Inorganic numbers

[matt grime]

Let S be a set, define S <= T if there is a n injection from S to T.

For finite sets let |S| denote the class of sets {T} where T<=S<=T

For each |S| where S is finite, let G, denote the set of all binary operations from pairs of elemnts of S to S satisfying the axioms we know from Group Theory.

Therefore we have made the "correct" paradigm of the natural numbers.

What do you reckon Organic?

[Organic]

Matt,

Please Define a set.

[matt grime]

It is the same notion of set as the ones you use to construct your objects too. We will work in the model of ZFC that you are most comfortable with.

[Organic]

You did not understand me.

Do you agree with that?

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset).

Some examples :

set: {a,b,c} = {b,a,c}

multiset: {a,a,a,a,b,b,c} = {b,a,a,c,a,b,a}

[matt grime]

Yes and no.

A class of objects is a "set" if it is a set in a model of ZFC.

[Organic]

Then what is a class of objects?

[matt grime]

Oh, that's just some collection of objects. You want to define collection? Go look it up in the dictionary. There naturally has to be some point at which we stop playing the silly recursive define every term in you definition game. Besides you as always fail to understand that when we say 'define symmetry degree' or something, we mean: tell us which objects might possess this property, what we look for when we look at one of these objects and how we then decide what its symmetry degree is (it is something that has units) or whether or not has a degree of symmetry. We don't want you to talk about symmetry breaking and the inherent beauty of nature. We just want to know how to use this damn thing.


For instance, using your nested set notation:

((1,1),(1,1))

suppose that is one of the valid objects in the set of quantity 4.

How can I tell if there is any uncertainty or redundancy in that? What is its vageness or symmetry degree.

[Organic]

We don't want you to talk about symmetry breaking and the inherent beauty of nature.

Why not?

Let us take the trees that included in quantity 3:

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

The first is a multiset and the last is a "normal" set.

For example let us say that there is a piano with 3 notes and we call it 3-system :

DO=D , RE=R , MI=M

The highest unclear information of 3-system is the most left information's-tree,
where each key has no unique value of its own, and vice versa.

<-Redundancy->
M M M ^<----Uncertainty
R R R | R R
D D D | D D M D R M
. . . v . . . . . .
| | | | | | | | |
3 = | | | |___|_ | |___| |
| | | | | | |
|___|___|_ |_______| |_______|
| | |


An example of 4-notes piano (DO=D , RE=R , MI=M , FA=F):

------------>>>

F F F F F F F F
M M M M M M M M
R R R R R R R R R R R R R R
D D D D D D D D D R D D D D D D
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |

4 =
M M M
R R R R R R R
D R D D D R D R D D D F D D M F
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |


D R M F
. . . .
| | | |
|__| | |
| | |
|_____| |
| |
|________|
|

[matt grime]

I don't need you to talk about symmetry breaking in order for me to be able to recognize symmetry degree. As it is your mathematical descriptions do not illuminate the objects you are talking about to other people. We are still completely unable to state what symmetry degree is, and how one identifies it or works it out, or even the types of object which might display this "thing".

I defined the degree of symmetry as what ever I did in the organic number thread. It is a simple rule that enables people to state the symetry degree of a tree. I don't need to explain the inherent ideas of symmetry in order to tell someone how to calculate it, or say, yes, the symmetry degree of that tree is 2/3. I've said it is defined for trees, and given the range of values it can take. If you want to justify it's name, then I suppose it has some relation to the subtrees and their properties.

You see you've labelled things in that diagram with uncertainty and redundancy. Why? How? What is it that lets you know when you can write those words above a diagram and when you can't?

[Organic]

Please refrash the screen, an look at my example in the previous post.

[matt grime]

I think I can now deduce from that that a diagram possess uncertainty if any of the columns of labels above one of the 'stems' has more than letter in it. Is that correct? We may if we choose then say that the number of stems that do not have a unique letter above them is the degree of uncertainty. That is just my artificial definition, you don't need to use it, or bother with it at all in fact.

That would allow for there to be two diagrams for '4' that do not have any uncertainty, is that correct? The last one and the one second row second from the left.

