Deeper than primes - Continuation

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Doron, I put the challenge again up to you: Do 1 thing that traditional science can not do and win the JREF MDC.

Everything else you do is just a waste of time and senseless arguing.
 
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S={00,10,01,11} looks as if it is complete, because by using Diagonalization on its elements, the result is an explicit element, which is already included in S.

Furthermore, according to this, so called, complete view of S, S can't have powerset (notated as P(S)).

But according to Traditional Mathematics (which is based on ZF(C)) every set has a power set according to the axiom of power set.

Since the existence of P(S) for any given S, is guaranteed by an axiom, then there can't be sets that do not have powersets, and as a result any given set must be incomplete, according to Traditional Mathematics.

Traditional Mathematics have missed its own reasoning, because it restricted the construction of the elements of P(S) to forms of subsets of S including the empty set and S itself.

By doing so, one can't realize that S and P(S) can have elements of the same construction, which actually enables to define elements of P(S), which are not elements of S, and enables one to conclude that S is incomplete.

Since the largest P(S) does not exists, this conclusion is extensible to P(S) as well.
 
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doronshadmi said:
S={00,10,01,11} looks as if it is complete, because by using Diagonalization on its elements, the result is an explicit element, which is already included in S.
Click to expand...
It looks complete because it is complete.

Furthermore, according to this, so called, complete view of S, S can't have powerset (notated as P(S)).
Click to expand...
Nonsense.
 
 
doronshadmi said:
S={00,10,01,11} looks as if it is complete, because by using Diagonalization on its elements, the result is an explicit element, which is already included in S.
Click to expand...

That diagonalization (well, more correctly, Doron-diagonalization) is a test for Doron-completeness is a bone-headed idea. That one set not having an element appearing in another set is a test for Doron-completeness is a bone-headed idea.

That all sets are Doron-incomplete is a worthless concept of no consequence.

Doron, why do you cascade bone-headed ideas that lead you to worthless concepts of no consequence?


And why does it surprise you so that a set of 16 elements will have at least one element not to be found in a set of 4?
 
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doronshadmi said:
According to your, so called, definition.


Please define the powerset of S={00,10,01,11}
Click to expand...

The definition of a power set is here. It's the set of all possible subsets of the set. The enumeration of the power set of S={00,10,01,11} is a little bit tiresome, but here it is:


P({00, 01, 10, 11}) = {{}, {00}, {01}, {00, 01}, {10}, {00, 10}, {01, 10}, {00, 01, 10}, {11}, {00, 11}, {01, 11}, {00, 01, 11}, {10, 11}, {00, 10, 11}, {01, 10, 11}, {00, 01, 10, 11}}
​
 
zooterkin said:
The definition of a power set is here. It's the set of all possible subsets of the set. The enumeration of the power set of S={00,10,01,11} is a little bit tiresome, but here it is:


P({00, 01, 10, 11}) = {{}, {00}, {01}, {00, 01}, {10}, {00, 10}, {01, 10}, {00, 01, 10}, {11}, {00, 11}, {01, 11}, {00, 01, 11}, {10, 11}, {00, 10, 11}, {01, 10, 11}, {00, 01, 10, 11}}
​
Click to expand...


Thank you for supporting the following argument:
doronshadmi said:
Traditional Mathematics have missed its own reasoning, because it restricted the construction of the elements of P(S) to forms of subsets of S including the empty set and S itself.

By doing so, one can't realize that S and P(S) can have elements of the same construction, which actually enables to define elements of P(S), which are not elements of S, and enables one to conclude that S is incomplete.

Since the largest P(S) does not exists, this conclusion is extensible to P(S) as well.
Click to expand...

Because of the limitations of your construction method, you can't know http://www.internationalskeptics.com/forums/showpost.php?p=8144371&postcount=942 .
 
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"defining concepts and being consistent with respect to meaning" can be developed, and being restricted to one and only one construction method of powersets, does not help very much to one's notion to be developed.

EDIT:

Furthermore, our notion must not be restricted only to sets, and this non-restriction is done by using the more general concept of non-empty collections of distinct elements.

For example:

C= a given non-empty collection with elements of a given type of forms.

P(C) = The power collection of any given C.


Argument:

Given any C, there is P(C), such that there is an explicit form that is an element of P(C), but it is not an element of any given C.

