Deeper than primes

[doronshadmi]

That P(S) Power sets are sets of subsets. You want to generalize it? Fine, but that means the subsets concept has to stay. Cast it aside, and then you aren't talking about power sets any more.

What you say is simply nonsense.

No subsets are needed in order to define the sizes of S and P(S), as shown in
http://www.internationalskeptics.com/forums/showpost.php?p=6769502&postcount=13937 .


You simply avoid http://www.internationalskeptics.com/forums/showpost.php?p=6769686&postcount=13939 because it does not fit to your dogma.

 

[epix]

By using <0,1>^k as a general 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 ( please see also http://www.internationalskeptics.com/forums/showpost.php?p=6756356&postcount=13911 ).

You can't get enough of that "Trojan Horse" in that power set. You think that the html tag <0,1> raised to the power of k defines some "general form." There is nothing else you can do though, coz altering the definition of the power set would require the symbolism which the set theory speaks with that you don't understand well enough to express your own ideas.

You are living in the dark box where french fries once lived, coz the power set is not the only mean to enumerate subset combinations. There are other sets like that

but the power set has been a choice for many not only for Georg Cantor, but also for John Henry Bonham.

You think that your Trojan Horse in the power set iffs Achilles Last Stand, but that's not really so.

Proof: http://crop720.tripod.com/

 

[jsfisher]

No subsets are needed in order to define the sizes of S and P(S)

What has this non sequitur to do with your nonsense generalization? We have not been discussing the cardinality of sets and their power sets. You were off on an entirely different tangent about something you claimed to be power sets but weren't.

Would you like to discuss cardinality now? Or is this just another example of you being unable to stay on topic?

By the way, while one can define what is meant by size of a set, you do not get to also define what is meant by the size of a power set. That's fixed by the first definition; you don't get to redefine it. Power sets are sets, after all.

 

[doronshadmi]

Would you like to discuss cardinality now?

Cut the nonsense.

We have S and P(S) such that P(S) members are not subsets of S because both P(S) and S are based on <0,1>^k (where k=0 to ∞) as their general form.

By using the diagonal method on the general form we discover that no set is complete exactly because both P(S) S are sets.

Furthermore, this incompleteness is the result of the different cardinality of S and P(S), which are both sets that have member's <0,1>^k common form.

Let us improve it.

By generalization, S and P(S) sizes are based on <0,1>^k form, where k = 0 to ∞.

By <0,1>^0
P(S)=
(
{0}
or
{1}
)
and
S=
(
{}
)

By <0,1>^1
P(S)=
{0,1}
and
S=
(
{0}
or
{1}
)

By <0,1>^2
P(S)=
{00,01,10,11}
and
S=
(
{
10,
11
} → 00
or
{
10,
00
} → 01
or
{
00,
11
} → 10
or
{
01,
10
} → 11
)

etc… ad infinitum … and as can be seen, no subsets are used.

It is more fundamental than your set\subset construction of S P(S)(your construction has no ability to comprehend the real meaning of the diagonal method) which rigorously demonstrates the incompleteness of any collection of distinct objects (without the need of any 1-to-1 mapping) exactly as Godel's first incompleteness theorem demonstrates ( please see also http://www.internationalskeptics.com/forums/showpost.php?p=6756356&postcount=13911 ).

Because of the lack of general form for both S and P(S), your traditional approach of this subject gets the illusionary dichotomy between enumerable and non-enumerable collections of distinct objects.

By the way, while one can define what is meant by size of a set, you do not get to also define what is meant by the size of a power set. That's fixed by the first definition; you don't get to redefine it. Power sets are sets, after all.

At the level of collections, given some cardinality there is another cardinality > than the given cardinality, ad infinitum, or in other words, no collection of distinct objects is complete.

 

[The Man]

Cut the nonsense.

Once again, by all means please, you first.

We have S and P(S) such that P(S) members are not subsets of S because both P(S) and S are based on <0,1>^k (where k=0 to ∞) as their general form.

OK so your “P(S)” is not the power set of “S” as that would require its members to be specifically all the “subsets of S” regardless of what you claim it is “based on” “as their general form”.

By using the diagonal method on the general form we discover that no set is complete exactly because both P(S) S are sets.

Please show any member of “S” that is not a member of “S” or any member of your “P(S)” that is not a member of your set “P(S)” otherwise both set are in fact, and by their deffinitions, complete.

Furthermore, this incompleteness is the result of the different cardinality of S and P(S), which are both sets that have member's <0,1>^k common form.

