The seven step series

[m.a.-2]

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My answers.    m.a.

 

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} True

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

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

Take your time, 

Bruno

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

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[Bruno Marchal]

On 06 Jul 2009, at 16:12, m.a. wrote (in bold):

My answers.    m.a.

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

OK.

 

{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}. 

Ask in case you are not OK with this, of course.

{3, 5} included-in { {3, 5} } False

OK. Very good.

{3, 5} belongs-to { {3, 5} } True

OK. {3, 5} is even the *only* element of  { {3, 5} }

No exercise today. Just a question, a suggestion, and a plan.

The question is: have you the feeling to learn something?

The suggestion: I think the best way to answer the preceding question consists in trying to explain what you learn to someone else. It is the best way to see if you remember and understand the definition. You could try to explain what you learn to some gentle "victim" in your neighborhood (wife, friend, child, parent, ...).

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". For this I need to explain what is a computable function. For this I need to explain what is a function, and for this I need to explain what is a set, given that functions can more easily be explained through sets relating sets. Once you will have a good grip of what is a universal machine, or what is a universal number, and what really means "universal", we will be able to tackle the notion of universal dovetailing, and especially the "mathematical universal dovetailing" (which is really important for the whole approach, and for the step eight). I am hesitating to work quickly on the notion of function, or to do some pieces of number theory and geometry to provide examples before.

As I said recently to John, the discovery of the notion universal machine is one of the most astonishing and gigantic discovery made by the humans, and what I do is just an exploitation of that discovery. Universes, cells, brains and computers are example of universal machine, and the notion of universal machine are a key to understand why eventually, once we say "yes to the doctor", and believe we can survive "qua computatio", we have to redefine physics as an invariant for the permutation of all possible observers, and how physics can be recovered from an invariant among all universal machines point-of-views ...

Feel free to slow me down, or to accelerate me, and to ask any question at whichever level of details you want. Feel free to ask any question that you have already asked.

Have a good day, and thanks for your effort and seriousness,

Bruno

PS. It should be obvious for everyone that if there are still questions, critics, objections, problems, feeling of dizziness, whatever, with the first six steps of the UDA, please, feel free to ask. And people should not hesitate to discuss other everything-like subject, I don't want to monopolize the list of course. But the UDA reasoning really changes the perspective on all possible TOEs, so I will feel free myself to point on UDA on each discussion where I find it relevant (of course also).

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

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[m.a.-2]

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Questions and comments interspersed below (in bold).

 

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

From: marchal@...

To: everything-list@...

Sent: Monday, July 06, 2009 12:14 PM

Subject: Re: The seven step series

On 06 Jul 2009, at 16:12, m.a. wrote (in bold):

My answers.    m.a.

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

OK.

 

{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.. 

Ask in case you are not OK with this, of course.

{3, 5} included-in { {3, 5} } False

OK. Very good.

{3, 5} belongs-to { {3, 5} } True

OK. {3, 5} is even the *only* element of  { {3, 5} }

No exercise today. Just a question, a suggestion, and a plan.

The question is: have you the feeling to learn something?

The suggestion: I think the best way to answer the preceding question consists in trying to explain what you learn to someone else. It is the best way to see if you remember and understand the definition. You could try to explain what you learn to some gentle "victim" in your neighborhood (wife, friend, child, parent, ...).

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.

 

For this I need to explain what is a computable function. For this I need to explain what is a function, and for this I need to explain what is a set, given that functions can more easily be explained through sets relating sets. Once you will have a good grip of what is a universal machine, or what is a universal number, and what really means "universal", we will be able to tackle the notion of universal dovetailing, and especially the "mathematical universal dovetailing" (which is really important for the whole approach, and for the step eight). I am hesitating to work quickly on the notion of function, or to do some pieces of number theory and geometry to provide examples before.

As I said recently to John, the discovery of the notion universal machine is one of the most astonishing and gigantic discovery made by the humans, and what I do is just an exploitation of that discovery. Universes, cells, brains and computers are example of universal machine, and the notion of universal machine are a key to understand why eventually, once we say "yes to the doctor", and believe we can survive "qua computatio", we have to redefine physics as an invariant for the permutation of all possible observers, and how physics can be recovered from an invariant among all universal machines point-of-views ...

Feel free to slow me down, or to accelerate me, and to ask any question at whichever level of details you want. Feel free to ask any question that you have already asked.

Have a good day, and thanks for your effort and seriousness,

Bruno

PS. It should be obvious for everyone that if there are still questions, critics, objections, problems, feeling of dizziness, whatever, with the first six steps of the UDA, please, feel free to ask. And people should not hesitate to discuss other everything-like subject, I don't want to monopolize the list of course. But the UDA reasoning really changes the perspective on all possible TOEs, so I will feel free myself to point on UDA on each discussion where I find it relevant (of course also).

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

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[Bruno Marchal]

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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[m.a.-2]

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Thanks, Bruno. I think I've got it now. Sorry to be such a slow learner.       

                                                                                                                   marty

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

From: marchal@...

