Deeper than primes

[doronshadmi]

Yes, there is an approach to establish an incompleteness theorem involving the diagonal method. So what? That has nothing to do with your continued confused equivocation of Gödel incompleteness with Doron incompleteness.

The confused person here is you, jsfisher. So what?

 

[The Man]

Yes it is.

Both Bozonic and Fermionic fields are derived form the unified field which is the common source of both physical brain and your thoughts.

What "Bozonic and Fermionic fields" are you referring to? What "unified field" are you referring to? You do understand that a vector field is just a representation of the magnitude and direction of some force at various points, thus just an expression itself and one specifically resulting from the physical brains and thoughts of numerous people over the centuries, don’t you (I guess not)? You seem to have it backasswords, as usual, with something we thought of to represent a distribution of vectors, a vector field, as somehow the “source of both physical brain and your thoughts”. Looks like we can add the word “field” to the list of words Doron does not understand and apparently simply refuses to try to understand.

You just Making up nonsensical crap and just stringing words together does not constitute thinking Doron, inside or outside of any box you would like to simply ascribe others to. Looks like I might have to rescind my pervious assertion that there may be others who know even less about science and physics than you.

You are thinker The Man.

At least I think Doron not like your "without any thoughts about it" reasoning.

 

[jsfisher]

The confused person here is you, jsfisher. So what?

Once again, Doron, you ascribe to others a defect that is completely yours.

You embarked on a whole tangential journey into what you claimed built power sets without indices (which you failed to do, but like that was unexpected), then you leap from no where to some conclusion about Gödel based entirely on you failure to understand some paper you stumbled across on diagonal proofs.

 

[zooterkin]

What "Bozonic and Fermionic fields" are you referring to?

I like "Bozonics" as an alternative title for "Doronetics".

 

[jsfisher]

Bozonics? That has a nice ring to it.

No matter. Before Doron cites even more papers he doesn't understand trying to align himself with Cantor and Gödel in ways no sensible person can fathom, perhaps we should go back to his most recent attempt to construct power sets.

The order of some collection of distinct objects has no influence on the number of the objects.

As Doron is inclined to do, he belabors the trivial, probably because he doesn't understand just how trivial it is.

The power set of, for example, {0,1,2} is {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}.

Here is some example of the translation of a power set's members into <0,1> form, such that no "{X}" or "{}" forms are used anymore.

Doron is about to introduce by way of hand-waving and incomplete explanation an alternate notation for sets of some finite collection of possible elements. A reference set for which a power set will be constructed provides the universe of elements. For the instant example, the universe consists of 0, 1, and 2.

By this generalization

Nothing has been generalized. Add "generalization" of the list of terms Doron does not understand. The only thing being done is a change in notation.

...the power set of some set is 2^(the number of the distinct objects of that set), for example:

{
000 ↔ {}
001 ↔ {0}
010 ↔ {1}
011 ↔ {2}
100 ↔ {0,1}
101 ↔ {0,2}
110 ↔ {1,2}
111 ↔ {0,1,2}
}

Intuitively, this is true, but Doron makes no attempt whatsoever to indicate how he'd generate the full list of three-character strings of 0's and 1's nor to establish formally that the list represents exactly the elements of the power set.

So by using <0,1>^X we can construct any power set,

...of a finite set, Doron left out that part, of a finite set...

...such that the members of the set and the members of power set are constructed by the same rule of <0,1> form,

I suppose this muddled sentence fragment refers to the relationship between the string of all 1's and the original set. It is never clear, though, what Doron is thinking, so it is usually best to just ask him what he was thinking but didn't think to write down.

...which enables us to use the diagonal method without any need of any extra set of indexes, because <0,1> form is its own index system.

Here we have a hint at another thing Doron may have meant but never wrote: His list of 0/1 strings has an established ordering based on interpreting the strings as binary numbers. Gee, and didn't Doron deny the use of concepts well above basic set theory?

Well, the point remains that the list is still indexed.

Please pay attention that if we use the diagonal method on any arbitrary set of X members (and in this case X=3), which are based on <0,1> form, we get the members of the power set that are not in the range of some X arbitrary members of that set

Now we get to the trivial revelation by Doron that a set of 3 elements can't have all the elements that are in a set of 8 elements. Doron revels in the trivial. Other than that, there is no reason Doron has dwelled on small subsets of the power set.

...<snip>...
By using the common constriction rule of <0,1> form, we are using the diagonal method also on the set of ∞ distinct members

Nope. This leap lacks foundation. Also, it is not true. Of couse, that was Cantor's point, now wasn't it? Doron, however, is on a different mission.

...as follows:

{
111… ,
100… ,
101… ,
…
}

Isn't that interesting. His example strings aren't implicitly ordered by value this time. So, we now have an ordering of the strings that is explicit, but Doron simply omits this detail because it would contradict his "without any need of any extra set of indexes" condition even though the condition has no relevence. Doron revels in the irrelevant.

