
physics have to be extracted from self-reference.

Later, much later, I could explain, if everyone is OK with UDA, how we
can already extract from self-reference the general shape of physics,
so that we can already refute empirically, or confirm, the comp. hyp.
And it appears that the empirical quantum mechanics,  currently,
confirms the comp. hyp. Quantum mechanics confirms the partial
indetermination of the outcomes of our possible experiences, and the
"high non booleanity" of the propositions describing those outcomes".

The object of the "seventh step thread' consists in making the seventh
step accessible to non mathematicians. So we have to start from zero.
I have decided to start from elementary "naive" set theory, without
which we cannot do anything in math. I will avoid all special
mathematical symbols, and use instead words with capital letters.

We have not yet done a lot. So I can sum up, with the new "notations".

Definition. A set is just a "many" considered, when clear enough, as a
"one". So a set is just a collection of objects, and those objects are
called the element, or the member, of the set. If some x is an element
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"
means that we give all elements of the set, enclosed in accolades.
When the set is not to complex (meaning big or infinite), we can use
the "...". We can give name to a set, to ease or talk about that set,
like we do all the times in mathematics. Most of the set we will
consider are set of mathematical object, mainly numbers in the
beginning, and then set of ... sets.

Example-exercise:

1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
the set {0, 1, 2, 3}. And local means that such a name is used in a
local context. One paragraph later "A" could designed another, so be
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".

OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
true

1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT
x is a positive integer little than 4}

2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

This makes 14 exercises, which should be easy. I intent to keep it
that way. I continue after I get either answers (correct or wrong), or
questions.

Everyone is welcome to participate. Yet, I ask those who are quick to
respect those who are slow. To be slow in the beginning usually help
for being deep in the sequel.

Best,

Bruno

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

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[m.a.-2]

Bruno,
            I'm still with you but I seriously wonder how far I can follow.
I have the sort of mind that groups of logical statement and propositions
cause to simply shut down. I am more than willing to accept that your proof
is consistent and I assume that others on the list will point out flaws if
there are any. What I would really appreciate would be a prose explanation
of the sequence of ideas that lead to the conclusion that physics can emerge
from and be (in some sense) actualized by math. If Kim and others wish to
continue the exposition of UDA, I'll try to keep up, but I won't ask you to
continue solely on my account. Best wishes,

                                                                             
                                                                             
                                                                             
                                    marty a.




----- Original Message -----
From: "Bruno Marchal" <marchal@...>
To: <everything-list@...>
Sent: Tuesday, June 30, 2009 6:45 AM
Subject: Re: The seven step series



Hi Johnathan,


On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:

>
> Bruno,
>
> I think you were off to a good start with your planned series of posts
> about the seven step argument.  I believe your first installment was a
> discussion of set theory as one of the mathematical preliminaries to
> the
> actual argument.
>
> I am looking forward to your next installment.

Well, thanks. I am not sure Kim and Marty are there, but I can provide
a summary, and recall the motivation.

Marty, did you come back from holiday? Kim? still interested in
electronical summer's school on mathematics.

The goal of the seven step thread is to make clear the seventh step of
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
make clear that the mind-body problem (or the consciousness/reality
problem, or the first person/third person) problem is reduced, when we
do the computationalist assumption, to a pure body appearance or
discourse problem. UDA shows that if we assume the comp. hyp. then we
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of
physics have to be extracted from self-reference.

Later, much later, I could explain, if everyone is OK with UDA, how we
can already extract from self-reference the general shape of physics,
so that we can already refute empirically, or confirm, the comp. hyp.
And it appears that the empirical quantum mechanics,  currently,
confirms the comp. hyp. Quantum mechanics confirms the partial
indetermination of the outcomes of our possible experiences, and the
"high non booleanity" of the propositions describing those outcomes".

The object of the "seventh step thread' consists in making the seventh
step accessible to non mathematicians. So we have to start from zero.
I have decided to start from elementary "naive" set theory, without
which we cannot do anything in math. I will avoid all special
mathematical symbols, and use instead words with capital letters.

We have not yet done a lot. So I can sum up, with the new "notations".

Definition. A set is just a "many" considered, when clear enough, as a
"one". So a set is just a collection of objects, and those objects are
called the element, or the member, of the set. If some x is an element
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"
means that we give all elements of the set, enclosed in accolades.
When the set is not to complex (meaning big or infinite), we can use
the "...". We can give name to a set, to ease or talk about that set,
like we do all the times in mathematics. Most of the set we will
consider are set of mathematical object, mainly numbers in the
beginning, and then set of ... sets.

Example-exercise:

1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
the set {0, 1, 2, 3}. And local means that such a name is used in a
local context. One paragraph later "A" could designed another, so be
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".

OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
true

1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT
x is a positive integer little than 4}

2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

This makes 14 exercises, which should be easy. I intent to keep it
that way. I continue after I get either answers (correct or wrong), or
questions.

Everyone is welcome to participate. Yet, I ask those who are quick to
respect those who are slow. To be slow in the beginning usually help
for being deep in the sequel.

Best,

Bruno

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






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[Bruno Marchal]

Hi John,

On 30 Jun 2009, at 14:32, John Mikes wrote:

Hi, Bruno

you know that I am in a different mindset, yet happy to read your train of thoughts. I consider a set a limited model of elements (and conclusions thereof are not applicable to wider domains) -

Well, as Thorgny illustrated correctly, the notion of SET does not apply to SETS. usually the collection of all sets is not considered as a set, for many good reasons, some of which appear already in the treatise on Numbers by Plotinus. To derive from this that infinite sets cannot exist, like Thorgny seems to believed is invalid.

when I read your

"A set can be described in extension or in intension. "in extension" means that we give all elements of the set, enclosed in accolades."

I was really happy with the next sentence:

"When the set is not to complex (meaning big or infinite), we can use the "...".  - "

(I missed here the exemption of the 'infinite' "set", really a contradiction, to which the 'set' considerations cannot apply - OR can they?

Of course they can. This is done in everyday mathematics all the time since Cantor discovered the notion of sets. It can be said that set have been exploited especially for the handling of infinities.

Typical infinite sets are the set of natural numbers {0, 1, 2, ...}, the set of odd numbers {1, 3, 5, 7, ...}, the set of prime numbers {2, 3, 5, 7, 11, ...} as already proved by Euclid, the set of decimal approximation of most real numbers, etc.

if you have something on that...)

Google on it. The notion of set (finite and especially infinite sets) is pervading all modern mathematics.

"Many" cannot be infinite (by MY definition).

Well, I prefer to use the word in their most used and standard sense. 

I loved your words on QM, the (linear) extension of the figment physical world as described in reductionist physical sciences.

I also cannot wait for something more about your approach on

the "self reference" - the basis of physics? -

This is the whole point of the UDA, and AUDA.

especially as to

'self' of what (who)? I hope the answer will not be "machine" or comp, because then I have to continue "and what is that?"

Of course it is comp, although I use comp because it is the simple way. Then, digging on mathematical logic, the same result can be retrieve from much weaker assumption. But comp is believed by all scientist and philosophers, except Penrose. Even John Searle is computationalist with my weak definition of it.