To: everything-list@...

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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[Bruno Marchal]

On 07 Jul 2009, at 16:18, m.a. wrote:

Thanks, Bruno. I think I've got it now. Sorry to be such a slow learner.       

                                                                                                                   marty

Slow? It seems to me that you are rather quick. A scientist friend on mine took 4 years to get the difference between "belongs-to" and "included-in". 

Of course it will take more time you get friendly familiar with such matter, but that is completely normal. 

You are not slow at all, and I appreciate your seriousness and your courage, because you can say publicly "I don't understand, explain again". It is the big virtue which will help you to proceed.

Now, understanding is but one phase. You will have to remember what you learn. For this I suggest you do by yourself some summaries, and as I said it could help to try to explain to some others. But there is no problem to ask any question, or to suggest me to recall definitions and examples.

At some point I will sum up myself.

Oh, let me sum up a few bit before I introduce a new notion, and three exercise (but you can take some holiday before, take your time).

---

What are sets? Sets are sort of boxes which can contains anything, like numbers, or sets. Most of the set we have encounter were set of numbers, or set of sets. 

You can perhaps intuit some use of set in logic. For example saying that being a human makes you mortal can be analysed by the statement that the set of humans is included in the set of mortal beings. The proposition "Julia is a human" is equivalent with the proposition that Julia belongs to the set of humans. If I let H be a name for the set of humans, M be the name of the set of mortal beings, and j be a name for Julia, the fact that Julia is human, can be translated in "set theory" by 

(j belongs-to H), 

and the fact that being a human makes you mortal, can be translated by

(H included-in M);

Remember that (H included-in M) means that all elements of H are element of  M, and so it means that if j belongs-to H then j belongs-to M.

A logician would say that with the axioms (j belongs-to H) and (H included-in M), you can deduce that (j belongs-to M).

A logician never care if the axioms are true or false, he cares only on the validity of the reasoning. 

Remark. Personally, I don't believe that in "real life" there are sets, like those we can meet in math. Take the set of humans. Do we have a really a set ? An anti-computationalist could classify Julia as an inhuman zombie the day she got her artificial brain, so H is already different for a computationalist and an anti-computationalist!  In real life, sets can be locally useful, but it would be a sort of occamization, to quote John, (inspired by Russell) to apply the notion of set so straightforwardly. I have the same opinion for the use of set in mathematics, concerning the long run, but then I understand how much they make thinks far easier and clearer. Indeed they pervade all branches of math: topology, probability, algebra, logic, and computer science is no exception.  (I think they will disappear, but this will take millenia!)

------

Now it is time to do some exercise.

Do you remember, I asked you to give me all the subsets of {1, 2}. That is, all the sets which are included in {1, 2}. You gave me the correct answer: those subsets are { }, {1}, {2}, {1, 2}. You see that the set {1, 2} has 2 elements, and 4 subsets. But then I asked to give me the set of all subsets of {1, 2}. 

{1, 2} has four subsets, and it is natural to make that many a one, by considering *the* set of all subsets of {1, 2}. The answer is:

{{ }, {1}, {2}, {1, 2}} 

Considering all subsets of a set is a rather important operation, which we will meet more than one times in the sequel. Given its importance mathematicians gave it a name. It is the power operation. Later I will be able to explain why it is called power. 

It is an UNARY operation, which means it applies on ONE set. (Intersection, and union are BINARY operations, they need two sets to work on).

So (power x) = {y such-that y is included in x}, by definition.

For example:

(power {1, 2}) = {{ }, {1}, {2}, {1, 2}} 

Here are the three promised exercises. Compute

(power {1}) = ?

(power {1, 2, 3}) = ?

(power { }) = ?

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.

 

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

From: marchal@...

To: everything-list@...

Sent: Tuesday, July 07, 2009 2:07 PM

Subject: Re: The seven step series

On 07 Jul 2009, at 16:18, m.a. wrote:

 

Bruno,

 

I'm not entirely sure of these answers, but I think I learn more from your corrections than from pondering the rules to the point where I confuse myself.   m.a.

Do you remember, I asked you to give me all the subsets of {1, 2}. That is, all the sets which are included in {1, 2}. You gave me the correct answer: those subsets are { }, {1}, {2}, {1, 2}. You see that the set {1, 2} has 2 elements, and 4 subsets. But then I asked to give me the set of all subsets of {1, 2}. 

{1, 2} has four subsets, and it is natural to make that many a one, by considering *the* set of all subsets of {1, 2}. The answer is:

{{ }, {1}, {2}, {1, 2}} 

Considering all subsets of a set is a rather important operation, which we will meet more than one times in the sequel. Given its importance mathematicians gave it a name. It is the power operation. Later I will be able to explain why it is called power. 

It is an UNARY operation, which means it applies on ONE set. (Intersection, and union are BINARY operations, they need two sets to work on).

So (power x) = {y such-that y is included in x}, by definition.