As a result any given C is incomplete.


Proof by construction:

The form is based on 0;1 symbols.

C=(0)

P(C)={0,1} where form 1 is an explicit member of P(C), but it is not an element of C.


C=(1)

P(C)={0,1} where form 0 is an explicit member of P(C), but it is not an element of C.


By using Diagonalization among the forms that are based on 0;1 symbols, we always are able construct an explicit type of form that is an element of P(C), but it is not an element of any given C.


C=
(
10,
11
)

P(C)=(00,01,10,11) where form 00 is an explicit element of P(C) that is not an element of C.


C=
(
10,
00
)

P(C)=(00,01,10,11) where form 01 is an explicit element of P(C) that is not an element of C.



C=
(
00,
11
)

P(C)=(00,01,10,11) where form 10 is an explicit element of P(C) that is not an element of C.


C=
(
01,
10
)

P(C)=(00,01,10,11) where form 11 is an explicit element of P(C) that is not an element of C.


...


Things do not change even if the 0;1 forms are infinite, as shown by Cantor.

Since the largest P(C) does not exist, we can conclude that no collection with given type of distinct forms is complete.

Q.E.D

Some claims that C is not an explicit collection, but according to the given argument, the given type of forms is the explicit property.
 
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doronshadmi said:
"defining concepts and being consistent with respect to meaning" can be developed, and being restricted to one and only one construction method of powersets, does not help very much to one's notion to be developed.
Click to expand...


I do enjoy the fact you insist on addressing me and my points obliquely. It suggests a fear of inadequacy on your part. Rightly so.

Be that as it may, in your most recent spate, Doron, you have yet to construct an actual power set. So lecturing us about being restricted with respect to power sets comes up empty. You continue to refer to something as a power set when it isn't. No one here, except you, is restricted to one and only one anything; that is just another straw man you have erected because you think it gives you credibility. (With whom is a mystery.)

Moreover, you have yet to define your Doron-complete concept and establish how it is in any way related to Doron-power sets.

Moreover, you have yet to demonstrate how any of this applies to non-finite sets. Your 2^|S| ceases to be meaningful when S is not a finite set.


What you insist are restrictions are nothing more than reality sneaking in in ways you find inconvenient. And that's why your fantasy math is useless, meaningless, and a horrible, horrible waste of, what is it now, 25 years of your life.
 
doronshadmi said:
Some claims that C is not an explicit collection, but according to the given argument, the given type of forms is the explicit property.
Click to expand...

I must admit... I really do not get that sentence.

To my eye it looks like it says:

Some people may claim that C is not a 'explicit' (?) collection, but according to the given specifications, this form is the 'explicit' (?) 'property' (?).

And I think it means:

C is a complete collection because, by definition, it is a complete collection, even though some may claim it is not.
 
zooterkin said:
Back to your old ways, I see. You edited that post you linked to over 90 minutes after you posted it. I, for one, will not be replying to it (not that one lot of gibberish is any better than another).
Click to expand...
If one gets something as gibberish, one can't reply anyway.
 
EDIT:

Decisions are taken and may influence on consequences,conclusions,results etc.

A decision: Set is a collection of distinguished objects, where order is insignificant.

According to this decision {a,a} = {a} and {a,b} = {b,a}.

According to this decision there is no difference between, for example, the given power sets:

P({00, 01, 10, 11}) = {{}, {00}, {01}, {00, 01}, {10}, {00, 10}, {01, 10}, {00, 01, 10}, {11}, {00, 11}, {01, 11}, {00, 01, 11}, {10, 11}, {00, 10, 11}, {01, 10, 11}, {00, 01, 10, 11}}

P({0000000000000000, 0100000000000000, 1000000000000000, 1100000000000000}) = {{}, {0000000000000000}, {0100000000000000}, {0000000000000000, 0100000000000000}, {1000000000000000}, {0000000000000000, 1000000000000000}, {0100000000000000, 1000000000000000}, {0000000000000000, 0100000000000000, 1000000000000000}, {1100000000000000}, {0000000000000000, 1100000000000000}, {0100000000000000, 1100000000000000}, {0000000000000000, 0100000000000000, 1100000000000000}, {1000000000000000, 1100000000000000}, {0000000000000000, 1000000000000000, 1100000000000000}, {0100000000000000, 1000000000000000, 1100000000000000}, {0000000000000000, 0100000000000000, 1000000000000000, 1100000000000000}}

because the two given examples are derived from decisions which cause that each explicit set has an explicit powerset, no matter what elements are included in the explicit set and its explicit powerset.

