Deeper than primes - Continuation

[laca]

jsfisher I do not ignore them.

You do realize that your posts are there for everyone to see that in fact you do, right?

On the contrary, I expand the meaning of things beyond their traditional meaning.

Add expand to the list of words you don't understand.

Since your communication style is done only in terms of the traditional meaning of things, there is no communication between us, and this is my claim about you all along this thread. You simply refuse to do any step beyond the traditional or (by using your expression) beyond the "established meanings".

There's no issue with new concepts here, doron. Just leave the existing ones alone. Go ahead and construct your own framework. Just keep in mind that unless it's at least consistent, coherent and useful (huge overlap, I know), nobody is going to take you seriously.

 

[laca]

I agree with you, it is indeed insane to reply to you, in the first place.

So now we have doron_insanity_index+++ and you are go back to my ignore list.


No bye laca.

Don't be such a drama queen. Either you have an actual argument, which I am more than willing to participate in, or you just can't take the bitter pill.

 

[realpaladin]

First off, the best wishes for 2014!

Doron, at the 30c3 I spoke with some renowned mathematicians about the problem of communicating mathematical ideas.

They pointed me to AGDA, which you can use to prototype new mathematical relations and construct proofs with.

I think it will be helpful if you can devote some time to this.

http://wiki.portal.chalmers.se/agda/pmwiki.php

 

[doronshadmi]

First off, the best wishes for 2014!

Doron, at the 30c3 I spoke with some renowned mathematicians about the problem of communicating mathematical ideas.

They pointed me to AGDA, which you can use to prototype new mathematical relations and construct proofs with.

I think it will be helpful if you can devote some time to this.

http://wiki.portal.chalmers.se/agda/pmwiki.php

First, I wish you realpaladin happy new year.

Thank you for your care about problems of communicating mathematical ideas, I am going to learn about AGDA in order to find better ways for communicating mathematical ideas.

Thank you.

 

[laca]

Good luck with that, doron.

 

[realpaladin]

First, I wish you realpaladin happy new year.

Thank you for your care about problems of communicating mathematical ideas, I am going to learn about AGDA in order to find better ways for communicating mathematical ideas.

Thank you.

You are welcome.

As about 90 percent of this thread is about linguistics rather than philosophy or mathematics, this should provide an elegant shortcut...

 

[doronshadmi]

You are welcome.

As about 90 percent of this thread is about linguistics rather than philosophy or mathematics, this should provide an elegant shortcut...

EDIT:

I disagree with you. This thread, among other things, deals also with the possible connections among Linguistics, Philosophy and Mathematics.

AGDA is based on the constructive philosophical point of view of Mathematics, which does not agree with actual infinity in terms of Formalism or Platonism, as defined by Cantor, Dedekind, Hilbert etc., which define actual infinity in terms of collections (even if these collections can't be constructed).

Organic Mathematics agrees with actual infinity, but it shows that it is beyond the power of collections (no matter if they are constructable or not).

Moreover, by expanding the definition of function, such that there is a function with an input but no output (as explained in details in http://www.internationalskeptics.com/forums/showpost.php?p=9732357&postcount=2979) one enables to show that |N| > |N| or |N| = |N| ^* , or in other words, the whole notion of transfinite cardinality is not mathematically well-defined.

Any attempt to deduce |N| > |N| by using the traditional definition of function, is doomed to fail, simply because "x in X has exactly one y in Y", is (by analogy) some kind of net that its holes are too big in order to catch fine things like |N| > |N|.

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

^* |N| > |N| or |N| = |N| examples are shows as follows:

An example of |N| = |N|:

(1,(1))
(2,(2))
(3,(3))
(4,(4))
(5,(5))
...


An example of |N| > |N|:

(1,(1))
(2,())
(3,(2))
(4,(3))
(5,(4))
...

 

[laca]

Whoa, now that's a word salad, sprinkled with plenty of gibberish. Yum.

 

[realpaladin]

EDIT:

I disagree with you. This thread, among other things, deals also with the possible connections among Linguistics, Philosophy and Mathematics.

You do realize that by saying that you have proven my point, don't you?

AGDA is based on the constructive philosophical point of view of Mathematics, which does not agree with actual infinity in terms of Formalism or Platonism, as defined by Cantor, Dedekind, Hilbert etc., which define actual infinity in terms of collections (even if these collections can't be constructed).