For example:

(power {1, 2}) = {{ }, {1}, {2}, {1, 2}} 

Here are the three promised exercises. Compute

(power {1}) = ? {{ }, {1}}

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}}

(power { }) = ?   {{ }}

  

Take your time,

Bruno

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

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[Bruno Marchal]

Marty,

On 08 Jul 2009, at 02:28, m.a. wrote:

 

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

From: marchal@...

To: everything-list@...

Sent: Tuesday, July 07, 2009 2:07 PM

Subject: Re: The seven step series

On 07 Jul 2009, at 16:18, m.a. wrote:

 

Bruno,

 

I'm not entirely sure of these answers, but I think I learn more from your corrections than from pondering the rules to the point where I confuse myself.   m.a.

Hmm... You are not slow, but you may be a bit lazy :)

Pondering the rules to the point where you confuse yourself is necessary to develop ... well, the art of pondering the rules to the point where you confuse yourself, which is part of the work of the researcher.

But that is OK Marty, given that the goal here is just to give the necessary passive understanding of math so as making you able to grasp the seventh step of UDA. 

Do you remember, I asked you to give me all the subsets of {1, 2}. That is, all the sets which are included in {1, 2}. You gave me the correct answer: those subsets are { }, {1}, {2}, {1, 2}. You see that the set {1, 2} has 2 elements, and 4 subsets. But then I asked to give me the set of all subsets of {1, 2}. 

{1, 2} has four subsets, and it is natural to make that many a one, by considering *the* set of all subsets of {1, 2}. The answer is:

{{ }, {1}, {2}, {1, 2}} 

Considering all subsets of a set is a rather important operation, which we will meet more than one times in the sequel. Given its importance mathematicians gave it a name. It is the power operation. Later I will be able to explain why it is called power. 

It is an UNARY operation, which means it applies on ONE set. (Intersection, and union are BINARY operations, they need two sets to work on).

So (power x) = {y such-that y is included in x}, by definition.

For example:

(power {1, 2}) = {{ }, {1}, {2}, {1, 2}} 

Here are the three promised exercises. Compute

(power {1}) = ? {{ }, {1}}

Excellent.

(power { }) = ?   {{ }}

Excellent. People are often wrong on this one!

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}}

Here you are a bit lazy, as I said. You miss many subsets. Is not {1, 2} a subset of {1, 2, 3}? Is not {2, 3} a subset, and where is {1, 2, 3}?

I have to go right now, so I let you search, meanwhile, for the complete solution by yourself. I give you a hint (power {1,2,3}) has 8 elements.

And I give you a little subject research: if a set x has n elements, how many elements are in (power x)?

Bruno

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

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[m.a.-2]

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Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

.

And I give you a little subject research: if a set x has n elements, how many elements are in (power x)? 

 

How's this for a wild guess? I have a feeling that it's missing the accolades, but I have no idea where to put them.

 

 

(power {x}) =  n(n-1) (n-2)...(n-x+1)

                                 x!

Bruno

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

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[Bruno Marchal]

On 08 Jul 2009, at 15:43, m.a. wrote:

Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

This is far better!  Not yet correct though.

I gave you the hint that there are 8 elements. Let us count: 

The empty set { }  ..................................1

Three singletons {1}, {2}, {3}................3

Two doubletons {1,2 }, {2,3 }................2

The biggest subset  {1,2,3}..................1

1 + 3 + 2 + 1 = 7

A subset is missing! Can you see which one?

And I give you a little subject research: if a set x has n elements, how many elements are in (power x)? 

 

How's this for a wild guess? I have a feeling that it's missing the accolades, but I have no idea where to put them 

 

(power {x}) =  n(n-1) (n-2)...(n-x+1)

                                 x!

Good intuition that there is a problem with the accolades. Although your expression is not *missing* accolades. Actually it has *too much* accolades. 

If x represents a set, for example the set {1,2}, it means that in the formula x can be substituted by {1,2}. we could write that x = {1, 2}, the accolades are in x, if you want. 

So on the left, you should have written (power x), like in the enunciation of the subject research, actually. 

If x is the set {1,2}, (power x) is (power {1, 2}). But (power {x}) is (power {{1,2,3}}), i.e. the powerset of {{1,2,3}}, which is {{ }, {{1,2,3}}}.

(Power {x}) looks like the powerset of an indeterminate singleton, a set with only one element.

You  could have written this:

When x has n elements, then (power x) = n(n-1) ... (n-x+1) / x!

Let us see. After all you already computes the powerset of { }, which is the set with 0 element, and you told me (power { })  = {{ }}. So it has one element, and your formula should confirm this, and ... well, your formula begins by n multiplied by something, if n = 0 then we will get 0,  because 0 times any number gives 0. But we have just seen that (power { }) = {{ }}, which is a set with 1 element. So your formula is already contradicted by the first example. Hmm...

May be that was bad luck, and sometimes in math the first example is also the trickiest, so let us look for n = 1. Let us take a set with one element, like {24}. Its power has 2 elements: {{ } {24}}, and you can guess that all singletons (set with two elements) have the same number of elements in their power set. So the answer is 2, in this case. If x has 1 element, the powerset of x, (power x) has two elements. Your formula should give 2, when n is equal to 1.

