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Deeper than primes

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jsfisher said:
Prove it.

Stop waving your hands. Stop assuming things that are not true. Stop spewing gibberish. Just formally prove it.
Click to expand...

Furthermore, given

11111111111111…
00000000000000…
00011100011100…

011xxxxxxxxxxx… (where x is 0 or 1) is not in the range of that collection.

It is formally proved exactly by using the diagonal method along any collection of <0,1> unique forms.

You still do not get the universal formality of <0,1> forms, because your reasoning is context-dependent.

EDIT:

The diagonal method along any collection of <0,1> unique forms, is a formal self evident truth.
 
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jsfisher said:
Rather than describing what might go into the proof, how about just providing the proof?



Here's a collection of "<0,1> unique forms": {0, 1}. Start with that as an example.
Click to expand...

By using the universality of <0,1> unique forms, {0,1} is translatable to 01xxxxxxxxxx… (where x is 0 or 1).

By using the diagonal method on 01xxxxxxxxxx…, 11xxxxxxxxxx… form has the same properties of 01xxxxxxxxxx…, but it is not in the range of the collection that has 01xxxxxxxxxx… form.

In other words, this collection is incomplete.

EDIT:

Another translation is

0xxxxx...
1xxxxx...

and also in this case 1xxxxxxxxx...(where x is 0 or 1) is not in the range of {0xxxxx..., 1xxxxx...}.
 
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doronshadmi said:
By using the universality of <0,1> unique forms, {0,1} is translatable to 01xxxxxxxxxx… (where x is 0 or 1).
Click to expand...


No, it was already translated. Leave the goal posts exactly where there were. This is a collection of "<0,1> unique forms": {0, 1}.

If that one is causing you endless confusion, you may substitute this one: {00, 11, 10, 01}, or even this one: {111, 101, 001, 010, 110, 011, 000, 100}
 
jsfisher said:
No, it was already translated. Leave the goal posts exactly where there were. This is a collection of "<0,1> unique forms": {0, 1}.

If that one is causing you endless confusion, you may substitute this one: {00, 11, 10, 01}, or even this one: {111, 101, 001, 010, 110, 011, 000, 100}
Click to expand...

Look at this:

x is 0 or 1,

{
00xxxxxxxx...,
11xxxxxxxx...,
10xxxxxxxx...,
01xxxxxxxx...
}

The <0,1> form that is not in the range of the collection of <0,1> unique forms is 10xxxxxxx... , where each bolded x has the inverse bit of the the 10xxxxxxxx..., and 01xxxxxxxx... forms of the given finite collection.
 
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If you still do not get the universality of <0,1> unique forms, then take that:

x is 0 or 1,

Finite case:

{
xxxxxxxx...,
xxxxxxxx...,
xxxxxxxx...,
xxxxxxxx...
}

Infinite case:

{
xxxxxxxx...,
xxxxxxxx...,
xxxxxxxx...,
xxxxxxxx...,
...
}

The <0,1> form that is not in the range of the collection of <0,1> unique forms is xxxxxxxx... , where each bolded x has the inverse bit of the forms of the given finite or infinite collection, such that xxxxxxxx... or xxxxxxxx... has the same properties of the finite or infinite collection, but it is not in the range of the finite or infinite collection.

In other words, the given collection (finite or not) is incomplete.
 
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doronshadmi said:
Look at this:
Click to expand...

You keep changing the collections. Leave them alone.

Well, what you are really doing is admitting your nonsense is nonsense for any finite case, but you continue to rely on hidden assumptions, hand-waving, contradiction, and gibberish to obscure your nonsense is nonsense for all infinite cases, too.

So, you are back to the simple task of providing proof of the infinite case. All you need do is prove that a collection of all possible strings of countably infinite 0's and 1's must be missing at least one string of countably infinite 0's and 1's.

Proceed.
 
jsfisher said:
You keep changing the collections. Leave them alone.

Well, what you are really doing is admitting your nonsense is nonsense for any finite case, but you continue to rely on hidden assumptions, hand-waving, contradiction, and gibberish to obscure your nonsense is nonsense for all infinite cases, too.

So, you are back to the simple task of providing proof of the infinite case. All you need do is prove that a collection of all possible strings of countably infinite 0's and 1's must be missing at least one string of countably infinite 0's and 1's.

Proceed.
Click to expand...
You still do not get the universality of <0,1> forms, please look at http://www.internationalskeptics.com/forums/showpost.php?p=6895000&postcount=14286 .
 
jsfisher said:
You keep changing the collections. Leave them alone.
Click to expand...
No jsfisher, you still do not get the translation of any given collection to infinitely long strings of <0,1> unique forms.
 
jsfisher said:
All you need do is prove that a collection of all possible strings of countably infinite 0's and 1's must be missing at least one string of countably infinite 0's and 1's.
Click to expand...
There is no such a thing like countably infinite or non-countably infinite collections.