Are you saying that AGDA can not be used to deal with infinity? Have you checked the support forums?

Organic Mathematics agrees with actual infinity, but it shows that it is beyond the power of collections (no matter if they are constructable or not).

I probably missed where this is shown. None of the previous posts have ever been enlightening enough so that leaves me with 2 options:

- The proof never was shown.
- You are not as good a teacher as you may think you are

And saying 'you don't get...' simply proves the second option.

Moreover, by expanding the definition of function, such that there is a function with an input but no output (as explained in details in http://www.internationalskeptics.com/forums/showpost.php?p=9732357&postcount=2979) one enables to show that |N| > |N| or |N| = |N| ^* , or in other words, the whole notion of transfinite cardinality is not mathematically well-defined.

Any attempt to deduce |N| > |N| by using the traditional definition of function, is doomed to fail, simply because "x in X has exactly one y in Y", is (by analogy) some kind of net that its holes are too big in order to catch fine things like |N| > |N|.

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

^* |N| > |N| or |N| = |N| examples are shows as follows:

An example of |N| = |N|:

(1,(1))
(2,(2))
(3,(3))
(4,(4))
(5,(5))
...


An example of |N| > |N|:

(1,(1))
(2,())
(3,(2))
(4,(3))
(5,(4))
...

So all you manage is to tack on a 'It depends whether you want to include the empty set in the cardinality or not', or 'It depends on how you look at it'.

Your |N| > |N| actually only manages to convey that your predicates are position dependent, meaning that the variable before an infix operator has it's own domain which is separate from the variable after the infix.

You could just as well write:

|N1| > |N2| where |N2| is |N1| and the empty set.

What's new?

 

[doronshadmi]

You do realize that by saying that you have proven my point, don't you?

Only if "90 percent of this thread is about linguistics rather than philosophy or mathematics" is equivalent to "This thread, among other things, deals also with the possible connections among Linguistics, Philosophy and Mathematics."

Since they are not equivalent then my answer is: no, I did not prove your point.

Are you saying that AGDA can not be used to deal with infinity? Have you checked the support forums?

The term is actual-infinity. Please read again what I wrote about it.

I probably missed where this is shown. None of the previous posts have ever been enlightening enough so that leaves me with 2 options:

- The proof never was shown.
- You are not as good a teacher as you may think you are

And saying 'you don't get...' simply proves the second option.

All you have is to understand the axiomatic state that sets with unbounded amount of objects are no more than some form of potential infinity, where actual infinity is related only to mathematical objects that are non-local by definition (x is non-local if it at AND beyond the domain of y, for example: a non-composed line or line-segment that is at AND beyond the position of a given point along it. On the contrary, the given point along the non-composed line or line segment is only at a given position).

So all you manage is to tack on a 'It depends whether you want to include the empty set in the cardinality or not', or 'It depends on how you look at it'.

Your |N| > |N| actually only manages to convey that your predicates are position dependent, meaning that the variable before an infix operator has it's own domain which is separate from the variable after the infix.

You could just as well write:

|N1| > |N2| where |N2| is |N1| and the empty set.

What's new?

Only the natural numbers are involved here (the internal () of the expression (2,()) is not equivalent to the empty set, exactly because we are using functions only between the names of the rooms and the names of the visitors in that rooms, and not between the names of the rooms and the rooms).

EDIT:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

is not the same as

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

Once again you are using the "some x in X has exactly one y in Y" traditional definition of function, instead of "x in X has at most one y in Y" non-traditional definition of function, where one of its options is input without any output (as seen in the case of 2 → ).

You, by mistake, think that there are two different cases which are

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

(called by you N1)

and

1 → (1)
2 → (2)
3 → (3)
4 → (4)
5 → (5)
...

(called by you N2)

 

[laca]

Nobody cares about how much idiocy you can conjure up, doron. We got plenty of that already. Got something else?

 

[doronshadmi]

I wish to correct some mistake that I made in http://www.internationalskeptics.com/forums/showpost.php?p=9717505&postcount=2905 , so I'll rewrite it:

Let's play with the pairs' game, by using an expression of the form (x,y) as follows:

The outer "(" and ")" define Hilbert's hotel environment.

x defines the name of a given room in that environment.

y defines a room in that environment such that it can be without any visitor (notated by ()) OR with exactly one visitor (notated by , where n is a placeholder for some visitor's name).