Let us see ...

it gives 1(1-1) ... (1 -

... but now, what could you mean by (n - x ...). ???

n is supposed to represent a number, x is supposed to represent a set, how could I, or you, subtract a set from a number?

So I'm afraid that your formula is senseless, although I will perhaps take the time, in some future, to explain why there *is* an atom of truth in it! The correct formula is much simpler, though.

Morality: if you have a theory, or a formula, test it on what you already know before submitting to publication!

To find the formula, you could try first the tedious brut force (if this is english). (Few mathematicians admit to do that, but all mathematicians do it!)

The number of sets included in { } = 1   (you have seen that). If x has 0 elements, (power x) = 1.

The number of sets included in  {a} = 2    (you have seen that). If x has 1 elements, (power x) = 2.

The number of sets included in {a, b} = 4  (you have seen that). If x has 2 elements (power x) = 4.

The number of sets included in {a,b,c} = 8 (cf the hint, exercise above). If x has 3 elements (power x) = 8

Could you compute and/or guess the number of sets included in {a,b,c,d} ?

And what about {a,b,c,d,e}?, and {a,b,c,d,e,f} ?, and ... ?

Bruno

PS Did I say that (power x) is called the powerset of x? It could be better to write (powerset x) instead of (power x), to emphasize that the powerset of x is the *set* of the sets included in x. OK? (Sorry).

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

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[m.a.-2]

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Here's my third try. I'll continue working on the (power x) problem.   m.a.

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

From: marchal@...

To: everything-list@...

Sent: Wednesday, July 08, 2009 1:31 PM

Subject: Re: The seven step series

On 08 Jul 2009, at 15:43, m.a. wrote:

Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

 

           Third try:

                                         =   {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}

This is far better!  Not yet correct though.

I gave you the hint that there are 8 elements. Let us count: 

The empty set { }  ..................................1

Three singletons {1}, {2}, {3}................3

Two doubletons {1,2 }, {2,3 }................2

The biggest subset  {1,2,3}..................1

1 + 3 + 2 + 1 = 7

A subset is missing! Can you see which one?

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[Bruno Marchal]

On 09 Jul 2009, at 03:05, m.a. wrote:

Here's my third try. I'll continue working on the (power x) problem.   m.a.

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

From: marchal@...

To: everything-list@...

Sent: Wednesday, July 08, 2009 1:31 PM

Subject: Re: The seven step series

On 08 Jul 2009, at 15:43, m.a. wrote:

Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

 

           Third try:

                                         =   {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}

You may be a bit saturated, because this is less correct than your preceding answer. You could, after some good night sleep, see this by yourself. Indeed if { { } 1, 2, 3} was a subset of { 1,2,3}, this would mean, by the definition of subset, that 1, 2, 3 and { } are elements of {1, 2, 3}. But { } is not an element of {1, 2, 3}.

The missing subset is just {1, 3}.

So instead of ...

I gave you the hint that there are 8 elements. Let us count: 

The empty set { }  ..................................1

Three singletons {1}, {2}, {3}................3

Two doubletons {1,2 }, {2,3 }................2

The biggest subset  {1,2,3}..................1

1 + 3 + 2 + 1 = 7

A subset is missing! Can you see which one?

... we have the much more symmetrical:

The empty set { }  .........................................1

Three singletons {1}, {2}, {3}.......................3

Three doubletons {1,2 }, {2,3 }, {1,3}..........3

The biggest subset  {1,2,3}..........................1

"1 3 3 1" is a row in the so called "Pascal triangle", and I have put those decomposition because the formula you gave me, was the formula giving the value of the number appearing in the Pascal triangle. 

In french we would say that you are searching the "little beast", making things more complex than they really are.

Let me do your, rather complex, reasoning:

I have a set A having n elements. A subset of A will have at most n elements. To find how many subsets are in A, I have to count the subset having 0 elements, 1 element, 2 elements; 3 elements, ... i elements, ... up to i = n. 

Searching in my memory, book or the net, I recall that the number of subsets having i elements taken in a set of n elements, let us write this (i;n),  is

n(n-1)(n-2) ... (n-i+1)/i!

So the number of elements in the powerset of A is the sum from i = 0 to n of (i; n) =

 sum from i = 0 to n of n(n-1)(n-2) ... (n-i+1)/i!

This is a very complex reasoning leading to a rather complex formula. It would have been rather not pedagogical from my part to give you a so difficult exercise, so you could have guessed a simpler reasoning, leading to a simpler formula.

Let us to do the "simpler" reasoning.

We have already seen that the powerset of a set with 0 element has 1 element. cf (powerset { }) = {{ }}

We have already seen that the powerset of a set with 1 element has 2 elements. cf (powerset {a}) = {{ }, {a}}

We have already seen that the powerset of a set with 2 element has 4 elements. cf (powerset {a, b}) = {{ }, {a} {b} {a, b}}

We have just seen that the powerset of a set with 3 element has 8 elements (cf above).