The diagonal member of the power set that is not in the range of set of ∞ distinct members, starts (in this case) with 010… <0,1> form, even if X=∞ (also in this case the members of the set and the members of power set are constructed by the same rule of <0,1> form).

Ok, so what do we have here? Doron has assumed he can construct a power set of an arbitrary infinite set using this 0/1 string method. From his construction, he finds a subset of the original set that isn't an element of the power set, at least not the one he's constructed.

Curious, that. A construction method for power sets that misses at least one subset. Hmmm, what will Doron conclude from this? What will he conclude?

Because the set and the power set are based on the same constriction rule of <0,1> form and there are always members of the power set that are not in the range of the set (whether X is finite or not), then no set is complete exactly because every set has a power set and every power set has also power set etc... ad infinitum ...

Everyone that predicted "something off the wall and bogus" for Doron's conclusion, award yourself full credit. The correct conclusion, of couse, is that the construction method doesn't work.

 

[epix]

Originally Posted by epix
which is a zero-dimensional object

Emptiness is not an object and it does not have a predecessor.

Did I say so? You cut my sentence in half and quoted the one without the subject. The entire sentence looks like this:

As far as I remember, you said that Emptiness was a predecessor to point, which is a zero-dimensional object.

It reads that Emptiness is a predecessor to point, as you have established it that way, not that Emptiness doesn't have a predecessor.

As far as the term "object" is concerned, if Emptiness is the predecessor to point, which is a 0-dimensional object, then there is nothing wrong to call Emptiness an object given the inescapable implication.

Nice try, but you are wrong.

No need to comment on your own cut&paste adventures.

But I think that it is your inability to comprehend a text that messes things up for you. Like saying that Godel's incomplete theorems addresses the power set. If you could comprehend the issue just slightly, you would find out that those theorems cannot and do not address some single axioms, like the axiom of power set that you have a problem with. Seeking support from Godel to support your vision of Trojan Horses, is another evidence that you don't have the slightest idea what's going on elsewhere. That's because you don't have the slightest idea what's going on in there -- in your "home improvement" of math.

Why don't you learn how to factor simple algebraic terms to avoid further embarrassment?

 

[doronshadmi]

Doron is about to introduce by way of hand-waving and incomplete explanation an alternate notation for sets of some finite collection of possible elements. A reference set for which a power set will be constructed provides the universe of elements. For the instant example, the universe consists of 0, 1, and 2.

This is the beauty of generalization, we can find a comprehensive method that is not influence by different concepts that actually have a common principle. One of the ways to expose the common principle of ,so called, different concepts, is to to express the different concepts by using a common form, and in this case the common form is the <0,1> form.

By ZFC axiom of powerset every set has a power set, where the concept of subset is not required ( see http://en.wikipedia.org/wiki/Axiom_of_power_set ), so "{x}" and "{}" forms are not required in order to express the members of the power set of some set, whether this set is finite or not.

By using <0,1> form as a common form for both sets and powersets, we actually discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines objects of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.

Nothing has been generalized. Add "generalization" of the list of terms Doron does not understand. The only thing being done is a change in notation.

Exactly the opposite, by your "{x}" and "{}" forms we get the illusion that members of set and power sets have are fundamentally different by their construction. By using <0,1> form I show that set's members and power set's members have the same form.

Intuitively, this is true, but Doron makes no attempt whatsoever to indicate how he'd generate the full list of three-character strings of 0's and 1's nor to establish formally that the list represents exactly the elements of the power set.

No intuition is used here. What is used here is a common <0,1> form for both set's members and power set's members.

...of a finite set, Doron left out that part, of a finite set...

No, the common <0,1> form for both set's members and power set's members holds whether the set (and it power set) is finite, or not.

I suppose this muddled sentence fragment refers to the relationship between the string of all 1's and the original set. It is never clear, though, what Doron is thinking, so it is usually best to just ask him what he was thinking but didn't think to write down.

I am vary clear about it: The common <0,1> form for both set's members and power set's members holds whether the set (and its power set) is finite, or not.

Here we have a hint at another thing Doron may have meant but never wrote: His list of 0/1 strings has an established ordering based on interpreting the strings as binary numbers. Gee, and didn't Doron deny the use of concepts well above basic set theory?

Well, the point remains that the list is still indexed.

The simple notion here is that the order of distinct members has no influence on the number of the distinct members, and by understanding this simple notion one immediately understands that the distinction of set's members is their built-in index (no 1-to-1 mapping is needed).

Now we get to the trivial revelation by Doron that a set of 3 elements can't have all the elements that are in a set of 8 elements. Doron revels in the trivial. Other than that, there is no reason Doron has dwelled on small subsets of the power set.