"and what is that"? It is the point of the present thread to explain that as slowly as possible so that good willing non mathematicians can understand the key points. 

(in more than a utilitarian explanation of what it does). ('it?')

What boils down to my ignorance as to the originating and maintaining to ANY action we speak about. The 'theos' of a non-assumed and non-supernatural factor (system?) yet involved in conducting all we just find natural and proceeding.

You may substitute 'numbers' for such, but so far did not reply (to my satisfaction at least) WHAT those 'numbers' may be.

Sorry, I am not of the religious kind.

If you can compute 34+89, or 65*87, then you know enough. My use of number in UDA is not religious, it is the same use as those who use number to send man on the moon. Too much and premature philosophizing makes it hard to proceed. "Religion" appears later, when you say "yes" to the digitalist surgeon. Comp asks for an act of faith.

*

Maybe my error is in 'believeing' that a REALITY may exist and 'we' have only access to part of it.

Neither science, nor philosophy, nor theology, could develop without such assumption. It is because we believe that there is a reality, that we can build theories to infer the part on which we have no access. science always consist in an attempt to see the invisible, be it atoms, far away galaxies, or mathematical constructions. So this is not an error.

Inventing for our comfort (the D. Bohmian idea) 'numbers' at the human level of pre-Platonian thinking. If 'reality' exists only by 'comp' or 'consequences' then I may be in a reversed error, due to brainwashing by in college imprinted  natural sciences - what I try to exceed yet it still sits there.

Our 'perceived reality' (ColinH) may also provide the numbers.

Sure, but this is independent of the fact that 17 is a prime numbers independently of me. That the set of prime numbers is infinite, independently of me, etc. 

Now that sounds heretical enough in this thread. Forgive me.

*

Waiting for the self-reference, (who's?)

Who's? But the universal number's self-reference of course. Even the Lobian one.

- with thanks so far

You are welcome,

Bruno

On Tue, Jun 30, 2009 at 6:45 AM, Bruno Marchal <marchal@...> wrote:

Hi Johnathan,

On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:

>
> Bruno,
>
> I think you were off to a good start with your planned series of posts
> about the seven step argument.  I believe your first installment was a
> discussion of set theory as one of the mathematical preliminaries to
> the
> actual argument.
>
> I am looking forward to your next installment.

Well, thanks. I am not sure Kim and Marty are there, but I can provide
a summary, and recall the motivation.

Marty, did you come back from holiday? Kim? still interested in
electronical summer's school on mathematics.

The goal of the seven step thread is to make clear the seventh step of
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
make clear that the mind-body problem (or the consciousness/reality
problem, or the first person/third person) problem is reduced, when we
do the computationalist assumption, to a pure body appearance or
discourse problem. UDA shows that if we assume the comp. hyp. then we
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of
physics have to be extracted from self-reference.

Later, much later, I could explain, if everyone is OK with UDA, how we
can already extract from self-reference the general shape of physics,
so that we can already refute empirically, or confirm, the comp. hyp.
And it appears that the empirical quantum mechanics,  currently,
confirms the comp. hyp. Quantum mechanics confirms the partial
indetermination of the outcomes of our possible experiences, and the
"high non booleanity" of the propositions describing those outcomes".

The object of the "seventh step thread' consists in making the seventh
step accessible to non mathematicians. So we have to start from zero.
I have decided to start from elementary "naive" set theory, without
which we cannot do anything in math. I will avoid all special
mathematical symbols, and use instead words with capital letters.

We have not yet done a lot. So I can sum up, with the new "notations".

Definition. A set is just a "many" considered, when clear enough, as a
"one". So a set is just a collection of objects, and those objects are
called the element, or the member, of the set. If some x is an element
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"
means that we give all elements of the set, enclosed in accolades.
When the set is not to complex (meaning big or infinite), we can use
the "...". We can give name to a set, to ease or talk about that set,
like we do all the times in mathematics. Most of the set we will
consider are set of mathematical object, mainly numbers in the
beginning, and then set of ... sets.

Example-exercise:

1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
the set {0, 1, 2, 3}. And local means that such a name is used in a
local context. One paragraph later "A" could designed another, so be
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".

OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
true

1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT
x is a positive integer little than 4}

2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

This makes 14 exercises, which should be easy. I intent to keep it
that way. I continue after I get either answers (correct or wrong), or
questions.

Everyone is welcome to participate. Yet, I ask those who are quick to
respect those who are slow. To be slow in the beginning usually help
for being deep in the sequel.

Best,

Bruno

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

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

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[Bruno Marchal]

Hi Marty,


On 30 Jun 2009, at 18:57, m.a. wrote:

>
> Bruno,
>            I'm still with you but I seriously wonder how far I can  
> follow.


Well, honestly, if you don't try to answer to the exercises, we will  
never know.
But I can imagine some shyness for doing so.



>
> I have the sort of mind that groups of logical statement and  
> propositions
> cause to simply shut down.


I disagree. You have already prove to me that you can handle such  
propositions. Your problem is that you don't memorize what you  
understand, so, especially after some break, you feel like you could  
shut down.
This just means that you have to work a little more, and to stop  
building negative self-prejudices, which are most of the time self-
fulfilling.
Now, I obvioulsy cannot ask you to do such a work, as life is rich and  
full of consuming time opportunities, noir can I really provide the  
motivation for doing so.
I appreciate very much your honesty, even if I suspect the reasons are  
bad.



> I am more than willing to accept that your proof
> is consistent and I assume that others on the list will point out  
> flaws if
> there are any.

The problem is that we have discussed it before. The only things which  
remains to be explained is what is a mathematical computation. This is  
not easy, and ask for some familiarity with basic mathematics.


> What I would really appreciate would be a prose explanation
> of the sequence of ideas that lead to the conclusion that physics  
> can emerge
> from and be (in some sense) actualized by math.

My problem is that UDA is exactly that. I get the AUDA in the early  
seventies, before UDA. And I have developed UDA in the late seventies,  
so as to provide some help for my friends. Since then I know that UDA  
is not so simple, especially the seventh and eighth steps. I can think  
about making a shorter attempt.



> If Kim and others wish to
> continue the exposition of UDA, I'll try to keep up, but I won't ask  
> you to
> continue solely on my account. Best wishes,




No problem Marty, at least you have tried. Kim? how do you do?  
Johnathan?

I know that to play the "candid" role in a public way, you need some  
courage, and it is OK to remain silent. But people have to understand  
than on those delicate matters, it is not really possible to proceed  
without asking question/exercise. We can only continue "the seven step  
series" thread if people provide answer to the exercises.
I will nevertheless try to explain things in a more "journalistic",  
and shorter, way. The problem is that such a thing can easily be  
misinterpreted. I will think about it. The "real thing" is more easy  
to explain, but admittedly longer when we start from quasi zero.