We see that the sequence of numbers on the right grows like

 1, 2, 4, 8, ....

And we have to guess the sequel, and find a general formula. Of course, if we don't see it yet we can still compute, tediously, the number of subsets of a set with four elements, like S = {a, b, c, d}. Let us do it:

The subset of S with 0 element { } .................................. 1

The subsets of S with 1 element {a}, {b}, {c}, {d} ...........4

The subsets of S with 2 elements {a,b} {a,c},{a,d} {b,c} {b,d}, {c,d} ...... 6 

The subsets of S with 3 elements {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d} .......4

The subset of S with 4 elements {a,b,c,d} ..........................................1

Thus there are 1+4+6+4+1 subsets of {a,b,c,d}, and this gives 16.

Now our sequence of answers is a bit longer:

 1, 2, 4, 8, 16, ....

Exercise: can you guess how many subsets there are in a set with 5 elements, 6 elements, ...?

After that I will help you to get the formula, and we will be able to soon approach a far reaching question: how many subsets are included in the infinite set N = {0, 1, 2, 3, ...}. But some vocabulary will have to introduced, some generalization will have to be done. After that we will come to the machines, and the question of computability, but let us go easy and slowly. We have already done a lot, and you can take some rest when you want. Bravo for the work you have already done. 

I just give you another little, not so obvious but very pleasant exercise: look at many Mandelbrot set videos on youtube, and try to discover the recurring sequence 1, 2, 4, 8, 16, ... appearing in the zooms. This is experimental mathematics! This is for your enjoyment only. Sort of dessert.

BTW, the sequence 1, 2, 4, 8, 16, ... appears also in some of the thought experiments in the comp setting. Perhaps you remember?

Bruno

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

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[Bruno Marchal]

Hi all,

I suddenly feel sorry putting too much burden on just one  
correspondent in the list, and I would appreciate if someone else  
could propose answers or any remarks to the exercises.

I am also a bit anxious about Kim, who is the one who suggested me the  
initial explanations, but who seems to have disappear right now.

There is also some sort of burden onto me, because it is hard to  
explain "the real thing" concerning the seventh step, without  
explaining or just illustrating at least some relevant portion of the  
mathematical reality: mainly the unexpected mathematical discovery of  
the universal functions, sets, numbers, systems, language, machine ...  
I don't mention the absence of drawing ability which does not help.

Given that the list raised from a critical approach toward Tegmark and  
Schmidhuber, I was usually assuming some knowledge in math and  
physics. What was harder for me in the beginning was to motivate the  
use of "philosophy of mind" notions, notably the key distinction  
between the first and third person point of view. Then UDA should make  
you realize how non obvious the relation between the first person and  
the third person can be once we assume comp (= work in the theory  
comp). My original goal was to illustrate that once we assume digital  
mechanism, we can build a "scientific formulation" of the mind-body  
problem or the consciousness-reality problem. We probably depart from  
Tegmark and Schmidhuber, or Wolfram, by taking into account that  
making comp explicit entails a delocalisation of the 1-person  
relatively to the third person computations, and makes the identity  
thesis, a most complex equivalence relations.

The knowledge of most people participating to the discussion is very  
varied, due to the extreme transdiciplinarity of the subject, and the  
interest it can evidently have for the layman (and indeed, for any  
universal machine).
Marty asked me to make an attempt toward a "journalistic" description  
of "how physics has to become  part of number theory".  This is very  
difficult, and risky due to inevitable misunderstanding.
And I feel like I have to explain in what deep sense the mathematical  
discovery of the "universal machine", made by Post, Turing, ... is  
already a quite utterly astonishing, yet subtle, discovery. Gödel  
himself took time to swallow it and he described Church thesis as an  
epistemological miracle.
My intention was to derive properly Cantor theorem, and then Kleene  
theorem, which was the object of my old "diagonalization posts".
I feel important that people understand how unbelievable Church thesis  
is, and why most startling propositions, including incompleteness, are  
easy consequences of it.

Typically I am happy to share my enthusiasm about all theorems in  
computer science which leads to the reversal, but knowing myself I  
know that I could accelerate too much and makes too much burden for  
the correspondent especially if he is alone.
So before becoming an harasser myself I invite Marty to let other  
people trying to answer the exercises.
Marty has fully agreed to this proposal and is happy the pressure is  
off him to represent all those who are following anonymously.
Eventually I can show the solution and proceed in addressing the post  
to everyone.
Note that this is what I have done with the combinators, feedback were  
made out-of-line, then. But this lead to difficulties too. I cannot  
solve all the exercise out-of-line.  By experience this ends up with  
finding myself writing too many posts with almost the same info to  
different people.
Yet I can imagine how much it is to be the only public target of what  
could look like an perpetual exam, and I really want to proceed in a  
cooler way.

Some people have encouraged me, out-of-line, to proceed, but now I  
think they should participate a little bit, if only to witness they  
are following the thread. I will probably stop to propose "easy" (a  
quite relative notion) exercise, but then it is important to stop me  
once anything is unclear. This is a problem with math, if you miss a  
piece, everything becomes senseless.