The following case of the pairs' game

(1,(1))
(2,())
(3,(2))
(4,(3))
(5,(4))
...

rigorously shows that even if there is 1-to-1 and onto from the names of the rooms and the names of the visitors into the set of all natural numbers, there is also at least one room beyond the range of the visitors.

The internal () of the expression (2,()) is not equivalent to the empty set, exactly because within the pairs' game framework we are using functions only between the names of the rooms and the names of the visitors in that rooms, and not between the names of the rooms and the rooms.

For example:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

(which are functions between the names of the rooms and the names of the visitors)

is not the same as

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

(which are functions between the names of the rooms and the rooms)

By using the "some x in X has exactly one y in Y" traditional definition of function, instead of "x in X has at most one y in Y" non-traditional definition of function, (where one of its options is input without any output (as seen in the case of 2 → )), one can't deduce that |N|>|N|.

 

[laca]

No, I said something other than gibberish, doron. What you wrote is still unadulterated gibberish.

 

[doronshadmi]

Here is a clearer version of the pairs' game framework.

Let's play with the pairs' game, by using an expression of the form (x,y) as follows:

The outer "(" and ")" define Hilbert's hotel environment.

x defines the name of a given room in that environment.

y defines a room in that environment such that it can be without any visitor (notated by ()) OR with exactly one visitor (notated by , where n is a placeholder for some visitor's name).

In the following pairs' game framework, where there is a function from rooms' names and visitors' names (such that both names are in 1-to-1 and onto from the names to the set of all natural numbers)

(1,(1))
(2,(2))
(3,(3))
(4,(4))
(5,(5))
...

is expressed by

1 → 1
2 → 2
3 → 3
4 → 4
5 → 5
...

which shows that |N| = |N|

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

In the following pairs' game framework, where there is a function from rooms' names and visitors' names (such that both names are in 1-to-1 and onto from the names to the set of all natural numbers)

(1,(1))
(2,( ))
(3,(2))
(4,(3))
(5,(4))
...

is expressed by

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

which shows that |N| > |N|

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

|N| > |N| is deducible only if the definition of function is:

x in X has at most one y in Y

Such definition enables to define function even if it has an input but not any output (as seen, for example, in 2 → ).

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

|N| > |N| is not deducible if the definition of function is:

x in X has exactly one y in Y

Such definition does not enable to define function unless it has input and output (for example, 2 → is not deducible by such definition).

 

[Little 10 Toes]

But you are forgetting the background of the Hotel. It already has people in the rooms.

Edit: either you have a guest without a room, or an existing room without and existing guest. Both violate the starting conditions.

 

[jsfisher]

In the following pairs' game framework....

I see you persist in using your own private meaning for 'function' -- quite unnecessarily, too, but what of that -- and you persist in misusing '1-to-1' and 'onto'.

Let's put that aside, though. Instead, I'd like to know what you mean by 'cardinality', since that appears to be a private doronism as well. In particular, how do you make relative comparisons of the cardinality of two sets?

 

[realpaladin]

Only if "90 percent of this thread is about linguistics rather than philosophy or mathematics" is equivalent to "This thread, among other things, deals also with the possible connections among Linguistics, Philosophy and Mathematics."

Since they are not equivalent then my answer is: no, I did not prove your point.

That's what you say! The rest of us can't make head nor tails out of what you say.

So, maybe more than 90% linguistics, but communications difficulties at the very least.

The term is actual-infinity. Please read again what I wrote about it.

^^^^ Q.E.D. ^^^^

All you have is to understand the axiomatic state that sets with unbounded amount of objects are no more than some form of potential infinity, where actual infinity is related only to mathematical objects that are non-local by definition (x is non-local if it at AND beyond the domain of y, for example: a non-composed line or line-segment that is at AND beyond the position of a given point along it. On the contrary, the given point along the non-composed line or line segment is only at a given position).


Only the natural numbers are involved here (the internal () of the expression (2,()) is not equivalent to the empty set, exactly because we are using functions only between the names of the rooms and the names of the visitors in that rooms, and not between the names of the rooms and the rooms).

EDIT:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

is not the same as

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

Once again you are using the "some x in X has exactly one y in Y" traditional definition of function, instead of "x in X has at most one y in Y" non-traditional definition of function, where one of its options is input without any output (as seen in the case of 2 → ).