Bruno

>
> ----- Original Message -----
> From: "Bruno Marchal" <marchal@...>
> To: <everything-list@...>
> Sent: Tuesday, June 30, 2009 6:45 AM
> Subject: Re: The seven step series
>
>
>
> Hi Johnathan,
>
>
> On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:
>
>>
>> Bruno,
>>
>> I think you were off to a good start with your planned series of  
>> posts
>> about the seven step argument.  I believe your first installment  
>> was a
>> discussion of set theory as one of the mathematical preliminaries to
>> the
>> actual argument.
>>
>> I am looking forward to your next installment.
>
>
> Well, thanks. I am not sure Kim and Marty are there, but I can provide
> a summary, and recall the motivation.
>
> Marty, did you come back from holiday? Kim? still interested in
> electronical summer's school on mathematics.
>
> The goal of the seven step thread is to make clear the seventh step of
> the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
> make clear that the mind-body problem (or the consciousness/reality
> problem, or the first person/third person) problem is reduced, when we
> do the computationalist assumption, to a pure body appearance or
> discourse problem. UDA shows that if we assume the comp. hyp. then we
> have to explain the appearance of matter from machine or number self-
> reference only. The proof is constructive, it shows *how* the laws of
> physics have to be extracted from self-reference.
>
> Later, much later, I could explain, if everyone is OK with UDA, how we
> can already extract from self-reference the general shape of physics,
> so that we can already refute empirically, or confirm, the comp. hyp.
> And it appears that the empirical quantum mechanics,  currently,
> confirms the comp. hyp. Quantum mechanics confirms the partial
> indetermination of the outcomes of our possible experiences, and the
> "high non booleanity" of the propositions describing those outcomes".
>
> The object of the "seventh step thread' consists in making the seventh
> step accessible to non mathematicians. So we have to start from zero.
> I have decided to start from elementary "naive" set theory, without
> which we cannot do anything in math. I will avoid all special
> mathematical symbols, and use instead words with capital letters.
>
> We have not yet done a lot. So I can sum up, with the new "notations".
>
> Definition. A set is just a "many" considered, when clear enough, as a
> "one". So a set is just a collection of objects, and those objects are
> called the element, or the member, of the set. If some x is an element
> of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
> A set can be described in extension or in intension. "in extension"
> means that we give all elements of the set, enclosed in accolades.
> When the set is not to complex (meaning big or infinite), we can use
> the "...". We can give name to a set, to ease or talk about that set,
> like we do all the times in mathematics. Most of the set we will
> consider are set of mathematical object, mainly numbers in the
> beginning, and then set of ... sets.
>
> Example-exercise:
>
> 1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
> the set {0, 1, 2, 3}. And local means that such a name is used in a
> local context. One paragraph later "A" could designed another, so be
> careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
>
> OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
> true
>
> 1) the number 2 is a member of A
> 2) the number 12 is a member of A
> 3) the number 12 is not a member of A
> 4) (3 BELONGS-TO A)
> 5) all members of A are numbers
> 6) one element of A is not a number
> 7) A can be defined in intension in the following way A = {x SUCH-THAT
> x is a positive integer little than 4}
>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}
>
> This makes 14 exercises, which should be easy. I intent to keep it
> that way. I continue after I get either answers (correct or wrong), or
> questions.
>
> Everyone is welcome to participate. Yet, I ask those who are quick to
> respect those who are slow. To be slow in the beginning usually help
> for being deep in the sequel.
>
> Best,
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
>
>
> >

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




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[m.a.-2]

Bruno,
            Your faith in my ability to master the logic is encouraging. I
remain determined to wrestle with the steps in whichever form in which you
decide to present them. Tally ho,
                                                                             
                                                                             
                                                                             
                                        m.a.


----- Original Message -----
From: "Bruno Marchal" <marchal@...>
To: <everything-list@...>
Sent: Wednesday, July 01, 2009 12:00 PM
Subject: Re: The seven step series



Hi Marty,


I know that to play the "candid" role in a public way, you need some
courage, and it is OK to remain silent. But people have to understand
than on those delicate matters, it is not really possible to proceed
without asking question/exercise. We can only continue "the seven step
series" thread if people provide answer to the exercises.
I will nevertheless try to explain things in a more "journalistic",
and shorter, way. The problem is that such a thing can easily be
misinterpreted. I will think about it. The "real thing" is more easy
to explain, but admittedly longer when we start from quasi zero.

Bruno







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






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[m.a.-2]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Hi Bruno,

                I'm responding to the quiz (see below). What does "high non booleanity" mean in the context of para.2?

> ----- Original Message -----

> From: "Bruno Marchal" <marchal@...>
> To: <everything-list@...>
> Sent: Tuesday, June 30, 2009 6:45 AM
> Subject: Re: The seven step series
>
>
>
> Hi Johnathan,
>
>
> The goal of the seven step thread is to make clear the seventh step of
> the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
> make clear that the mind-body problem (or the consciousness/reality
> problem, or the first person/third person) problem is reduced, when we
> do the computationalist assumption, to a pure body appearance or
> discourse problem. UDA shows that if we assume the comp. hyp. then we
> have to explain the appearance of matter from machine or number self-
> reference only. The proof is constructive, it shows *how* the laws of
> physics have to be extracted from self-reference.
>
> Later, much later, I could explain, if everyone is OK with UDA, how we
> can already extract from self-reference the general shape of physics,
> so that we can already refute empirically, or confirm, the comp. hyp.
> And it appears that the empirical quantum mechanics,  currently,
> confirms the comp. hyp. Quantum mechanics confirms the partial
> indetermination of the outcomes of our possible experiences, and the
> "high non booleanity" of the propositions describing those outcomes".
>
> The object of the "seventh step thread' consists in making the seventh
> step accessible to non mathematicians. So we have to start from zero.
> I have decided to start from elementary "naive" set theory, without
> which we cannot do anything in math. I will avoid all special
> mathematical symbols, and use instead words with capital letters.
>
> We have not yet done a lot. So I can sum up, with the new "notations".
>
> Definition. A set is just a "many" considered, when clear enough, as a
> "one". So a set is just a collection of objects, and those objects are
> called the element, or the member, of the set. If some x is an element
> of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
> A set can be described in extension or in intension. "in extension"
> means that we give all elements of the set, enclosed in accolades.
> When the set is not to complex (meaning big or infinite), we can use
> the "...". We can give name to a set, to ease or talk about that set,
> like we do all the times in mathematics. Most of the set we will
> consider are set of mathematical object, mainly numbers in the
> beginning, and then set of ... sets.
>
> Example-exercise:
>
> 1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
> the set {0, 1, 2, 3}. And local means that such a name is used in a
> local context. One paragraph later "A" could designed another, so be
> careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
>
> OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
> true
>
> 1) the number 2 is a member of A   True
> 2) the number 12 is a member of A  False
> 3) the number 12 is not a member of A  True
> 4) (3 BELONGS-TO A)    True: but you haven't told us whether the parenthesis cancels the locality of brackets.
> 5) all members of A are numbers  True
> 6) one element of A is not a number  False: we've established that zero is a number.
> 7) A can be defined in intension in the following way A = {x SUCH-THAT
> x is a positive integer little than 4}    True...if zero is considered a positive integer.
>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

1. True

2. True

3. False

4. True: same question as 4 above.

5. True

6. False: zero is a number

7. False

>
> This makes 14 exercises, which should be easy. I intent to keep it
> that way. I continue after I get either answers (correct or wrong), or
> questions.
>
> Everyone is welcome to participate. Yet, I ask those who are quick to
> respect those who are slow. To be slow in the beginning usually help
> for being deep in the sequel.
>
> Best,
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/

>
>
>
>
>
>
> >

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




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[John Mikes]

Hi, Bruno,

and thanks for taking the time to eviscerate my ignorance. I asked for it <G>. Just a bit in detail:

I did not pay enough attention to Torgny's sets vs set, I see now.