The ability to translate any finite or infinite collection to <0,1> unique forms, and then to show (by using the diagonal method) that there is an object, which has the properties of the collection of <0,1> unique forms, but it is not in the range of the given collection, clearly shows the self evident truth of the incompleteness of any collection that is translatable to <0,1> unique forms.
 
jsfisher said:
How's that proof coming?
Click to expand...

By the ability to translate any given collection to <0,1> infinitely long unique forms, we are able to show the incompleteness of any given collection of unique objects.


Some example of <0,1> translation:


For example, here is the natural numbers' <0,1> translation (where by Peano axioms, 0 is a natural number ( http://en.wikipedia.org/wiki/Peano_axioms )):

00000000000... ↔ 0
10000000000... ↔ 1
01000000000... ↔ 2
00100000000... ↔ 3
00010000000... ↔ 4
00001000000... ↔ 5
…
etc. ad infinitum ...

After the translation all we care about is the <0,1> unique forms, and in this case the <0,1> form that is not in the range of the collection of <0,1> unique forms, is 111111... in this case.

You may say that 111111… is not a natural number, but again, this is the whole idea of the universality of <0,1> unique forms, which according to it all we care about is the uniqueness of <0,1> forms.

Please look again at http://www.internationalskeptics.com/forums/showpost.php?p=6894492&postcount=14277 for better understanding of the universality of <0,1> unique forms.
 
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doronshadmi said:
There is no such a thing like countably infinite or non-countably infinite collections.
Click to expand...


Really? Mr. Cantor and others would disagree with that statement of yours. Then again, they had Mathematics on their side. What do you have, Doron?
 
doronshadmi said:
By the ability to translate any given collection to <0,1> infinitely long unique forms, we are able to show the incompleteness of any given collection of unique objects.
Click to expand...


So show it already. So far, you have asserted a lot, but proven nothing.
 
doronshadmi said:
Some example of <0,1> translation:
Click to expand...

render-taj-jigsaw.jpg


The proof that the jigsaw puzzle is incomplete is apparent: there is an empty space between J and M: TAJ MAHAL.

...And that concludes our discourse on the genius of professor Shadmi.
 
jsfisher said:
Attempting to show the set of natural numbers doesn't include something that isn't a natural numbers is proof of what, exactly?
Click to expand...

<0,1> forms which exist between 000... and 111... (where 000... and 111... are included) are unique codes that are common for X , P(X) , P(P(X)) , P(P(P(X))) , ...P(P(P(X)))... no matter if X is finite or infinite collection.

In other words, {000..., ... , 111...} is universal and it is shown that it is incomplete because the inverse of the diagonal of such a set is not in the range of that set.

If you try to understand {000..., ... , 111...} in terms of particular set (which is a context-dependent reasoning), you miss its universality (which is a cross-contexts reasoning).

jsfisher said:
Mr. Cantor and others would disagree with that statement of yours. Then again, they had Mathematics on their side.
Click to expand...

Mr. Cantor and others used and still are using only context-dependent reasoning.


jsfisher said:
What do you have, Doron?
Click to expand...
I have also cross-contexts reasoning in addition to context-dependent reasoning.

Universality is always at the level of cross-contexts reasoning, but since you are using only context-dependent reasoning you can't get it.
 
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epix said:
[qimg]http://docs.gimp.org/pl/images/filters/examples/render-taj-jigsaw.jpg[/qimg]

The proof that the jigsaw puzzle is incomplete is apparent: there is an empty space between J and M: TAJ MAHAL.

...And that concludes our discourse on the genius of professor Shadmi.
Click to expand...
It is complete only if it is not a collection.

Given any collection, if it is translatable to <0,1> unique codes, then it is incomplete.
 
doronshadmi said:
In other words, {000..., ... , 111...} is universal and it is shown that it is incomplete because the inverse of the diagonal of such a set is not in the range of that set.
Click to expand...

No, you haven't done that yet. You've asserted it, but you haven't proven it. All you have demonstrated in all of this meaningless smoke and mirrors is that sets do not include elements they do not include. We remain unimpressed.

The fact is, you cannot prove you bald assertion because it is false. I welcome you to try, though.

Heck, a big first step would be to actually formally describe how you'd construct the inverse of the diagonal. That alone might help expose some of the assumptions you are making. But you can't do that can you, Doron? You are not a details person; you have direct perception that lets you simply assume a result, no matter how inane.
 
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