Now we get jsfisher's trivial notion that can't get <0,1> as a common form for both set's members and power set's members, whether the set (and its power set) is finite, or not.

Nope. This leap lacks foundation. Also, it is not true. Of couse, that was Cantor's point, now wasn't it? Doron, however, is on a different mission.

Cantor did not understand the ability to define a common form for both some given set's members and its power set's members.

Isn't that interesting. His example strings aren't implicitly ordered by value this time. So, we now have an ordering of the strings that is explicit, but Doron simply omits this detail because it would contradict his "without any need of any extra set of indexes" condition even though the condition has no relevence. Doron revels in the irrelevant.

Since jsfisher gets power sets only in terms of subsets of a given set he misses (again) this:

By ZFC axiom of powerset every set has a power set, where the concept of subset is not required ( see http://en.wikipedia.org/wiki/Axiom_of_power_set ), so "{x}" and "{}" forms are not required in order to express the members of the power set of some set, whether this set is finite or not.

Ok, so what do we have here? Doron has assumed he can construct a power set of an arbitrary infinite set using this 0/1 string method. From his construction, he finds a subset of the original set that isn't an element of the power set, at least not the one he's constructed.

Again,The concept of subset is not is not required. We have a common form for both some given set's members and its power set's members.

Curious, that. A construction method for power sets that misses at least one subset. Hmmm, what will Doron conclude from this? What will he conclude?

Everyone that predicted "something off the wall and bogus" for Doron's conclusion, award yourself full credit. The correct conclusion, of couse, is that the construction method doesn't work.

The correct conclusion is this:

By using <0,1> form as a common form for both sets and powersets, we actually discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines objects of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.

 

[jsfisher]

This is the beauty of generalization....

Let's see. Doron creates a method to do something, and then he proceeds to prove the method does not work. This he calls a generalization.

How novel.

So, in Doron's vocabulary, we can all agree Doronetics is a generalization.

 

[epix]

By ZFC axiom of powerset every set has a power set, where the concept of subset is not required ( see http://en.wikipedia.org/wiki/Axiom_of_power_set ), so "{x}" and "{}" forms are not required in order to express the members of the power set of some set, whether this set is finite or not.

By using <0,1> form as a common form for both sets and powersets, we actually discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines objects of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.

Hmm... You know, you may be right about the Trojan Horse in that power set. The languge that introduces the power set includes letter 'C', which could mean set C, subset C, but also an initial. See, younger male horses are called "colts" - youknowwhatimean.

 

[doronshadmi]

Let's see. Doron creates a method to do something, and then he proceeds to prove the method does not work.

It works perfectly by proving that every non-empty collection is incomplete.

It is devastating for any one who gets Math only at the level of collections and also claims about the completeness of these collections.

 

[jsfisher]

It works perfectly by proving that every non-empty collection is incomplete.

No, it fails spectacularly to correctly construct power sets.

 

[doronshadmi]

You seem to have it backasswords,

At the level of the unified field there is no direction of any kind.

The unified field is the natural source of both space and time, so at this calm and fundamental level there are no backwards, forwards or vectors of any kind (which are all based of directions).

Your rough perception can't get this unified level because you are stuck at the level of expressions without the awareness of their unified source.

 

[doronshadmi]

No, if fails spectacularly to correctly construct power sets.

Prove it.

 

[jsfisher]

Prove it.

You already did. It cannot construct the power set for any infinite set.

 

[doronshadmi]

You already did. It cannot construct the power set for any infinite set.

No, jsfisher, you simply can't get <0,1> as a common form for S P(S), P(P(S)) , ... etc. ad infinitum, whether S is finite or not.


EDIT:

Again.

The correct conclusion is this:

By using <0,1> as a common form for both sets and powersets, we discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines some object of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.

 

[jsfisher]

No, jsfisher, you simply can't get <0,1> as a common form for S P(S), P(P(S)) , ... etc. ad infinitum, whether S is finite or not.

Repeating your faulty conclusion doesn't make your premise any more correct. Your 0/1 string notation does not work for anything beyond an enumerable set.

 

[doronshadmi]

Repeating your faulty conclusion doesn't make your premise any more correct. Your 0/1 string notation does not work for anything beyond an enumerable set.

EDIT:

No jsfisher, the notion of enumerable set is simply the result of the 1-to-1 mapping between distinct objects that can't be tested by some infinitely long object, and distinct objects that can be tested by some infinitely long object.

This is just an arbitrary construction that can't be used in order to conclude that the collection of distinct objects that can't be tested by some infinitely long object, is enumerable, and the collection of distinct objects that can be tested by some infinitely long object, is non-enumerable.

Jsfisher, your reasoning about enumerable and non-enumerable sets is nothing but an illusion that is derived from the asymmetric condition of your 1-to-1 mapping between collections of distinct objects.