Question: There are 'rules' applicable how to handle 'a' set, operations and values to obtain etc. Are these rules also valid and applicable to Torgny's "sets", the infinite set of all sets? Can you do with that anything (practical in some sense)? even 'think' of it?  (BTW I did not mean that 'infinite' sets do not exist, only that we need a different consideration for them from the finite bunch usually understood as 'a' set - maybe also 'naming' them differently). 

 

I did not find the 'intension' sets (description), only the 'extension' part.  

 

(JM)"Many" cannot be infinite (by MY definition).(BM):Well, I prefer to use the word in their most used and standard sense. 

-----which one is that? Can, or cannot?-----

 

I did not identify "self-referenced" with computationalist. And that 'many' think so is no satisfying evidence in my view: scientific thinking is not a democratic voting formula. Even the "BIG" names... Al Gore and Jimmy Carter received Nobels.

There is no "scientific" statement which could not be contrasted by two opposite ones of 'other' scientists.

Now the delicate part:

BM:

"If you can compute 34+89, or 65*87, then you know enough."

IF!! I accept that a number means a math-book and writing a number is not 'that' number without applying 'rules'. Like:

34 is a 3 and a 4 unless you defined the space and comma if applicable. (I have something on that 'comma' I forgot to write about. Maybe now I will resort to it at the end). To make the 3 !!! and the 4 !!!! a !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! is far fetched.

Then again your + and * are NOT numbers. It looks to me (in a personalized overwhelming exaggeration, of course) that by 'numbers' you mean the 'science of math expressed in symbols of a huge vocabulary' (maybe books ->library!) Ph.D.

I don't deny the practicality of applying 'numbers-based' science in sending a man to Mars, but it is NOT the numbers that does the job. It is the complexity of the state of the art we reached, which includes science, technology, skills, ideas AND of course numbers-application. Bohm's idea - as I understood it - was that searching nature, you do not bounce into numbers,  you can observe 3-leaf or 4legged and manyshaped things, big and small, YOU (the human) can 'count them' if you invented the symbols 1 2 3 4 etc. but these refer to quantities and it required lots of abstracting in mental evolution to arrive in a numbers-based math - how humans think about nature.

That's what I referred to as the pre-Platonistic times.

*

Let me postpone the 'comma' part in the sets for next time.

Thanks again and my mind works in crooked ways, if you can excuse me for that. It seems I need too much learning to catch up.

 

John M

--------------------------------------------------------------------------

On Wed, Jul 1, 2009 at 10:20 AM, Bruno Marchal <marchal@...> wrote:

Hi John,

On 30 Jun 2009, at 14:32, John Mikes wrote:

Hi, Bruno

you know that I am in a different mindset, yet happy to read your train of thoughts. I consider a set a limited model of elements (and conclusions thereof are not applicable to wider domains) -

Well, as Thorgny illustrated correctly, the notion of SET does not apply to SETS. usually the collection of all sets is not considered as a set, for many good reasons, some of which appear already in the treatise on Numbers by Plotinus. To derive from this that infinite sets cannot exist, like Thorgny seems to believed is invalid.

when I read your

"A set can be described in extension or in intension. "in extension" means that we give all elements of the set, enclosed in accolades."

I was really happy with the next sentence:

"When the set is not to complex (meaning big or infinite), we can use the "...".  - "

(I missed here the exemption of the 'infinite' "set", really a contradiction, to which the 'set' considerations cannot apply - OR can they?

Of course they can. This is done in everyday mathematics all the time since Cantor discovered the notion of sets. It can be said that set have been exploited especially for the handling of infinities.

Typical infinite sets are the set of natural numbers {0, 1, 2, ...}, the set of odd numbers {1, 3, 5, 7, ...}, the set of prime numbers {2, 3, 5, 7, 11, ...} as already proved by Euclid, the set of decimal approximation of most real numbers, etc.

if you have something on that...)

Google on it. The notion of set (finite and especially infinite sets) is pervading all modern mathematics.

"Many" cannot be infinite (by MY definition).

Well, I prefer to use the word in their most used and standard sense. 

I loved your words on QM, the (linear) extension of the figment physical world as described in reductionist physical sciences.

I also cannot wait for something more about your approach on

the "self reference" - the basis of physics? -

This is the whole point of the UDA, and AUDA.

especially as to

'self' of what (who)? I hope the answer will not be "machine" or comp, because then I have to continue "and what is that?"

Of course it is comp, although I use comp because it is the simple way. Then, digging on mathematical logic, the same result can be retrieve from much weaker assumption. But comp is believed by all scientist and philosophers, except Penrose. Even John Searle is computationalist with my weak definition of it.

"and what is that"? It is the point of the present thread to explain that as slowly as possible so that good willing non mathematicians can understand the key points. 

(in more than a utilitarian explanation of what it does). ('it?')

What boils down to my ignorance as to the originating and maintaining to ANY action we speak about. The 'theos' of a non-assumed and non-supernatural factor (system?) yet involved in conducting all we just find natural and proceeding.

You may substitute 'numbers' for such, but so far did not reply (to my satisfaction at least) WHAT those 'numbers' may be.

Sorry, I am not of the religious kind.

If you can compute 34+89, or 65*87, then you know enough. My use of number in UDA is not religious, it is the same use as those who use number to send man on the moon. Too much and premature philosophizing makes it hard to proceed. "Religion" appears later, when you say "yes" to the digitalist surgeon. Comp asks for an act of faith.

*

Maybe my error is in 'believeing' that a REALITY may exist and 'we' have only access to part of it.

Neither science, nor philosophy, nor theology, could develop without such assumption. It is because we believe that there is a reality, that we can build theories to infer the part on which we have no access. science always consist in an attempt to see the invisible, be it atoms, far away galaxies, or mathematical constructions. So this is not an error.

Inventing for our comfort (the D. Bohmian idea) 'numbers' at the human level of pre-Platonian thinking. If 'reality' exists only by 'comp' or 'consequences' then I may be in a reversed error, due to brainwashing by in college imprinted  natural sciences - what I try to exceed yet it still sits there.

Our 'perceived reality' (ColinH) may also provide the numbers.

Sure, but this is independent of the fact that 17 is a prime numbers independently of me. That the set of prime numbers is infinite, independently of me, etc. 

Now that sounds heretical enough in this thread. Forgive me.

*

Waiting for the self-reference, (who's?)

Who's? But the universal number's self-reference of course. Even the Lobian one.