Understanding implies some self-implication in the reasoning. So,  
either someone else try to participate, or I continue impersonally and  
eventually I will try some "non technical summary". I recall that one  
of the goal consists in explaining the difference between a  
computation and a description of a computation (beyond just doing the  
step 7).

Any remark to improve the communication or to design a better  
methodology is welcome,

Best regards to all of you, and thanks for letting me know your  
interests,

Bruno

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




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[Johnathan Corgan]

On Fri, 2009-07-10 at 22:24 +0200, Bruno Marchal wrote:

> I suddenly feel sorry putting too much burden on just one  
> correspondent in the list, and I would appreciate if someone else  
> could propose answers or any remarks to the exercises.

Bruno--you're doing great.  I think it is the case where silence means
"I understand, continue", rather than disinterest.

> There is also some sort of burden onto me, because it is hard to  
> explain "the real thing" concerning the seventh step, without  
> explaining or just illustrating at least some relevant portion of the  
> mathematical reality: mainly the unexpected mathematical discovery of  
> the universal functions, sets, numbers, systems, language, machine ...  
> I don't mention the absence of drawing ability which does not help.

The derivation of your thesis from first principles is a very compelling
idea.  The somewhat startling and unorthodox conclusions you espouse are
bound to cause confusion unless all their underpinnings are well
understood.  The arguments from others then can have a much more
specific target than the top-level conclusions; instead they will come
out earlier in the derivation process and at the time of introduction of
the controversial subject.

> The knowledge of most people participating to the discussion is very  
> varied, due to the extreme transdiciplinarity of the subject, and the  
> interest it can evidently have for the layman (and indeed, for any  
> universal machine).

While I do have training in math and physics, I still benefit from your
targeting the motivated layman.  Personally, I'm not interested in doing
the exercises on the list, but they are still useful to check my
understanding.

> Best regards to all of you, and thanks for letting me know your  
> interests,

By all means, proceed.  Personally, if I don't understand something or
have an objection, you'll hear about it on the list, but I think you
should take silence as assent.

Johnathan Corgan



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[John Mikes]

Dear Bruno,

when I looked at the set-analysis it immediately popped up that {1,3} was missing, - YET - this fantastic<G> discovery of mine did not bring me closer to the idea "what are numbers".

It seems I can win the battle and still lose the war.

John

On Wed, Jul 8, 2009 at 9:05 PM, m.a. <marty684@...> wrote:

Here's my third try. I'll continue working on the (power x) problem.   m.a.

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

From: marchal@...

To: everything-list@...

Sent: Wednesday, July 08, 2009 1:31 PM

Subject: Re: The seven step series

On 08 Jul 2009, at 15:43, m.a. wrote:

Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

 

           Third try:

                                         =   {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}

This is far better!  Not yet correct though.

I gave you the hint that there are 8 elements. Let us count: 

The empty set { }  ..................................1

Three singletons {1}, {2}, {3}................3

Two doubletons {1,2 }, {2,3 }................2

The biggest subset  {1,2,3}..................1

1 + 3 + 2 + 1 = 7

A subset is missing! Can you see which one?

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[Bruno Marchal]

On 10 Jul 2009, at 22:52, Johnathan Corgan wrote:

>
> On Fri, 2009-07-10 at 22:24 +0200, Bruno Marchal wrote:
>
>> I suddenly feel sorry putting too much burden on just one
>> correspondent in the list, and I would appreciate if someone else
>> could propose answers or any remarks to the exercises.
>
> Bruno--you're doing great.  I think it is the case where silence means
> "I understand, continue", rather than disinterest.

Well, thanks, OK, perhaps.

>
>
>> There is also some sort of burden onto me, because it is hard to
>> explain "the real thing" concerning the seventh step, without
>> explaining or just illustrating at least some relevant portion of the
>> mathematical reality: mainly the unexpected mathematical discovery of
>> the universal functions, sets, numbers, systems, language,  
>> machine ...
>> I don't mention the absence of drawing ability which does not help.
>
> The derivation of your thesis from first principles is a very  
> compelling
> idea.  The somewhat startling and unorthodox conclusions you espouse  
> are
> bound to cause confusion unless all their underpinnings are well
> understood.

There are two things. Understanding the conclusions, and understanding  
how we get to them.
Many variations are possible in between are possible for varied  
audience.




> The arguments from others then can have a much more
> specific target than the top-level conclusions; instead they will come
> out earlier in the derivation process and at the time of  
> introduction of
> the controversial subject.