What the heck are you on about now? The different cardinalities only demonstrates that the sets have different sizes not that either is incomplete. You seem to be confusing (again perhaps deliberately) the size of a set with the completeness of a set. All sets are complete by their definitions (as has already been explained to you) they have only and all the members they are defined to have. You seem to be inferring that any set that doesn’t include absolutely everything would be “incomplete”.

Let us improve it.

Well that would be a refreshing deviation from your normal trend of just making your nonsense worse or simply repeating it.

By generalization, S and P(S) sizes are based on <0,1>^k form, where k = 0 to ∞.

By <0,1>^0
P(S)=
(
{0}
or
{1}
)
and
S=
(
{}
)

By <0,1>^1
P(S)=
{0,1}
and
S=
(
{0}
or
{1}
)

By <0,1>^2
P(S)=
{00,01,10,11}
and
S=
(
{
10,
11
} → 00
or
{
10,
00
} → 01
or
{
00,
11
} → 10
or
{
01,
10
} → 11
)

etc… ad infinitum … and as can be seen, no subsets are used.

So once again, by your own assertion no power sets were used. What happened to your purported ‘improvement’?

It is more fundamental than your set\subset construction of S P(S)(your construction has no ability to comprehend the real meaning of the diagonal method) which rigorously demonstrates the incompleteness of any collection of distinct objects (without the need of any 1-to-1 mapping) exactly as Godel's first incompleteness theorem demonstrates ( please see also http://www.internationalskeptics.com/forums/showpost.php?p=6756356&postcount=13911 ).

Ah, I see you’ve simply returned to your standard practice of simply repeating your previous nonsense and claiming it is “more fundamental”

Because of the lack of general form for both S and P(S), your traditional approach of this subject gets the illusionary dichotomy between enumerable and non-enumerable collections of distinct objects.

Doron you’re the one claiming your “S and P(S)” have different cardinally thus different sizes so you still get what you call that “illusionary dichotomy between enumerable and non-enumerable collections of distinct objects”

At the level of collections, given some cardinality there is another cardinality > than the given cardinality, ad infinitum, or in other words, no collection of distinct objects is complete.

So you are deliberately trying to confuse the cardinality of a set with the completeness of a set in order to perpetuate your nonsense. Color me, once again, unsurprised. Looks like you’ve thrown in a little of your tendency to make things worse for yourself as a parting gift at the end there (you should have stuck with just repeating your nonsense).

 

[epix]

At the level of collections, given some cardinality there is another cardinality > than the given cardinality, ad infinitum, or in other words, no collection of distinct objects is complete.

And what else is new? You don't need the power set to demonstrate that for any n there is n+1. But when you talk about two sets, then things may get tight.

Cantor's theorem that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostility than any other mathematical argument, before or since, with the exception of Hilbert's introduction of completely non-constructive proofs of existence some decades later.

So if the size of a finite set of naturals is N, then the size of a particular combination (binomial) of the members of the set is 2^N-1. If such sequence isn't bounded, then the negative unit is purged by the rules of calculus. Since sets are about counting and nothing else, Cantor added an empty set into the power set to satisfy [lim → ∞]2^N-1 = 2^N.

What are you discovering this time, Doron?

Btw, Haldegard says that the symbol for the power set is P__S elsewhere in the universe and wonders why is it so that you, as an earthling, get so concerned about the size of it.

I guess it got something to do with Revelation 22:13.
I am the Alpha and the Omega, the First and the Last, the Beginning and the End.

So the full name of God is Jehovah Dickinson. But don't bet on it. You know the weasels from Andromeda.

 

[jsfisher]

Doron, if you are trying to say there is no set of everything, please just do so so we can move on to your next trivial revelation.

 

[doronshadmi]

OK so your “P(S)” is not the power set of “S”

P(S) is the power set of S, and no subsets are needed, exactly as shown in http://www.internationalskeptics.com/forums/showpost.php?p=6771897&postcount=13944 , such that the diagonal method demonstrates the incompleteness of any collection of non-empty distinct objects, and it is also equivalent to Godel's first incompleteness theorem.

The Man, your reasoning does not hold water about this fine subject.

 

[doronshadmi]

Doron, if you are trying to say there is no set of everything, please just do so so we can move on to your next trivial revelation.