- with thanks so far

You are welcome,

Bruno

On Tue, Jun 30, 2009 at 6:45 AM, Bruno Marchal <marchal@...> wrote:

Hi Johnathan,

On 29 Jun 2009, at 17:22, Johnathan Corgan wrote:

>
> Bruno,
>
> I think you were off to a good start with your planned series of posts
> about the seven step argument.  I believe your first installment was a
> discussion of set theory as one of the mathematical preliminaries to
> the
> actual argument.
>
> I am looking forward to your next installment.

Well, thanks. I am not sure Kim and Marty are there, but I can provide
a summary, and recall the motivation.

Marty, did you come back from holiday? Kim? still interested in
electronical summer's school on mathematics.

The goal of the seven step thread is to make clear the seventh step of
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
make clear that the mind-body problem (or the consciousness/reality
problem, or the first person/third person) problem is reduced, when we
do the computationalist assumption, to a pure body appearance or
discourse problem. UDA shows that if we assume the comp. hyp. then we
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of
physics have to be extracted from self-reference.

Later, much later, I could explain, if everyone is OK with UDA, how we
can already extract from self-reference the general shape of physics,
so that we can already refute empirically, or confirm, the comp. hyp.
And it appears that the empirical quantum mechanics,  currently,
confirms the comp. hyp. Quantum mechanics confirms the partial
indetermination of the outcomes of our possible experiences, and the
"high non booleanity" of the propositions describing those outcomes".

The object of the "seventh step thread' consists in making the seventh
step accessible to non mathematicians. So we have to start from zero.
I have decided to start from elementary "naive" set theory, without
which we cannot do anything in math. I will avoid all special
mathematical symbols, and use instead words with capital letters.

We have not yet done a lot. So I can sum up, with the new "notations".

Definition. A set is just a "many" considered, when clear enough, as a
"one". So a set is just a collection of objects, and those objects are
called the element, or the member, of the set. If some x is an element
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"
means that we give all elements of the set, enclosed in accolades.
When the set is not to complex (meaning big or infinite), we can use
the "...". We can give name to a set, to ease or talk about that set,
like we do all the times in mathematics. Most of the set we will
consider are set of mathematical object, mainly numbers in the
beginning, and then set of ... sets.

Example-exercise:

1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
the set {0, 1, 2, 3}. And local means that such a name is used in a
local context. One paragraph later "A" could designed another, so be
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".

OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
true

1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT
x is a positive integer little than 4}

2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

This makes 14 exercises, which should be easy. I intent to keep it
that way. I continue after I get either answers (correct or wrong), or
questions.

Everyone is welcome to participate. Yet, I ask those who are quick to
respect those who are slow. To be slow in the beginning usually help
for being deep in the sequel.

Best,

Bruno

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

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

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[Bruno Marchal]

Hi Marty,

On 01 Jul 2009, at 18:57, m.a. wrote:

Hi Bruno,

                I'm responding to the quiz (see below). What does "high non booleanity" mean in the context of para.2?

We will need more math for the details, but "boolean" refer to classical or even platonist logic, when applied in a frame where we can agree that propositions are either true or false.Like arithmetic: you surely agree that a postive integer is either even, or not even.

 Classical logic is the one used implicitly or explicitly in most "scientific" discourses, especially in (classical) mathematics. "Boolean" comes from Boole who wrote a book "The laws of thought", and which plays some role in the birth of mathematical logic and computer science. But classical logic has a very long story, both in East and West.

>
> Example-exercise:
>
> 1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
> the set {0, 1, 2, 3}. And local means that such a name is used in a
> local context. One paragraph later "A" could designed another, so be
> careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
>
> OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
> true
>
> 1) the number 2 is a member of A   True

OK.

> 2) the number 12 is a member of A  False

OK.

> 3) the number 12 is not a member of A  True

OK.

> 4) (3 BELONGS-TO A)    True: but you haven't told us whether the parenthesis cancels the locality of brackets.

OK.

You can suppress anything in the notations as far as you, and those who will read the text, can figure out what you mean. Here I could have simply 

4) 3 BELONGS-TO A.

> 5) all members of A are numbers  True

OK.

> 6) one element of A is not a number  False: we've established that zero is a number.

OK. (I am not sure that we have established that, nor even what you mean by "established", but we surely welcome 0 in the numbers. Note that in the beginning 0 was rejected. And both the numbers 1 and 2 takes time to be accepted as number, for the reason that "number" means originally numerous. The "num" of "numerous" has the same origin as "number".

> 7) A can be defined in intension in the following way A = {x SUCH-THAT
> x is a positive integer little than 4}    True...if zero is considered a positive integer.

OK.

>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}

1. True

2. True

3. False

4. True: same question as 4 above.

5. True

6. False: zero is a number

7. False

OK. 

OK.

OK. 

OK.

OK. 

OK.

OK.

Very good. A "sans faute" we would say in french. In cyclism, you would be suspected taking drug! Bravo!

next lesson: (but take your time)

Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):

(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}

(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}

Can you compute

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?

{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ?

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?

{1} UNION { } = ?

{1, 2, 3} UNION {1, 2, 3} = ?

{ } UNION { } = ?

{1, 2} INTERSECTION { } = ?

{1} INTERSECTION { } = ?

{1, 2, 3} INTERSECTION {1, 2, 3} = ?

{ } INTERSECTION { } = ?

Now, an important distinction which will follow us through ... forever.  I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?

Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false. 

Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.

By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.

For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.

Exercise: in the following, what is true or false?

45 LESS-THAN 67

0 LESS-THAN 1

999 LESS-THAN 4

{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8}

{1} INCLUDED-IN {1, 2}

oops, I must go. You are lucky ;) 

Bruno

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

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[m.a.-2]

Some parts of this message have been removed. Learn more about Nabble's security policy.

 

 

Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):

(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}

(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}

Can you compute

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?  1,2,7,789, 5678

{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789

 

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?  1,2

{1} UNION { } = ? { }

{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3

{ } UNION { } = ? { }

{1, 2} INTERSECTION { } = ? { }

{1} INTERSECTION { } = ? { }

{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3

{ } INTERSECTION { } = ?  { }

Now, an important distinction which will follow us through ... forever.  I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?

Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false. 

Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.

By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.

For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.

Exercise: in the following, what is true or false?

45 LESS-THAN 67  true

0 LESS-THAN 1   true

999 LESS-THAN 4  false

{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true

{1} INCLUDED-IN {1, 2} true

oops, I must go. You are lucky ;) 

Bruno

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

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[Bruno Marchal]

You are quick!

On 02 Jul 2009, at 18:42, m.a. wrote:

 

 

Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):

(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}

(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}

Can you compute

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?  1,2,7,789, 5678

Almost OK. 

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = {1,2,7,789, 5678}. 

Don't forget the accolades, which means that you have as result the SET {1,2,7,789, 5678}

{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789

Not correct. To belong to A INTERSECTION B, the element must belong to A, *and* must belong to B. 1, 2 and 7 does belong indeed to A and to B, in this case, with A = {1, 2, 7, 789}, and B = {1, 2, 7, 5678}), but neither 789, nor 5678 do belong to both A and B.

So {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = {1, 2, 7}

Just tell me if you agree.

 

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?  1,2

OK, but don't forget the accolades. 