A decision: Set is a collection of distinguished or undistinguished objects, where order is significant or insignificant.

According to this decision ({a,a} = {a}) OR ({a,a} = {a,a}) and ({a,b} = {b,a}) OR ({a,b} ≠ {b,a}).

According to this decision one can get conclusions\results that can't be achieved by the previous decision (also please pay attention that by the previous decision, there is no difference between the previous and current decision, since order (previous,current) is insignificant by the previews decision, and as a result the previews decision can't comprehend anything that is related to the current decision, but this is not the case of the current decision w.r.t the previous decision, because according to the current decision order can be significant and\or objects can be undistinguished of each other).

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

Let us use the current decision of Set:

Set is a collection of distinguished or undistinguished objects, where order is significant or insignificant.

By this decision powersets are not necessarily constructed by collections of subsets, for example:

Given non-empty sets and their power set, one enables to construct their elements according to the same type of form, for example:

The type of form is based on 0;1 symbols.

The non-empty sets can be:
{
10,
11
} → 00
or
{
10,
00
} → 01
or
{
00,
11
} → 10
or
{
01,
10
} → 11

and their power set is {00,01,10,11}

Some claims that according to the current decision of Set there is no explicit set that have an explicit powerset (as shown according to the previous decision of Set) but according to the current decision, the given type of forms is the explicit common property among sets and their powerset.

The current decision of Set enables to conclude that no given set with a considered type of forms of given elements, is complete, since there is always an explicit element that is defined by using Diagonalization (with this considered type of forms), which is an element of the powerset, but it is not an element of any of the sets that their elements are constructed by the considered type of forms (also since the largest powerset does not exist, the conclusion of incompleteness is extensible also to powesets).

By using the current decision of Set let us demonstrate how the distinguishable and the indistinguishable are shared with each other, by using significant order of the forms that are based on 0;1 symbols:

S=
{
0,
1,
}

S=
{
00,
10,
01,
11,
}

S=
{
0000,
1000,
0100,
1100,
0010,
1010,
0110,
1110,
0001,
1001,
0101,
1101,
0011,
1011,
0111,
1111,
}

It is clearly seen how sets are included in other sets, where the distinguishable and the indistinguishable (which is characterized by at least the redundancy among the symbols of the given type of forms) are shared with each other.
 
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doronshadmi said:
Decisions are taken and may influence on the conclusions\results.

A decision: Set is a collection of distinguished objects, where order is insignificant.
Click to expand...


As per usual, Doron, you get things backwards. There is/was no such decision made. Sets are characterized by a set of axioms, and those axioms lead to a consequence where ordering and replication of elements are undetectable.

Of course, it is doubtful you will understand this because it requires (a) taking things in their correct order, (b) following the basic rules of logic, and (c) accepting definitions. Oh! so many horrible restrictions. How could anyone accept such draconian limitations?

According to this decision {a,a} = {a} and {a,b} = {b,a}.
Click to expand...

The sets {a,a} and {a} are indistinguishable, yes, but because of this decision you allege, no.

According to this decision there is no difference between, for example, the given power sets:

P({00, 01, 10, 11}) = {{}, {00}, {01}, {00, 01}, {10}, {00, 10}, {01, 10}, {00, 01, 10}, {11}, {00, 11}, {01, 11}, {00, 01, 11}, {10, 11}, {00, 10, 11}, {01, 10, 11}, {00, 01, 10, 11}}

P({0000000000000000, 0100000000000000, 1000000000000000, 1100000000000000}) = {{}, {0000000000000000}, {0100000000000000}, {0000000000000000, 0100000000000000}, {1000000000000000}, {0000000000000000, 1000000000000000}, {0100000000000000, 1000000000000000}, {0000000000000000, 0100000000000000, 1000000000000000}, {1100000000000000}, {0000000000000000, 1100000000000000}, {0100000000000000, 1100000000000000}, {0000000000000000, 0100000000000000, 1100000000000000}, {1000000000000000, 1100000000000000}, {0000000000000000, 1000000000000000, 1100000000000000}, {0100000000000000, 1000000000000000, 1100000000000000}, {0000000000000000, 0100000000000000, 1000000000000000, 1100000000000000}}
Click to expand...

