Deeper than primes

[jsfisher]

Who is this fictional "we" to which you refer?

"we" is the common jargon for formal writing style.

You don't see the contradiction in that sentence, do you?

Define "incomplete". So far, your usage is been that incomplete means the set doesn't contain an element it doesn't contain.

...<irrelevance snipped>...

So, when asked to define "incomplete", you respond with something that is not the definition of "incomplete".

 

[doronshadmi]

So, when asked to define "incomplete", you respond with something that is not the definition of "incomplete".

It is given by theorem A in http://www.internationalskeptics.com/forums/showpost.php?p=6904202&postcount=14339 .

 

[jsfisher]

Theorem A:
"Given any S collection of <0,1> X different objects, such that each object has X bits, S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection."

Is that supposed to be a theorem or a defintion, Doron?

No matter. Let's filter out the smoke by translating it back to sets without the bit-map nonsense:

"Given any S collection of <0,1> X different objects" = Let S be a set where |S| = X
"such that each object has X bits" = selected from a domain U, where |U| = 2^X"S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection." = then if there exists Y in U and Y not in S, then S is doron-incomplete.

Since we know X < 2^X for all X, we can conclude that all sets are doron-incomplete, and the entire "theorem" is trivial.

 

[epix]

So, when asked to define "incomplete", you respond with something that is not the definition of "incomplete".

The particular answer is a collection of words translatable to the <0,1> unique codes and therefore, by the Theorem of Incompleteness, such an answer is incomplete and the incomplete part unfortunately includes the definition of "incomplete."

 

[epix]

Definition A:
"Let X be a placeholder for any finite or infinite cardinality."

Theorem A:
"Given any S collection of <0,1> X different objects, such that each object has X bits, S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection."

This is a fantasmagoric idea that attempts to prove the last Euclid's Common Notion that says, "The whole is greater than the part."

 

[dafydd]

Emptiness is weaker than the existence of collections.

Fullness is stronger than the existence of collections.

I find your gibberish very entertaining.

 

[epix]

That's the last Fellini's flick, isn't it?

No it's not.

I think it is. He didn't shoot anything after 1990.

According to professor Shadmi, he did.

Definition A:
"Let X be a placeholder for any finite or infinite cardinality."

Theorem A:
"Given any S collection of <0,1> X different objects, such that each object has X bits, S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection."

Proof:
1) Let S be any collection of X different objects, such that each object is <0,1> form of X bits (we get a matrix of X^2 bits).

2) Each different object of P(S) is <0,1> form of X bits (we get a list of 2^X different objects, where each object has X bits).

3) According to (1) and (2) there is at least one <0,1> form of X bits, which is not in the range of a given S.

Conclusion: S is incomplete.

 

[doronshadmi]

Is that supposed to be a theorem or a defintion, Doron?

It is a theorem, and it explicitly provides the needed terms for incompleteness, as an inseparable aspect of the theorem.

No matter. Let's filter out the smoke by translating it back to sets without the bit-map nonsense:

"Given any S collection of <0,1> X different objects" = Let S be a set where |S| = X
"such that each object has X bits" = selected from a domain U, where |U| = 2^X"S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection." = then if there exists Y in U and Y not in S, then S is doron-incomplete.

Since we know X < 2^X for all X, we can conclude that all sets are doron-incomplete, and the entire "theorem" is trivial.

Without this <0,1> bit-map "nonsense" you can't show that there exists Y in U and Y not in S, in order to conclude that S is incomplete.

 

[doronshadmi]

[qimg]http://media.screened.com/uploads/0/34/65808-la_voce_della_luna_large.jpg[/qimg]

That's the last Fellini's flick, isn't it?

No it's not.

I think it is. He didn't shoot anything after 1990.

According to professor Shadmi, he did.

Wrong, by generalization, Fellini's last film is not the last film of the set of films, and actually by using the diagonal method, it is shown that the set of films is ever expending, where Fellini's films are some partial case of the set of ever expending films.

 

[doronshadmi]

Since we know X < 2^X for all X, we can conclude that all sets are doron-incomplete, and the entire "theorem" is trivial.

Another mistake. There is no such a thing like "all sets" (of any kind), and this is exactly the reasoning of why the vary concept of Set is incomplete (the concept of Set has an internal ever expending property).

 

[doronshadmi]

Looking beyond the Matrix II

Since we know X < 2^X for all X, ...

Let X be a placeholder for any finite or infinite cardinality.

Let us play a further game with the <0,1> forms.

By the previous game we have shown that S AND 2^S have objects with the same structural property (both have X bits).

The previous game:

By translating any given S to X^2 <0,1> bit's matrix, and by using the diagonal method along any given X^2 <0,1> bit's matrix, we are able to provide at least one <0,1> object with X bits (the inverse object of a given diagonal), which is not in the range of the objects that construct any given X^2 <0,1> bit's matrix.


The further game:

Now, instead of defining the mapping between natural numbers and the objects that construct any given X^2 <0,1> bit's matrix, we define the mapping between any given natural number and any given inverse <0,1> object of the diagonal of any given X^2 <0,1> bit's matrix.

By doing that, we discover that there is a bijection between the natural numbers and P(S) which its cardinality is 2^X.

But this is a one case of ever increasing cases, because, by the same construction method, there is a bijection between natural numbers and P(P(S)) which its cardinality is 2^(2^X) (where the number of <0,1> bits of each object of P(P(S)), is 2^X).