{1, 2} UNION { } = ?  {1,2}

{1} UNION { } =  { }

You are too quick here, you forget to type the 1. 

{1} UNION { } =  {1 }

{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3

OK (my mind adds the accolades)

{ } UNION { } = ? { }

Very good. You could eliminate the "?".

{1, 2} INTERSECTION { } = ? { }

Excellent.

{1} INTERSECTION { } = ? { }

Bravo.

{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3

Exact. (well, I continue to add the accolades, and eliminate the "?")

{ } INTERSECTION { } = ?  { }

Exact. In this case you see how much it is important to not forget the accolades!

Now, an important distinction which will follow us through ... forever.  I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?

Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false. 

Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.

By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.

For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.

Exercise: in the following, what is true or false?

45 LESS-THAN 67  true

OK.

0 LESS-THAN 1   true

OK.

999 LESS-THAN 4  false

OK.

{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true

OK.

{1} INCLUDED-IN {1, 2} true

OK.

oops, I must go. You are lucky ;) 

I'm back!  I give you two last exercises to ponder about, just  in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution ! 

1°) In the two relational formula below, one is true, the other is false. Which one are what?

a)    { } INCLUDED-IN { }

b)    { } BELONGS-TO { } 

2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?

Good job, Marty. 

Bruno

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

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[Bruno Marchal]

On 02 Jul 2009, at 00:22, John Mikes wrote:

> I don't deny the practicality of applying 'numbers-based' science in  
> sending a man to Mars, but it is NOT the numbers that does the job.  
> It is the complexity of the state of the art we reached, which  
> includes science, technology, skills, ideas AND of course numbers-
> application. Bohm's idea - as I understood it - was that searching  
> nature, you do not bounce into numbers,  you can observe 3-leaf or  
> 4legged and manyshaped things, big and small, YOU (the human) can  
> 'count them' if you invented the symbols 1 2 3 4 etc. but these  
> refer to quantities and it required lots of abstracting in mental  
> evolution to arrive in a numbers-based math - how humans think about  
> nature.

I know well that theory. It is based on the idea that some primary  
Nature exists. A common "superstition" among christians and atheists.  
Which could be true, actually. I don't know.

But what I am almost completely sure, is that if comp is true, then it  
is has to be supersitution. And that is what I try to explain.




> Thanks again and my mind works in crooked ways, if you can excuse me  
> for that. It seems I need too much learning to catch up.


You are welcome. If you have the time and courage, I really encourage  
you to follow the thread. You may be surprised ... soon!

Bruno

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




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[m.a.-2]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Bruno,

              Comments and questions are interspersed below.  

                                                                                                      marty

----- Original Message -----

From: marchal@...

To: everything-list@...

Sent: Thursday, July 02, 2009 1:44 PM

Subject: Re: The seven step series

You are quick!

On 02 Jul 2009, at 18:42, m.a. wrote:

 

 

Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):

(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}

(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}

Can you compute

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?  1,2,7,789, 5678

Almost OK. 

{1, 2, 7, 789} UNION {1, 2, 7, 5678} = {1,2,7,789, 5678}. 

Don't forget the accolades, which means that you have as result the SET {1,2,7,789, 5678}

{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789

Not correct. To belong to A INTERSECTION B, the element must belong to A, *and* must belong to B. 1, 2 and 7 does belong indeed to A and to B, in this case, with A = {1, 2, 7, 789}, and B = {1, 2, 7, 5678}), but neither 789, nor 5678 do belong to both A and B.

So {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = {1, 2, 7}

Just tell me if you agree.     I agree and can't understand how I could have been so careless.

 

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?  1,2

OK, but don't forget the accolades.    Are accolades brackets?

{1, 2} UNION { } = ?  {1,2}

{1} UNION { } =  { }

You are too quick here, you forget to type the 1. 

{1} UNION { } =  {1 }             Yes, I mistook the {1} for the number of the question...not part of the equation. I tend to overlook the fine points.

{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3

OK (my mind adds the accolades)

{ } UNION { } = ? { }

Very good. You could eliminate the "?".

{1, 2} INTERSECTION { } = ? { }

Excellent.

{1} INTERSECTION { } = ? { }

Bravo.

{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3

Exact. (well, I continue to add the accolades, and eliminate the "?")

{ } INTERSECTION { } = ?  { }

Exact. In this case you see how much it is important to not forget the accolades!

Now, an important distinction which will follow us through ... forever.  I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!

 

                                                                                                                                                                                                                          Why not the sets {1,2,7} if INTERSECTION means BOTH?

Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false. 

Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.

By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.

For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.

Exercise: in the following, what is true or false?

45 LESS-THAN 67  true

OK.

0 LESS-THAN 1   true

OK.

999 LESS-THAN 4  false

OK.

{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true

OK.

{1} INCLUDED-IN {1, 2} true

OK.

oops, I must go. You are lucky ;) 

I'm back!  I give you two last exercises to ponder about, just  in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution ! 

1°) In the two relational formula below, one is true, the other is false. Which one are what?

a)    { } INCLUDED-IN { }True

b)    { } BELONGS-TO { } True

2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?

 

                                                                         {1} {2} {1,2} {2,1}     why not {3} ?

Good job, Marty. 

Bruno

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

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[Bruno Marchal]

Hi Marty,

On 03 Jul 2009, at 00:18, m.a. wrote:

Bruno,

              Comments and questions are interspersed below.  

                                                                                                      marty

<snip>

Just tell me if you agree.     I agree and can't understand how I could have been so careless.

So you have to introspect a little bit. 

 

Do you remember the empty set? Can you compute:

{1, 2} UNION { } = ?  1,2

OK, but don't forget the accolades.    Are accolades brackets?

Due to Dirac, in Quantum Mechanics, I tend to believe that brackets are "<" and ">". parentheses are "(" and ")". I call "{" and "}" accolades, but perhaps they are called bracket. The terms are not important as far as we understand each other. How would you call "[" and "]" ?

You are too quick here, you forget to type the 1. 

{1} UNION { } =  {1 }             Yes, I mistook the {1} for the number of the question...not part of the equation. I tend to overlook the fine points.

It is good that you are aware of that facts. It is what matter. It is what will make it possible to develop the familiarity with the important fine points which can arise from time to time.

Now, an important distinction which will follow us through ... forever.  I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.

INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.

Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!

 

                                                                                                                                                                                                                          Why not the sets {1,2,7} if INTERSECTION means BOTH?

Ah, but the word "both" alone is ambiguous. You could say that the UNION of two sets is the merging of BOTH set, and the intersection is the given of the elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7}, which indeed merges BOTH sets. But for computing the intersection, you must ask yourself, does this *element* belongs to BOTH set? So, for the intersection of {1, 2} and {2, 7}, you have to ask yourself the following question: does 1 belong to both set? well, the answer is NO. the 1 belongs to the first set but not to the second, and so 1 does not belong to the intersection. Does 2 belongs to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}. Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but does not belong  to the first set, so 7 is not in the intersection.

Tell me if you are OK with this.

I'm back!  I give you two last exercises to ponder about, just  in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution ! 