No, those two power sets are different and can therefore be distinguished. The first, for example, contains {00} as a member; the second does not.

A decision: Set is a collection of distinguished or undistinguished objects, where order is significant or insignificant.
Click to expand...

Sounds like a Doron decision. So, let's refer to these things as Doron-collections which hale from the land of Doronetics. Also, while Doron-collections may be either ordered or unordered or may be multisets or not, which of those four possibilities is set for each instance of a collection. (My guess is that Doron will eventually equivocate on this point.)

According to this decision ({a,a} = {a}) OR ({a,a} = {a,a}) and ({a,b} = {b,a}) OR ({a,b} ≠ {b,a}).

According to this decision one can get conclusions\results that can't be achieved by the previous decision (also please pay attention that by the previous decision, there is no difference between the previous and current decision, since order (previous,current) is insignificant by the previews decision, and as a result the previews decision can't comprehend anything that is related to the current decision, but this is not the case of the current decision w.r.t the previous decision, because according to the current decision order can be significant and\or objects can be undistinguished of each other).

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

Let us use the current decision of Set:

Set is a collection of distinguished or undistinguished objects, where order is significant or insignificant.

By this decision powersets are not necessarily constructed by collections of subsets
Click to expand...

You must mean Doron-power sets, I guess. The set-theory concept of power set easily extends to multisets and/or ordered sets.

...for example:

Given non-empty sets and their power set, one enables construct their elements according to the same type of form
Click to expand...

If I already have the power set, there is no need to construct it. Well, maybe in Doronetics, but not so much in Mathematics.

This is also the place where Doron abandons anything resembling power sets for the whole purpose of being surprised the set {0} doesn't contain 1 as an element.

Nothing to see here. Even illogic gets abused.
 
Look at this phrase:

"Sets are characterized by a set of axioms, and those axioms lead to a consequence where ordering and replication of elements are undetectable."

The person that wrote this phrase is not aware of the fact that decisions were taken in order to determine the set of axioms, which lead to certain consequences,conclusions,results etc.

Another phrase of this person:

"...those two power sets are different and can therefore be distinguished."

In this case this person does not understand that the two given examples are derived from decisions which cause that each explicit set has an explicit powerset, no matter what elements are included in the explicit set and its explicit powerset (in that case he does not get that elements that are constructed by 16 0;1 symbols, are shared by sets with at least 16 elements each and their powerset, where their powerset has 2^16 elements with 16 0;1 symbols for each element).

Another phrase of this person:

"If I already have the power set, there is no need to construct it."

Here this parson wrongly thinks that construction is some kind of serial process, exactly because this person gets things only by serial step-by-step reasoning, and any decision that he takes is closed under this kind of reasoning.

No wonder that he simply misses http://www.internationalskeptics.com/forums/showpost.php?p=8157521&postcount=1023.
 
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The person gets its subjective aspects as objective.

I tried to ignore its subjective aspects in order to help him to get its objective aspect, which is beyond thoughts (where definitions are done only at the level of thoughts, and therefore they are naturally changeable) but he still can't comprehend that.
 
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Another phrase of this parson:

"This is also the place where Doron abandons anything resembling power sets for the whole purpose of being surprised the set {0} doesn't contain 1 as an element."

Since this person can't comprehend a given powerset, unless it is addressed as a collection of subsets of a given set, he can't grasp, for example, the following powerset, that is based on 0;1 symbols:

P(S)=
{
0,
1,
}

which shares the type of its elements with set {0} and set {1} (which are also based on 0;1 symbols), which are incomplete, since there is an explicit element of the same type of form, which is an element of P(S) but it is not an element of set {0}(element 1, in this case) or an element of set {1}(element 0, in this case).

And again, since the largest powerset does not exist, the conclusion of incompleteness is extensible also to powersets.
 
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doronshadmi said:
Another phrase of this parson:
Click to expand...

Parson? I was assuming it was more like Voldemort (you know, he whose name must not be spoken), then worried that Doron identified himself with Harry Potter. That would be just wrong on so many levels. But no, I'm a parson.