I am very clear about what I say:

Any non-empty collection of distinct objects is incomplete, and the diagonal method (without any need of 1-to-1 mapping or using subsets) rigorously proves it, and it is also equivalent to Godel's first incompleteness theorem.

jsfisher, your reasoning is based on the particular structure of subsets and can't comprehend the fact that this particular structure is nothing but the <0,1>^k(k=0 to ∞) form, which is itself a particular case of <0,1,3,...>^k(k=0 to ∞) that goes beyond P(S) S.

 

[doronshadmi]

Please show any member of “S” that is not a member of “S” or any member of your “P(S)” that is not a member of your set “P(S)” otherwise both set are in fact, and by their definitions, complete.

The Man, your limited reasoning about P(S) as a collection of subsets of S, prevents from you to get the fact that the diagonal number (which is an object of P(S) ) is not in the range of any S, and the diagonal number (which is an object of P(P(S)) ) is not in the range of any P(S) etc... ad infinitum, whether S is finite or not.

You can't stop at some arbitrary S P(S) state and then conclude that P(S) is complete, exactly because P(S) is the set of P(P(S)), P(P(S)) is the set of P(P(P(S))) ...etc. ... ad infinitum ... and the <0,1>^k(k=0 to ∞) common form for both S and P(S) simply prevents such an arbitrary stop.

Furthermore, your reasoning is based on the particular structure of subsets and can't comprehend the fact that this particular structure is nothing but the <0,1>^k(k=0 to ∞) form, which is itself a particular case of <0,1,3,...>^k(k=0 to ∞) that goes beyond P(S) S.

 

[epix]

P(S) is the power set of S, and no subsets are needed, exactly as shown in http://www.internationalskeptics.com/forums/showpost.php?p=6771897&postcount=13944 ,such that the diagonal method demonstrates the incompleteness of any collection of non-empty distinct objects, and it is also equivalent to Godel's first incompleteness theorem.

I don't agree with your statement, especially with the part for which there is a poof of it being false:
"...the diagonal method demonstrates the incompleteness of any collection of non-empty distinct objects..."

The Man, your reasoning does not hold water about this fine subject.

That would be sad if it did.

 

[epix]

You can't stop at some arbitrary S P(S) state and then conclude that P(S) is complete, exactly because P(S) is the set of P(P(S)), P(P(S)) is the set of P(P(P(S))) ...etc. ... ad infinitum ... and the <0,1>^k(k=0 to ∞) common form for both S and P(S) simply prevents such an arbitrary stop.

There is clearly a clinical aspect involved. Let F be a finite set of fruit

F = {apple, orange, lemon}

Doron's power set with no subset distinction is

P(F) = {apple, orange, lemon, apple, orange, apple, lemon, orange, lemon, apple, orange, lemon}

The Florida fruit growers would love it.

 

[doronshadmi]

I don't agree with your statement

First you have to show that you understand <0,1>^k(k=0 to ∞) form.

Only then you can air some view about it.

Since you do not get it ( see your wrong F P(F) construction in http://www.internationalskeptics.com/forums/showpost.php?p=6774857&postcount=13952 ) it does no matter what you say about it.

 

[doronshadmi]

There is clearly a clinical aspect involved. Let F be a finite set of fruit

F = {apple, orange, lemon}

Doron's power set with no subset distinction is

P(F) = {apple, orange, lemon, apple, orange, apple, lemon, orange, lemon, apple, orange, lemon}

The Florida fruit growers would love it.

You miss the <0,1>^k(k=0 to ∞) general form of both P(F) and F, you are still closed under subsets, as follows:

{
000 ↔ {}
001 ↔ {apple}
010 ↔ {orange}
011 ↔ {lemon}
100 ↔ {apple,orange}
101 ↔ {apple,lemon}
110 ↔ {orange,lemon}
111 ↔ {apple, orange, lemon}
}

As can be seen <0,1>^3 is the general form of both P(F) and F, in this case, as follows:

P(F)=
{
000,
001,
010,
011,
100,
101,
110,
111
}

F=
(
{
100,
110,
111
} → 000
or
{
101,
010,
000
} → 001
or

... etc. and we have at least 8 distinct F collections of 3 distinct objects each, where for each given F collection of 3 distinct objects we define a diagonal object that is not in the range of that F.

)

In other words, <0,1>^k(k=0 to ∞) is a general form for both P(F) and F, ... etc. ... ad infinitum ...

 

[sympathic]

First you have to show that you understand <0,1>^k(k=0 to ∞) form.

Only then you can air some view about it.

Since you do not get it ( see your wrong F P(F) construction in http://www.internationalskeptics.com/forums/showpost.php?p=6774857&postcount=13952 ) it does no matter what you say about it.