But what is controversial? I have never heard about something  
controversial seen in the reasoning. The conclusion are astonishing,  
and certainly annoying for someone who believes "religiously" in both  
physicalism and digital mechanism.
The subject matter was controversial a long time ago, but today, it is  
no more, I think. Well, it depends on which circle. That something  
appears in the academy (like studies on consciousness, does not mean  
that all academicians understand the questioning there, alas).
I have heard that the first person indeterminacy, which is my mean  
early contribution, is controversial, but I have never seen any  
controversy on it, just sometimes, some discussion on the vocabulary  
or definition, which does not change any conclusion.
The subject matter is difficult, so it easier for the "religious"  
people (like convinced atheists, to be clear) to speculate about some  
difficulties they don't even try to single out.
I proceed in the scientific way, which means that I just ask  
questions, and anyone can verify what follows from what, or interrupt  
and present an objection. Up to now, none of the "real" objections  
presented were fatal, and eventually those reduce also to a problem of  
vocabulary.

>
>
>> The knowledge of most people participating to the discussion is very
>> varied, due to the extreme transdiciplinarity of the subject, and the
>> interest it can evidently have for the layman (and indeed, for any
>> universal machine).
>
> While I do have training in math and physics, I still benefit from  
> your
> targeting the motivated layman.  Personally, I'm not interested in  
> doing
> the exercises on the list, but they are still useful to check my
> understanding.

OK.


>
>
>> Best regards to all of you, and thanks for letting me know your
>> interests,
>
> By all means, proceed.  Personally, if I don't understand something or
> have an objection, you'll hear about it on the list, but I think you
> should take silence as assent.

If only silence could be assent!
But I am willing to take yours as such and I will proceed.

Best,

Bruno



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




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[Bruno Marchal]

On 12 Jul 2009, at 17:28, John Mikes wrote:

Dear Bruno,

when I looked at the set-analysis it immediately popped up that {1,3} was missing, -

Good!

YET - this fantastic<G> discovery of mine did not bring me closer to the idea "what are numbers".

And, assuming comp, you never will. None of us will. Sets can provide a nice representation of the numbers, which I have already given some times ago, but for now, I prefer to take the numbers as basic entities. Sets are more complex entities than numbers, in general.

What are the numbers? I can give you only some hints, like zero is the number of matches in a empty box, one is the number of matches in a box having one match in, and so on ... 

About the "and so on ... " I can only hope you have been enough trained in high school to have an idea.

I cannot explain really what are numbers, but, if you assume the comp theory,  I can explain "completely", from it,  why numbers develop beliefs in galaxies, get conscious, makes a big variety of dreams, and eventually realize why they will never knows what numbers are, and what they are capable of.

Number are the necessary mystery, from which we start. What I like in comp, is that it explains why the numbers have to be a mystery.

Bruno

It seems I can win the battle and still lose the war.

John

On Wed, Jul 8, 2009 at 9:05 PM, m.a. <marty684@...> wrote:

Here's my third try. I'll continue working on the (power x) problem.   m.a.

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

From: marchal@...

To: everything-list@...

Sent: Wednesday, July 08, 2009 1:31 PM

Subject: Re: The seven step series

On 08 Jul 2009, at 15:43, m.a. wrote:

Second try:

(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

 

           Third try:

                                         =   {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}

This is far better!  Not yet correct though.

I gave you the hint that there are 8 elements. Let us count: 

The empty set { }  ..................................1

Three singletons {1}, {2}, {3}................3

Two doubletons {1,2 }, {2,3 }................2

The biggest subset  {1,2,3}..................1

1 + 3 + 2 + 1 = 7

A subset is missing! Can you see which one?

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

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[Miroslav Dobsicek]

Hi Bruno,

I'd like to let you know that I'm following the serie of your letters.
While I have the background you are covering right now, I still enjoy
your insights.

I joined the list like two years ago and from that time I've read most
of your key papers. Honestly, it is not the easiest stuff to read
style-wise. You try to precise, define well, etc. yet it cannot really
be compared to the quality of, let us say, Physical Review Letters and
alike articles. In my opinion, that is why it is hard to either agree or
disagree with your thesis.

I can imagine that right now you are tempted to write something along
the lines
a\ I just propose to take Church thesis seriously
b\ All I ask you is 'Do you say yes to the doctor?
While valid proposal and question, there is really not much to agree
with/disagree with/critize unless one is willing to undertake long
discussions, clarifications and position adjustments.


Anyway, your papers and letters are really a great source of ideas to
think about and that is exactly what I do. From the day one on the list
I keep myself busy with the question of "Why should I believe in the
Church thesis" (you see, I don't write "Why do I ..."). I've got into
the writing of Bernard Bolzano (I consider his work cruicial in order to
keep an open mind about the Cantor diagonal argument) ..
 - and now back to the beginning of my letter -
Bolzano (Cantor), your insights and thinking about alternatives at any
moment make me pretty happy. Thanks!

Mirek

PS: I'd love to read a book by Bruno Marchal.



Bruno Marchal wrote:
> Hi all,
>
> I suddenly feel sorry putting too much burden on just one  
> correspondent in the list, and I would appreciate if someone else  
> could propose answers or any remarks to the exercises.

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[Kim Jones-2]

On 11/07/2009, at 6:24 AM, Bruno Marchal wrote:

>
> I am also a bit anxious about Kim, who is the one who suggested me the
> initial explanations, but who seems to have disappear right now.



OK - I'm back. Since May 27 to two days ago I have been without  
Internet access.