Since ...P(P(S))... is ever increasing AND there is a bijection of natural numbers with the ever increasing ...P(P(S))..., then:

1) there is no such a thing like a complete collection of different <0,1> objects.

2) The collection of natural numbers is incomplete (its cardinality is ever increasing).

 

[epix]

Wrong, by generalization, Fellini's last film is not the last film of the set of films, and actually by using the diagonal method, it is shown that the set of films is ever expending, where Fellini's films are some partial case of the set of ever expending films.

Read again.

Originally Posted by epix
http://media.screened.com/uploads/0/...luna_large.jpg

That's the last Fellini's flick, isn't it?

Only an idiot would mistake the last Fellini's film for the last movie shot by Homo sapiens ever.

 

[doronshadmi]

Read again.

Only an idiot would mistake the last Fellini's film for the last movie shot by Homo sapiens ever.

And you are not an idiot, isn't it epix?

Please read again http://www.internationalskeptics.com/forums/showpost.php?p=6907413&postcount=14349 .

 

[The Man]

Wrong, by generalization, Fellini's last film is not the last film of the set of films, and actually by using the diagonal method, it is shown that the set of films is ever expending, where Fellini's films are some partial case of the set of ever expending films.

"the set of films"? The set of what films? Fellini's last film is certainly the last of the set of Fellini's films. Even if an unknown last Fellini film is discovered tomorrow, the last is still the last. Perhaps you mean 'the set of all films'? Which although expanding (as long as films are produced) is still closed under the operation of succession. Unless you can find a film that is not a film. I'm sure every poster and most lurkers on this thread will take note of your deliberate and futile avoidance of the explicate assertion of 'the set of all films' while you simply attempt to rely on its inference.

 

[doronshadmi]

Perhaps you mean 'the set of all films'? Which although expanding (as long as films are produced) is still closed under the operation of succession.

In that case it is incomplete (its cardinality is ever increasing).

The set of Fellini's films is a partial case of the increasing set of films.

If one is closed under a given set's cardinality (finite or not) he\she can't show its incompleteness.

I'm sure every poster and most lurkers on this thread will take note of your deliberate and futile avoidance of the explicate assertion of 'the set of all films' while you simply attempt to rely on its inference.

In order to understand better the notion of increasing set, please read http://www.internationalskeptics.com/forums/showpost.php?p=6907490&postcount=14351 .

 

[jsfisher]

Without this <0,1> bit-map "nonsense" you can't show that there exists Y in U and Y not in S

Sure I can...and quite easily. Cantor did it already.

...in order to conclude that S is incomplete.

No, S continues to be complete either way. You must have meant that worthless term you invented best called doron-incomplete -- a triviality no one other than you could take seriously.

 

[jsfisher]

Another mistake. There is no such a thing like "all sets" (of any kind), and this is exactly the reasoning of why the vary concept of Set is incomplete (the concept of Set has an internal ever expending property).

Only in doronetics where the sole proprietor fails to demonstrate basic language skills. Real math has no trouble with the concept.

 

[The Man]

In that case it is incomplete (its cardinality is ever increasing).

Still trying to make cardinality as well as incomplete into whatever "cross-context" nonsense you want.

The set of Fellini's films is a partial case of the increasing set of films.

Set of what films? All films? Perhaps you are inferring some other set of films? Since you still simply and deliberately refuse to explicitly state to what "set of films" you are referring.

If one is closed under a given set's cardinality (finite or not) he\she can't show its incompleteness.

Closed under the operation of succession Doron, the successor of any member of the set is also a member of the set. We have been over this before. No one said "closed under a given set's cardinality" other than just you. Still according to you a set is complete only if it has a member that is not a member of that set. You've cross-contexted yourself into your usual self-contradictory corner.

In order to understand better the notion of increasing set, please read http://www.internationalskeptics.com/forums/showpost.php?p=6907490&postcount=14351 .

Read it already and evidently it hasn't made you "understand better the notion of increasing set" nor one closed under the operation of succession .


Tell us what is the current "cardinality" of your set of natural numbers and how long will it be before you include another natural number to increase your current "cardinality"? Please tell us specifically what natural number was not a natural number before you increased your "cardinality"? You are still confusing (at this point it can only be deliberate) a list with a set.

 

[jsfisher]

Let X be a placeholder for any finite or infinite cardinality.

You really like to be perverse in your wording, don't you? Meanwhile you leave out important detail, like, for example, cardinality of what? Did you perhaps mean to say something more intelligible like: For a given set S, let X = |S|. Is that what you meant?

...Now, instead of defining the mapping between natural numbers and the objects that construct any given X^2 <0,1> bit's matrix

This being something you failed to do in the previous "game." You simply assumed it was possible even though it is not.

The rest of your post suffers the same defect. You assume things that are not true.

 

[epix]

If one is closed under a given set's cardinality (finite or not) he\she can't show its incompleteness.

Of course he/she/it can:

Definition A:
"Let X be a placeholder for any finite or infinite cardinality."

Theorem A:
"Given any S collection of <0,1> X different objects, such that each object has X bits, S is incomplete if there is <0,1> object with X bits, which is not in the range of the given S collection."

Your condition "closed under" never appeared in the arguments of your initial far-reaching discovery of any set being actually incomplete.

1 <--> 01
2 <--> 10

(Inverse left-right diagonal = 11) => (11bin = 3dec) => (S={1, 2} is incomplete)