1°) In the two relational formula below, one is true, the other is false. Which one are what?

a)    { } INCLUDED-IN { }True

Very good. All elements of { } are among the elements of { }. This is sometimes said to be "trivially" true, because { } has no elements at all.

This is an example of an important "fine point".  Examples:

To verify if the set {1, 2, 3} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check THREE things: does 1 belongs to the second set, does 2 belongs to the second set, does 3 belongs to the second set.

To verify if the set {1, 2} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check TWO things: does 1 belongs to the second set, does 2 belongs to the second set.

To verify if {1} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check ONE things: does 1 belongs to the second set.

To verify if { } is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to verify ZERO thing! So it is automatically true. That is why logicians say it is trivially true. 

From this you should understand that the empty set, { }, is included to any set.

So { } is included in all the sets:  { }, {1}, {1, 2}, ... {0, 1, 2, ...}, ....

In particular, as you said correctly, { } is included in { }. Put in another way, ({ } INCLUDED-IN { }) = true.

b)    { } BELONGS-TO { } True

NOT correct. Remember that the empty set is empty, so nothing belongs to it. All formula like (x belongs to { }) will be false. You can conceive a set as an empty box { }, in which you can fill elements. So the set {a, b} is the empty set in which you put the elements a, and then, the element b. The accolades "{" and "}" represents the box itself, and what is in between the accolades represents the elements of the set. You could have guessed the solution because I was helping you when saying that one of the proposition is true and the other is false, and this means that, like many beginners, you read the enunciation of the problem too much quickly. That is why I suggest you take your time, and read often, at different time, the enunciation of the problems, and actually all explanations as well.

The moral is:

"x belongs to { }"     is never true, or is always false, whatever x represents.

"{ } included in x"    is always true, or never false, whatever x represents.

2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?

 

                                                                         {1} {2} {1,2} {2,1}     why not {3} ?

Not too bad. 3/4 correct:  

{1} is included in {1, 2}.  Indeed.

{2} is included in {1, 2}. Indeed.

{1, 2} is included in {1, 2}. Indeed.

{2, 1} is included in {1, 2}. Indeed, that is true, but you have to remember what you have already agree on: the set {1, 2} is equal to the set {2, 1}, so this is not a new solution. It is the preceding one in disguised!

Why not {3}? {3} is not included in {1, 2} just because 3 does not belong to {1, 2}. Reread the definition of inclusion. A is included in B if all the elements of A belongs to B. OK?

So you have found three subsets, among the four. Reading today's explanations I think you could find the missing subset. I let you search a little bit.

So just one exercise: what is the missing subset?

I apologize if all of this is a bit boring, but soon enough it will be highly rewarding. You will see.

Bruno

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[Brent Meeker-2]

Bruno Marchal wrote:
> ...
> Due to Dirac, in Quantum Mechanics, I tend to believe that brackets
> are "<" and ">". parentheses are "(" and ")". I call "{" and "}"
> accolades, but perhaps they are called bracket. The terms are not
> important as far as we understand each other. How would you call "["
> and "]" ?

I've never seen "{" and "}" denoted "accolades" but I like it; they are
more commonly called "braces".   I don't know a specific term for "["
and "]", I generally refer to them as "square brackets".

Brent


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[m.a.-2]

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New comments in italics.

 For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!

 

                                                                                                      Why not the sets {1,2,7} if INTERSECTION means BOTH?

Ah, but the word "both" alone is ambiguous. You could say that the UNION of two sets is the merging of BOTH set, and the intersection is the given of the elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7}, which indeed merges BOTH sets. But for computing the intersection, you must ask yourself, does this *element* belongs to BOTH set? So, for the intersection of {1, 2} and {2, 7}, you have to ask yourself the following question: does 1 belong to both set? well, the answer is NO. the 1 belongs to the first set but not to the second, and so 1 does not belong to the intersection. Does 2 belongs to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}. Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but does not belong  to the first set, so 7 is not in the intersection.

Tell me if you are OK with this.

                                                 Not OK. You previously defined UNION as one OR the other. Now you seem to be giving me the same definition for INTERSECTION.

I'm back!  I give you two last exercises to ponder about, just  in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution ! 

1°) In the two relational formula below, one is true, the other is false. Which one are what?

a)    { } INCLUDED-IN { }True

Very good. All elements of { } are among the elements of { }. This is sometimes said to be "trivially" true, because { } has no elements at all.

This is an example of an important "fine point".  Examples:

To verify if the set {1, 2, 3} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check THREE things: does 1 belongs to the second set, does 2 belongs to the second set, does 3 belongs to the second set.

To verify if the set {1, 2} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check TWO things: does 1 belongs to the second set, does 2 belongs to the second set.

To verify if {1} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to check ONE things: does 1 belongs to the second set.

To verify if { } is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to verify ZERO thing! So it is automatically true. That is why logicians say it is trivially true. 

From this you should understand that the empty set, { }, is included to any set.

So { } is included in all the sets:  { }, {1}, {1, 2}, ... {0, 1, 2, ...}, ....

In particular, as you said correctly, { } is included in { }. Put in another way, ({ } INCLUDED-IN { }) = true.

b)    { } BELONGS-TO { } True

NOT correct. Remember that the empty set is empty, so nothing belongs to it. All formula like (x belongs to { }) will be false. You can conceive a set as an empty box { }, in which you can fill elements. So the set {a, b} is the empty set in which you put the elements a, and then, the element b. The accolades "{" and "}" represents the box itself, and what is in between the accolades represents the elements of the set. You could have guessed the solution because I was helping you when saying that one of the proposition is true and the other is false, and this means that, like many beginners, you read the enunciation of the problem too much quickly. That is why I suggest you take your time, and read often, at different time, the enunciation of the problems, and actually all explanations as well.

The moral is:

"x belongs to { }"     is never true, or is always false, whatever x represents.

"{ } included in x"    is always true, or never false, whatever x represents.

2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?

 

                                                                         {1} {2} {1,2} {2,1}     why not {3} ?

Not too bad. 3/4 correct:  

{1} is included in {1, 2}.  Indeed.

{2} is included in {1, 2}. Indeed.

{1, 2} is included in {1, 2}. Indeed.

{2, 1} is included in {1, 2}. Indeed, that is true, but you have to remember what you have already agree on: the set {1, 2} is equal to the set {2, 1}, so this is not a new solution. It is the preceding one in disguised!

Why not {3}? {3} is not included in {1, 2} just because 3 does not belong to {1, 2}. Reread the definition of inclusion. A is included in B if all the elements of A belongs to B. OK?

So you have found three subsets, among the four. Reading today's explanations I think you could find the missing subset. I let you search a little bit.

So just one exercise: what is the missing subset?

 

Is the missing subset   { }    ?

I apologize if all of this is a bit boring, but soon enough it will be highly rewarding. You will see.

Bruno

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[Bruno Marchal]

On 04 Jul 2009, at 04:31, m.a. wrote:

New comments in italics.

 For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!

 

                                                                                                      Why not the sets {1,2,7} if INTERSECTION means BOTH?