I made the mistake of upgrading my Broadband plan to add Internet  
phone. It took two telcos a month to complete this ridiculously basic  
operation with mistakes made and attendant extra waiting times.

Then, just as the connection was restored at the beginning of July,  
the plumbing in this block of apartments fell apart and a major  
excavation work went ahead and this time the plumbers cut the phone  
cable and didn't realise it which meant I wasted another week trying  
to get the problem diagnosed.

So now finally everything is back to normal. I have just started  
reading this thread and can see that the class is a very exclusive  
one! I will try my best to follow through on the exercises and the  
comments, corrections. I feel I have access to the correct  
mathematical symbols on my Mac now but *time* is the thing that I  
don't have much of anymore, so I feel a bit depressed about the level  
of effort I can devote to it. If only we didn't have to work for a  
living things would be vastly easier.

The notion of sets is indeed a tricky one. I am just now going over  
the initial exercises again. Do not wait for me. I am also trying to  
catch up on about 4,000 emails.

Bruno - my sincerest apologies for this hiatus. You seem eager to get  
to the seventh and eighth steps. Why wouldn't you be.

regards,

Kim



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[Bruno Marchal]

On 13 Jul 2009, at 00:46, Mirek Dobsicek wrote:



>
> I'd like to let you know that I'm following the serie of your letters.
> While I have the background you are covering right now, I still enjoy
> your insights.

Thanks for letting me know.

>
>
> I joined the list like two years ago and from that time I've read most
> of your key papers. Honestly, it is not the easiest stuff to read
> style-wise. You try to precise, define well, etc. yet it cannot really
> be compared to the quality of, let us say, Physical Review Letters and
> alike articles.

I work on a subject which is not usually approach in any reviews of  
physics.
And then my english is sometimes a bit hazardous. But I have never get  
referee or any feedback about the rigors, which in my field (machine  
theology) is far more developed than usual. It is part of the problem  
for some people I think. It is just unusual.



> In my opinion, that is why it is hard to either agree or
> disagree with your thesis.

I disagree. It is simple. Just say a number between 1 and 8, with a  
justification of what you don't understand.
Perhaps between 0 and 8, if you have a problem with the definition of  
comp.


>
>
> I can imagine that right now you are tempted to write something along
> the lines
> a\ I just propose to take Church thesis seriously
> b\ All I ask you is 'Do you say yes to the doctor?

yes, for the sake of the argument. A non computationalist can just  
consider someone else saying yes to the doctor. A bit more is needed,  
and it is necessary to recall the definition of comp: it exists a  
level of description of my (generalized) such that I survive through a  
digital functional substitution made at that level.


>
> While valid proposal and question, there is really not much to agree
> with/disagree with/critize unless one is willing to undertake long
> discussions, clarifications and position adjustments.

Indeed, there is nothing to disagree. Only to understand, and who  
knows? to clarify. But then it is up to those who try to understand to  
say what they don't understand, besides the intrinsic difficulty with  
the subject. In the seventies some people argued that any sentence  
containing the word "consciousness" was automatically crackpot. Of  
course this is an illustration of the complete absence of  
understanding of the axiomatic method. We never know what we are  
talking about, we can only agree on starting propositions and method  
of reasoning, and then see if the conclusion follows from what we have  
admitted.


>
>
>
> Anyway, your papers and letters are really a great source of ideas to
> think about and that is exactly what I do.


I am happy with that.


> From the day one on the list
> I keep myself busy with the question of "Why should I believe in the
> Church thesis" (you see, I don't write "Why do I ...").

Good question. A lot of my work consists in showing that CT is a very  
strong principle. It is far stronger than most computer scientist  
imagine.



> I've got into
> the writing of Bernard Bolzano (I consider his work cruicial in  
> order to
> keep an open mind about the Cantor diagonal argument) ..
> - and now back to the beginning of my letter -
> Bolzano (Cantor), your insights and thinking about alternatives at any
> moment make me pretty happy. Thanks!


You are welcome.

> PS: I'd love to read a book by Bruno Marchal.

I have already written three books, and one was ordered by a publisher  
after getting a price. The two other one were disputed by different  
publishers, and then suddenly, without explanation, all those projects  
were abandoned. I have lost my trust in that kind of world I'm afraid.  
I don't think the reason of that abandon has any relationship with my  
work which is really of the type: "find the error".  I will surely  
write one paper and one book. Recently I have submitted a paper, and  
the referees were quite enthusiast, but the paper has been refused for  
being out of the topic, which it was not (unless you don't believe  
that observers are person).
We will see. My work is simple in two senses: UDA is simple because  
you need nothing more than a very tiny amount of understanding on  
numbers, set, computable functions and consciousness/kowledge. AUDA is  
relatively "simple" because you need only to understand Solovay's  
theorem and the Theaetetical definitions of knowledge. It makes comp  
hard to believe, no doubt, but here the work of Everett in quantum  
mechanics can provide a big help. Ah, ok, all this at once needs works  
and time, but nothing more.



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




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