[Organic]

DO=D , RE=R , MI=M

The highest unclear information of 3-system is the most left information's-tree,
where each key has no unique value of its own, and vice versa.

<-Redundancy->
M M M ^<----Uncertainty
R R R | R R
D D D | D D M D R M
. . . v . . . . . .
| | | | | | | | |
3 = | | | |___|_ | |___| |
| | | | | | |
|___|___|_ |_______| |_______|
| | |

Well, I like to look at the connection between redundancy_AND_uncertainty as "a cloud of possibilities", you know like the possibilities that we have to some Quantum element.

For example:

Let C be a closed door.

Let O be an opened door.

Let # be XOR.


C C
# #
O O
. .
| |
|___|_
|

O C
. .
| |
|___|
|

[matt grime]

The questions I asked both needed a simple yes or no answer. So, yes or no? I don't need you to repost those pictures for the umpteenth time.

[Organic]

Matt,

Do you understand my previous post?

Please answer by yes or no.

[matt grime]

No.

And I don't understand why you wrote it because I asked a simple question that requires a yes or no answer. Nor do I understand the assertions about a quantum element. Nor do I care about that right now. I asked a simple question. Is it too much to ask for you to answer it? Apparently so.

[matt grime]

Actually I 'd like to elaborate on that as: No, I don't know if I understand it. I asked a question about this materail which pertained to my understanding of it and you refused to tell me if I was correct or incorrect.

[Organic]

Please open the new thread that I opened on this subject here:

http://www.physicsforums.com/showthread.php?p=170802#post170802

[matt grime]

Why are those little words the hardest for you to utter. YES or NO.

[Organic]

You still do not understand it.

Please see my new thread on this subject.

[matt grime]

So the answer to my questions were both no; a tree does not display uncertainty and redundancy simply because there is more than one letter in each column above each stem. And that the diagram for 4 on the second row second from the left does have uncertainty. So what is it then about the diagram which implies uncertainty and redundancy?

[Organic]

Matt,

Please open the new thread that I opened on this subject here:

http://www.physicsforums.com/showthread.php?p=170802#post170802

The anser to your qeustion is there.

[matt grime]

I looked and I didn't see any such answer. Try explaining it here too. Seeing as I didn't understand that other thread perhaps you ought to consider a different tack? There are two different diagrams which have only one row of letters above them. In one case there are four different entries, in the other there is repetition. So ought I to say that there is no uncertainty and reduncdancy iff there is exactly one row of n different letters in the labelling of a tree for a diagram of quantity n?

[Organic]

to say that there is no uncertainty and reduncdancy iff there is exactly one row of n different letters in the labelling of a tree for a diagram of quantity n?

Yes this description is right.

Now, look at this:

through this structural/quantitative point of view 1*5 not= 1+1+1+1+1 not= 1*3+2 not= 1+4 and so on, because each arithmetical expression has a unique information form.

I did not find any mathematical branch that distinguishes between arithmetic operations according to their level of clarity, for example:

1*5 = *5 = {1,1,1,1,1}
1*3+2 = *3+2 = {{{1,1,1},1},1}
1*3;+2= *3;+2 = {{1,1,1},{{1},1}}
1*3;1*2= *3;*2 = {{1,1,1},{1,1}}
1+1+1+1+1 = +5 = {{{{1},1},1},1},1}

and so on ...

Shortly specking, any use of N to describe an information building-block, is at least some n AND addition or multiplication operations.

A proof that cannot be done by using standard N members.

Theorem: 1*5 not= 1+1+1+1+1

Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

[matt grime]

Then why didn't you say so earlier? I asked about this and it was one step removed to say this, but no, you couldn't do so. And you still expect people to bother with anything you say? This is my last one for a while, ciao. Oh, and to reiterate, to say 1*5 is not 1+1+1+1+1 requires you to redefine 1 and + and =, but I expect that won't worry you unduly.

[Organic]

= and 1 are the same as standard Math.

+ and * has two univereses, external( the standard one) and internal(the new one).

Please open the new thread that I opened on this subject here:

http://www.physicsforums.com/showthread.php?p=170802#post170802