Ah, but the word "both" alone is ambiguous. You could say that the UNION of two sets is the merging of BOTH set, and the intersection is the given of the elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7}, which indeed merges BOTH sets. But for computing the intersection, you must ask yourself, does this *element* belongs to BOTH set? So, for the intersection of {1, 2} and {2, 7}, you have to ask yourself the following question: does 1 belong to both set? well, the answer is NO. the 1 belongs to the first set but not to the second, and so 1 does not belong to the intersection. Does 2 belongs to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}. Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but does not belong  to the first set, so 7 is not in the intersection.

Tell me if you are OK with this.

                                                 Not OK. You previously defined UNION as one OR the other. Now you seem to be giving me the same definition for INTERSECTION.

Let us take the set a = {1, 2} and the set b = {2, 3}.

Let us recall the definition (I abandon the capital letters because they are ugly, and I feel talking louder!)

1) intersection

(a intersection b) = {x such-that (x belongs-to a) AND (x belongs-to b) }.

So, some x, to belong to the intersection, has to belong simultaneously to the two sets involved. Only when x is equal to 2, is that condition verified.

2 belongs to (a intersection b) because 2 belongs to a, AND, 2 belongs to b.

That condition is not verify for x = 1, nor for x = 3. 3 belongs to b, but not to a. So 3 is not in the intersection. Nor 1, because it does not belong to a.

So (a intersection b) =  ( {1, 2} intersection {2, 3} ) = {2}.

2) Union

(a union b) = {x such-that (x belongs-to a) OR (x belongs-to b).

Does 1 belong to the union of a and b?  That is do we have that 1 belongs-to (a union b)? With same a and b as above.

Let us see.

Does 1 belongs to a union b?  Does 1 verify the condition written in the definition?  Do we have that (1 belongs-to a) OR (1 belongs to b))? 

A proposition shaped like P OR Q is true in the case one or both of P and Q is true. It is true that 1 belongs to {1, 2} OR to {2, 3}. A bit like "any number is odd or is not odd" is always true. So 1 is in the union.

2 is in the union, because it is true that 2 belongs to a or 2 belongs to b. Indeed 2 belongs to both of them. And 3 is in the union too, because iit belongs to one of them again, actually {2, 3}.

So (union b) = {1, 2, 3}.

OK?

Don't hesitate to tell me if it is not OK.

2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?

 

                                                                         {1} {2} {1,2} {2,1}     why not {3} ?

Not too bad. 3/4 correct:  

{1} is included in {1, 2}.  Indeed.

{2} is included in {1, 2}. Indeed.

{1, 2} is included in {1, 2}. Indeed.

{2, 1} is included in {1, 2}. Indeed, that is true, but you have to remember what you have already agree on: the set {1, 2} is equal to the set {2, 1}, so this is not a new solution. It is the preceding one in disguised!

Why not {3}? {3} is not included in {1, 2} just because 3 does not belong to {1, 2}. Reread the definition of inclusion. A is included in B if all the elements of A belongs to B. OK?

So you have found three subsets, among the four. Reading today's explanations I think you could find the missing subset. I let you search a little bit.

So just one exercise: what is the missing subset?

 

Is the missing subset   { }    ?

Correct.

So the subsets of {1, 2} are { }, {1}, {2}, {1, 2}. 

Could you find all subsets of {1, 2, 3}?

And now I give you an exercise which is so much easy that you could panic, and so I will provide the solution. I have seen often that too much easy question can make a student panic, and then the prey of out-of place mockery, and useless loss of confidence.

The easy exercise: could you give me the set of subsets of {1, 2} ?

Solution: You already told me that the subsets of {1, 2} are { }, {1}, {2}, {1, 2}.  So, the set of subsets of {1, 2} is 

{ { }, {1}, {2}, {1, 2} }

OK? It is just the solutions you give me, enclosed by braces (accolades) "{",  "}". Look at the expression with a spectacle.

If we except the set of books on Brent Meeker's shell, up to now, we have met mainly set of numbers, like

{1}

{1, 2}

{0, 2, 4, 6, ...}

{0, 1, 2, 3, ...}

Here we met a set of sets.

The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that  {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:

Which of the following are true

{3, 5} included-in {3, 5}

{3, 5} belongs-to {3, 5}

{3, 5} included-in { {3, 5} }

{3, 5} belongs-to { {3, 5} }

Take your time, 

Bruno

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

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[m.a.-2]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Bruno,

          Can you provide definitions of  "belongs-to"  and  "included-in" that distinguish them from "union" and "intersection"?

 

 

 

 

 

 

 

Here we met a set of sets.

The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that  {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:

Which of the following are true

{3, 5} included-in {3, 5} True

{3, 5} belongs-to {3, 5} 

{3, 5} included-in { {3, 5} }

{3, 5} belongs-to { {3, 5} }

Take your time, 

Bruno

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

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[Bruno Marchal]

On 04 Jul 2009, at 15:17, m.a. wrote:

Bruno,

          Can you provide definitions of  "belongs-to"  and  "included-in" that distinguish them from "union" and "intersection"?

"Belongs-to" and "included-in" are relations. Their value are true or false.

1) (x belongs-to A) means that the object x belongs to the set A.

Examples: 3 belongs-to {3, 4}, because 3 is an element of the set {3, 4}, or, put in another way (which can be useful for later):

 (3 belongs-to {3, 4}) = true

2) Similarly (x included-in y) is a relation bearing on sets, and (x included-in y) = true, means that x is included-in y, and this means that all elements of x are elements of y. You don't need to know more, but if you want you can define (x included-in y) by

For any z ((z belongs-to x) -> (z belongs-to y)). But I intended to introduce "->" later, so don't worry.

Example {3, 4} is included in {3,5,4} because all elements of {3, 4} are in {3, 5, 4}. We can write ({3,4} included-in {3,5,4}) = true. Another example is ({3, 4} included-in {3,6,7,9,567}) is false, because not all elements of {3,4} are in {3,6,7,9,567} (indeed 4 belongs to {3,4} and not to  {3,6,7,9,567}.

Union and intersection are not relation, but operation.

({3,4} union {4,5}) is not equal to true or to false, it is equal to a set: actually the set you get by the union of {3,4} with {4,5}, and this is {3,4,5}.

Likewise, the intersection of {3,4} with {4,5}, that is ({3,4} union {4,5}) is not true or false, but is equal to {4}. 

So:

({3,4} included-in {3,4,5}) = true

({3,4} union {3,4,5}) = {3,4,5}

You see the difference. It is the difference between "the brother of Paul", which denotes a human. and "Paul is greater than Julia", which is true or false.

Or, on the numbers, less-than (<)  is a relation, and addition and multiplication are operation:

(3 <  7) = true

(3+4) = 7

(7 <  3) = false

(7*3) = 21

({ } included-in {3,4}) = true

({1,2} intersection {2,7}) = {2}.

Does this help?

Here we met a set of sets.

The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that  {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:

Which of the following are true

{3, 5} included-in {3, 5} True

This is correct.

{3, 5} belongs-to {3, 5} 

{3, 5} included-in { {3, 5} }

{3, 5} belongs-to { {3, 5} }

Take your time, 

Bruno

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

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

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