 

[jsfisher]

No jsfisher, the notion of enumerable set is simply the result of the 1-to-1 mapping between distinct objects that can't be tested by some infinitely long object, and distinct objects that can be tested by some infinitely long object.

What are you on about now? I can represent enumerable sets just fine with this trivial 0/1 notation you hail as some sort of second coming. I can, for example, represent any set of positive integers if I define the positive integers as my universe and use their natural ordering to establish a correspondence between each integer and a position in a magical 0/1 string.

Then, for example,

100000... is the set {1},
101000... is the set {1,3},
111111... is the set of all integers, and
010101... is the set of even numbers.

Whoop-de-do!! Enumerable sets can be represented in this silly notation. You weren't trying to say I couldn't were you? It is so hard to decode your writings most of the time.

This is just an arbitrary construction that can't be used in order to conclude that the collection of distinct objects that can't be tested by infinitely long object is enumerable and the, and collection of distinct objects that can be tested by infinitely long object is non-enumerable.

Is this gibberish supposed to be coherent English. It isn't.

Jsfisher, your reasoning about enumerable and non-enumerable sets is nothing but an illusion that is derived from the asymmetric condition of your 1-to-1 mapping between collections of distinct objects.

Really? Please demonstrate how your 0/1 string notation would represent, oh, how about the power set of the set of integers? Or maybe the set of reals along the interval, [0,1]?

 

[epix]

By using <0,1> as a common form for both sets and powersets, we discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines some object of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.

Did you figure what that C in
http://en.wikipedia.org/wiki/Axiom_of_power_set
stands for, or are you going to twist around just the part that has been translated? That C is really Trojan Horse -- in your game of pretense.

Why don't you bless the forum with scholarly discourse on the difference between the version stated in Wiki and the one which is listed here:
http://mathworld.wolfram.com/AxiomofthePowerSet.html



What cardinality is it that, erm, set of yours? Pardon me? Let me see . . . Well, perhaps this could be right for you.

(wild horses, couldn't drag me away . . .)

 

[doronshadmi]

Then, for example,

100000... is the set {1},
101000... is the set {1,3},
111111... is the set of all integers, and
010101... is the set of even numbers.

EDIT:

You produced a particular correlation between subsets and <0,1> form.

The <0,1> is general exactly because it is not based on any particular correlation between <0,1> form and the internal structure of some subset, because subsets are not considered at all, and all we care is the size of the considered collections.

Since subsets are not considered at all, the members of some S set are actually a partial case of the members of P(S), for example:

The members of P(S) are:
{
00,
01,
10,
11
}
and the members of S are partial collections of any possible 2 distinct members that have a common <0,1> form with P(S) distinct members, for example:

S =
{
10,
11
}
or
{
00,
10
}


etc ... , where given any S version of 2 members , there is a diagonal object that is based on <0,1> form ( which is common for both S an P(S) ) that is not in the range of S, but it is in the range of P(S).

But also P(S) is a set that has a common <0,1> form with P(P(S)), for example:

0 1 0 1
0 0 1 1
-------
0 0 0 0
1 0 0 0
0 1 0 0
1 1 0 0
0 0 1 0
1 0 1 0
0 1 1 0
1 1 1 0
0 0 0 1
1 0 0 1
0 1 0 1
1 1 0 1
0 0 1 1
1 0 1 1
0 1 1 1
1 1 1 1

and in this case some 4 P(S) objects ( which are partial case of P(P(S)) ) are:

P(S)=
{
0 1 1 0,
1 1 1 0,
0 0 0 1,
1 0 0 1
}
or
{
0 1 0 0,
0 0 1 1,
1 1 1 1,
0 0 1 0
}

etc ... , where given any P(S) version of 4 members , there is a diagonal object that is based on <0,1> form ( which is common for both P(S) and P(P(S)) ) that is not in the range of P(S), but it is in the range of P(P(S)).

Really? Please demonstrate how your 0/1 string notation would represent, oh, how about the power set of the set of integers? Or maybe the set of reals along the interval, [0,1]?

The same reasoning works also if S has an infinite size, as follows:

S=
{
.0 1 1 0 ...,
.1 1 1 0 ...,
.0 0 0 1 ...,
.1 0 0 1 ...,
...
}
or
{
.0 1 0 0 ...,
.0 0 1 1 ...,
.1 1 1 1 ...,
.0 0 1 0 ...,
...
}

etc ... are partial cases of P(S) where the diagonal member is not in the range of any S version ( although both S and P(S) have a common <0,1> form ).

By using <0,1> as a common form for both sets and powersets (finite or not), we discover that the ZFC axiom of powerset is actually a "Trojan horse" , which defines some object of ZFC that have the properties of a given set (it obeys the construction rules of the given framework) but it is not in the range of the given set (but can't be proved within this framework), exactly as Godel's first incompleteness theorem demonstrates.