"This is also the place where Doron abandons anything resembling power sets for the whole purpose of being surprised the set {0} doesn't contain 1 as an element."
Click to expand...

Yep, that was an accurate statement by Parson Jsfisher.

Since this person can't comprehend a given powerset, unless it is addressed as a collection of subsets of a given set
Click to expand...

Oh, gee. There I go again, insisting the words have meaning.

...he can't grasp, for example, the following powerset, that is based on 0;1 symbols:

P(S)=
{
0,
1,
}
Click to expand...

Yep, I cannot grasp that that is a power set. That may be because it isn't a power set. It is also not an orange, and it is not a gravitational theory.

...which shares the type of its elements with set {0} and set {1} (which are also based on 0;1 symbols), which are incomplete, since there is an explicit element of the same type of form, which is an element of P(S) but it is not an element of set {0}(element 1, in this case) or an element of set {1}(element 0, in this case).

And again, since the largest powerset does not exist, the conclusion of incompleteness is extensible also to powersets.
Click to expand...

Wow. So much gibberish. Be that as it may, though, Doron, please tell us all how incompleteness can be established based on this pseudo-relationship between these ill-conceived sets of (0|1)* strings?

Moreover, since {1} is a complete set by the normal meaning of complete, how does Doron define complete in Doronetics?

Also, let us know just how this property, doron-completeness, is at all useful.

Also, let us know how you leap the gap from the finite set to the non-finite.
 
Sometimes, we need to be reminded of our roots. According to DoronShadmi:
  • A set is equal to the union of its members.
  • 2 may not be a member of {2}.
  • The construct A if B, the construct A only if B, and the construct A if and only if B are all equivalent.
  • There is no such thing as a number in standard Mathematics.
  • Cardinality cannot exceed 1. The number of elements in any set is between 0 and 1.
  • The 1/4 and 0.25 are different numbers.
  • Zero is a positive number.
Should we add to the list that the power set of {00, 01} is {00, 01, 10, 11}?


-Tom Marvolo Riddle
 
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Voldemort is still closed under an explicit set with an explicit powerset point of view.
 
doronshadmi said:
Voldemort is still closed under an explicit set with an explicit powerset point of view.
Click to expand...


{00, 01} is an explicit set, but only you in your fantasy Doronetics world allege {00, 01, 10, 11} to be its explicit power set.

I'll add it to the list.



-Tom Marvolo Riddle
 
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Oh, dear! I have neglected to include this classic:
  • 2 is not part of { 2,3,4 } much like a severed finger is not part of your body.


-Tom Marvolo Riddle
 
I now predict a series of convoluted 'insults', 'jibs' and 'jives' until there is so much text between the original statement and the end of the thread that everyone falls silent.
Then we will be presented with another clue on how Doronetics differs from reality.
Then the circus starts again.

Doron, the burden of proof is on the claimant. You claim Doronetics can do something. The burden of proof is upon you. It is on nobody else.
 
jsfisher said:
Oh, dear! I have neglected to include this classic:

[*]2 is not part of { 2,3,4 } much like a severed finger is not part of your body.


-Tom Marvolo Riddle
Click to expand...

You don't get it. Amen.
 
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doronshadmi said:
You and voldemort do not do anything on order to clean your balls
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Definitely sig material. Be that as it may, we should start a tally per realpaladin's prediction. We are now at 1 to 0 (insult to substance).

By the way, Doron, I have a forum identity, and it is not Voldemort. Please use my actual id.
 
doronshadmi said:
The burden of get the proof is on the observer (he must clean his ball in order to get it).

You and voldemort do not do anything on order to clean your balls ( http://www.internationalskeptics.com/forums/showpost.php?p=8158054&postcount=1027 ).

There is a result for this neglect.
Click to expand...

Quoted for posterity. The burden of get the proof is on the observer... the wacko's over at the CT and BF threads would thank you for it.

But no, you can claim, say, wish, babble whatever you want...

You have made a claim and it is upon you to come with a proof that is not rejected.

So far, all of your proofs have been rebutted, rejected or have been diverted into oblivion.

Do 1 single thing that you can do practically with Doronetics that traditional math can not do practically.

I mention 'practical' specifically because building constructs with arbitrarily chosen criteria is something anyone can.

Solve a known math problem, solve a known physics problem... anything.
 
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