You, by mistake, think that there are two different cases which are

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

(called by you N1)

and

1 → (1)
2 → (2)
3 → (3)
4 → (4)
5 → (5)
...

(called by you N2)

It does not matter what *you* call it or how *you* *describe* it (linguistics again), but what *you* *do*.

*You* *do* the following:

(Label A) == (Label A) AND (Label A) > (Label A)

No matter how you fill the (Label A) or whatever you call it, all you have succeeded in doing is giving the infix operator a different meaning. That is all.

If you seriously want your concept to work then you must start out with a dictionary of what predicates and operators mean.

Let me give you a start:

Operators:

• == Infix equality. This means that the predicates before and after are considered equal if and only if they both have the exact same content.
• > Infix greater than. The predicate before the infix operator has a greater value (dependent on type) than the predicate after the operator.

Predicates:

• |<predicate>| The complete collection of all elements of <predicate>

If you start by that, then and only then, perhaps you could start convincing anyone.

Remember, the above is just an example on how to construct any logic (even faulty logic) so I'd say you can start getting this thing off the linguistics track.

 

[doronshadmi]

• |<predicate>| The complete collection of all elements of <predicate>

The names of the rooms and the names of the visitors are actually one and only one thing, which is the all members of the the set of natural numbers, notated as N (<predicate> in your language) so there is no linguistic problem here.

The function is from the names of the rooms to the names of the visitors, and by using the definition of function "x in X has at most one y in Y", we get a room name that is beyond the range of the visitors' names (where both of them are actually the all members of the the set of natural numbers, notated as N (<predicate> in your language), so we get |N| > |N|.

So we get two different cases with OR condition between them, which are:

1 → 1
2 → 2
3 → 3
4 → 4
5 → 5
...

|N| = |N|

OR

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

|N| > |N|

So, by using your expression, we have (Label A) == (Label A) OR (Label A) > (Label A), which is actually |N| = |N| OR |N| > |N|.

 

[realpaladin]

The names of the rooms and the names of the visitors are actually one and only one thing, which is the all members of the the set of natural numbers, notated as N (<predicate> in your language) so there is no linguistic problem here.

The function is from the names of the rooms to the names of the visitors, and by using the definition of function "x in X has at most one y in Y", we get a room name that is beyond the range of the visitors' names (where both of them are actually the all members of the the set of natural numbers, notated as N (<predicate> in your language), so we get |N| > |N|.

So we get two different cases with OR condition between them, which are:

1 -> 1
2 -> 2
3 -> 3
4 -> 4
5 -> 5
...

|N| = |N|

OR

1 -> 1
2 ->
3 -> 2
4 -> 3
5 -> 4
...

|N| > |N|

So, by using your expression, we have (Label A) == (Label A) OR (Label A) > (Label A), which is actually |N| = |N| OR |N| > |N|.

Are you really the king of incompetence?

I didn't ask for a clarification on the case, I simply explained that nothing you state or show has any meaning until you first define what *your * symbols mean.

If you fail doing that then the conclusion can only be that even philosophy is too difficult a hobby for you.

So, please try to read the question and define your symbols without examples.

We are polite enough to wade through your words, show that you have at least a modicum of civilization by trying to understand ours.

 

[doronshadmi]

Are you really the king of incompetence?

I didn't ask for a clarification on the case, I simply explained that nothing you state or show has any meaning until you first define what *your * symbols

<predicate> means N.

The definition of function is "x in X has at most one y in Y", which enables also functions with input that do not have any output.

By using such a function within the framework of Hilbert's hotel, one of the possible results is

1 → 1
2 →
3 → 2
4 → 3
5 → 4
...

and since only all N members are involved (the empty set is not involved, exactly because the function is form the names of the rooms to the names of the visitors, and not from the names of the rooms to the rooms), we get |N| > |N| case.

You, wrongly use the function from the names of the rooms to the rooms, as follows:

1 → (1)
2 → ()
3 → (2)
4 → (3)
5 → (4)
...

and by doing that we are missing the fact that there is function 2 → of room's name input 2, which is beyond the range of all the rooms' names outputs.

-----------

EDIT:

Once again, the names of the rooms and the names of the visitors in that hotel are actually all the members of the set of all natural numbers (notated by N), and no members other than natural numbers are used in Hilbert's Hotel framework after moving visitors (such that no visitor left the hotel) AND before visitors' new reception.