We say the same thing about all your nonsense and your "I do not agree" ramblings. First YOU have to show you understand basic concepts in Math, otherwise "it does not matter what you say about it."

 

[doronshadmi]

First YOU have to show you understand basic concepts in Math, otherwise "it does not matter what you say about it."

No sympathic. All you show is some limited agreed notion about S P(S) that is based on 1-to-1 mapping and subsets, which leads you to the nonsense of enumerable and non-enumerable complete collections of distinct objects, so?

 

[doronshadmi]

... your "I do not agree" ramblings.

The "I do not agree" ramblings is epix's ramblings in this case.

First read before you reply, sympathic.

 

[jsfisher]

P(S) is the power set of S, and no subsets are needed

Then it is not a power set of S. Please stop redefining things. You always get it wrong.

Any non-empty collection of distinct objects is incomplete

So, please show us which member is missing from {1,2,3} that makes it incomplete.

The Man, your limited reasoning about P(S) as a collection of subsets of S

Yeah, shame on you, The Man, for actually knowing what power set means. Weren't you paying attention? Doron has redefined power set and incompleteness and from there drawn all sorts of illogical conclusions.

 

[epix]

P(S) is the power set of S, and no subsets are needed...

You must take into your consideration the very basic definitions that apply to set. One of them is that a set cannot include members identical to each other, otherwise you would count them twice. See, each distinct member of the set, like @ # $ % ..., is assigned these symbols: 1, 2, 3, 4... So when you create the fruity power set without subsets,

my quote:

F = {apple, orange, lemon}

P(F) = {apple, orange, lemon, apple, orange, apple, lemon, orange, lemon, apple, orange, lemon}

then the members are not distinct. But once you create subsets, such as {apple, orange}, these subsets are basically baskets, each of them holding a distinct combination of fruit so you can deploy 1, 2, 3, ... In other words, you count the baskets/subsets and not the individual fruit.

Cantor found a tree which grew less fruit then it turned out in the baskets. That's why he ran into a strong opposition from the community of mathematicians. But Cantor maintained that the fruit in those baskets was irrevelant -- there were those baskets that mattered. Of course, only God can pull such a trick and fill the baskets with more fruit then the tree grows. But can He?

One year later, he [Cantor] was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since it had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God.

See? He was so upset that he briefly questioned the divine skills of God.

{
000 ↔ {}
001 ↔ {apple}
010 ↔ {orange}
011 ↔ {lemon}
100 ↔ {apple,orange}
101 ↔ {apple,lemon}
110 ↔ {orange,lemon}
111 ↔ {apple, orange, lemon}
}

Here you go. You must repent all the time.

 

[sympathic]

The main idea is to use a language that does not reduce the many possible points of view of the researched into a one and only one agreed point of view about a given notion.

Standard Mathematics is exactly an education method that does its best in order to reduce any given notion into a one and only one agreed description (what is callad "a well-defined ..."). Any other view that does not agree with the already agreed definition, is automatically marked as nonsense or "out of Mathematics" gibberish, cranky etc. …

I agree with you, but if you tell it to jsfisher or The Man they will try to convince you that the researcher is not a significant factor of any result (abstract or not).

Furthermore, they will tell you that any result where the researcher is involved, is not a valid result and should be ignored. They are using a naïve approach about the subjective influence of the researcher on the objective results. Their approach is naïve because their school of thought did not develop any comprehensive method that systematically researches the non-trivial interactions between the researcher and the researched, and this is exactly where Organic Mathematics gets on the scientific stage, and instead of eliminating the researcher as a subjective "white noise" (as Standard Math does for many years), it develops tools in order to research the possible interactions between the researched and the researcher, for at least to purposes:

1) To rigorously avoid subjective tendencies that may harm the validity of the results.

2) To rigorously research the researcher's objective properties (the properties that are not changed from person to person) and how the researcher's objective properties are related to the researched subject.


To teach students that each one of us is like a cell of a one organism (branches of the same tree, if you like), such that at the basis of our different points of view and many levels of awareness, there is a common ground for both Ethics and Logics that can be developed into a one scientific method, which under its wings people will no longer use the language of Mathematics in order to develop destructive technologies, that their aims is to kill other people that get things (abstract or not) different than them.

You are lying. You consistently use the "I do not agree with this definition" approach, and then you try to redefine things. Only you can comment and disagree with things you do not understand. How convenient.