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.
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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}
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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.
 
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