Deeper than primes - Continuation

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doronshadmi said:
EDIT:

Suppose there are three options of the expressions =,>,< between A and B.

Now suppose you tell me, "|A|=|B| AND |A|>|B| AND |A|<|B|". Does this give me the information I need?
Click to expand...

Yes, it does. It tells you that the relationships were ill-defined and conflicting. Instead, I would have expected something like: "not (|A| = |B|) AND |A| > |B| AND not(|A| < |B|)" for example.

(Please use the corrected version http://www.internationalskeptics.com/forums/showpost.php?p=9756363&postcount=3116).
Click to expand...

I'd prefer to await a corrected post following this, one that corrects the conjunction issue and also addresses the issue I raised a few posts up.
 
BenjaminTR said:
No, it does not. However, in the posts in question you are not disjoining three atomic statements such as '|A|=|B|', you are disjoining three biconditionals. This is not what you want to do.
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This is exactly what I want to do, to define the cardinality for each = OR > OR <, that are used between A AND B.

BenjaminTR said:
You are basically saying, "Either [definition 1] is the definition of equality, or [definition 2] is the definition of greater than, or [definition 3] is the definition of less than." If we take your use of 'or' seriously, this does not establish the definition of any of the three.
Click to expand...
I am basically saying:

Definition 1 provides the equality |A|=|B| or Definition 2 provides the inequality |A|>|B| or Definition 3 provides the inequality |A|<|B|
 
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jsfisher said:
Yes, it does. It tells you that the relationships were ill-defined and conflicting. Instead, I would have expected something like: "not (|A| = |B|) AND |A| > |B| AND not(|A| < |B|)" for example.
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jsfisher, is the expression ≤ means = OR < (or < OR =)?

Please answer by yes or no?
 
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doronshadmi said:
This is exactly what I want to do, to define the cardinality for each = OR > OR <, that are used between A AND B.
Click to expand...

No, you are not.

Statement #1 establishes whether the relationship is equal or not equal.
...AND...
Statement #2 establishes whether the relationship is greater than or not greater than.
...AND...
Statement #3 is extraneous.
 
doronshadmi said:
I agree with you, the definitions in http://www.internationalskeptics.com/forums/showpost.php?p=9754527&postcount=3099 hold also between different sets.


I disagree with you.


For example, this case:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
…

is

|N| > |N|


and this case:

1 ← 1
← 2
2 ← 3
3 ← 4
4 ← 5
…

is

|N| < |N|

exactly because both names (of rooms and visitors in Hilbert's Hotel http://www.internationalskeptics.com/forums/showpost.php?p=9742045&postcount=3022) are actually one and only one set, which is the set of all natural number (which is notated as N).
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No Doron, they are not one and only one set.

'No result', 'nothing', 'zero' etc are all part of that set. Any given element, even a 'No result' is part of the resultant set.

If you say you are playing with Natural numbers, then 'No result' is not part of that set.

So by stating '← 2' you move away from natural numbers and the 'No result' is part of the set.

Therefore, |N| < |N| can only mean that you have redefined the < operator so that your expression means that the sets do not have to be in the same domain. Period.
 
doronshadmi said:
Wrong. The definition of doron-cardinality is statement #1 (A=B) OR statement #2 (A>B) OR statement #3 (A<B) as follows:

Given two sets, A and B, the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).

For example:

1 ↔ 1
2 ↔ 2
3 ↔ 3
4 ↔ 4
5 ↔ 5
…



OR

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (also there is at least one a in A that does not have any b in B).

For example:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
…



OR

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from B to A such that (for each b in B there is exactly one a in A) AND (also there is at least one b in B that does not have any a in A).

For example:

1 ← 1
← 2
2 ← 3
3 ← 4
4 ← 5
…
Click to expand...

Peyo?
 
realpaladin said:
'No result', 'nothing', 'zero' etc are all part of that set. Any given element, even a 'No result' is part of the resultant set.
Click to expand...
'No result', 'nothing', 'zero' are some words, and no one of them is what they describe, exactly as the word 'silence' is not what it describes.
 
EDIT:

So we have A=B OR A>B OR A<B.

The generalization of the three definitions in http://www.internationalskeptics.com/forums/showpost.php?p=9756363&postcount=3116 is:

X and Y are placeholders for domain or codomain , such that (if X=domain then Y=codomain) OR (if X=codomain then Y=domain).

x in X has at most one y in Y (where the term at most enables also doron-functions that have an input but not any output, as seen in definitions #2 (A>B) or #3 (A<B).
 
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doronshadmi said:
Statement #1 establishes |X|=|Y|
Click to expand...

No, it establishes which of |X| = |Y| or |X| ^= |Y| is true.

...OR...
Statement #2 establishes |X|>|Y|
Click to expand...

No, it establishes which of |X| > |Y| or |X| ^> |Y| is true (the later being equivalent to |X| <= |Y|).

...OR...
Statement #3 establishes |X|<|Y|
Click to expand...

Statement #3 is extraneous, irrelevant, unnecessary, superfluous, redundant.
 
realpaladin said:
But this means you actually have nothing.
Click to expand...
Not exactly.

I have a name of a room that is not related to any name of some visitor.

Yet both the names of the rooms and the names of the visitors are actually one and only one thing, which is the set of all natural numbers.

By using the definition:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).

it is shown that

1 ↔ 1
2 →
3 ↔ 2
4 ↔ 3
5 ↔ 4
…

or in other words, |N|>|N|.
 
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jsfisher said:
No, it establishes which of |X| = |Y| or |X| ^= |Y| is true.



No, it establishes which of |X| > |Y| or |X| ^> |Y| is true (the later being equivalent to |X| <= |Y|).



Statement #3 is extraneous, irrelevant, unnecessary, superfluous, redundant.
Click to expand...
EDIT:

Thank you for your reduction (which is statement or its negation).

Yet, it does not change the fact of statement #1 OR statement #2, and it does not change the validity of http://www.internationalskeptics.com/forums/showpost.php?p=9757927&postcount=3135 and the generalization of #1 or #2 (as defined in http://www.internationalskeptics.com/forums/showpost.php?p=9756363&postcount=3116) into "x in X has at most one y in Y", as follows:

x in X has at most one y in Y (where the term at most enables also doron-functions that have an input but not any output, as seen in definition #2 (A>B).
 
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doronshadmi said:
Not exactly.

I have a name of a room that is not related to any name of some visitor.

Yet both the names of the rooms and the names of the visitors are actually one and only one thing, which is the set of all natural numbers.

By using the definition:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).

it is shown that

1 ↔ 1
2 →
3 ↔ 2
4 ↔ 3
5 ↔ 4
…

or in other words, |N|>|N|.
Click to expand...

As soon as you have a room with a non-relation, that non relation is, no matter how you *try* to define your set, part of your set.

Like saying your car has three wheels because you define the left-front as a rapuctor.

It does not matter what you say or state, the non-relation *actually* is part of your resultant set.

In your examples this is blatantly visible as 2 →

You *show* that the righthand set *is* different.

No matter of wordwrangling will change that.
 
EDIT:

realpaladin said:
As soon as you have a room with a non-relation, that non relation is, no matter how you *try* to define your set, part of your set.
Click to expand...
It is some room's name without any relation. The room's name is indeed a part of the set, but the non-relation (the lack of output) is not part of this set.


realpaladin said:
It does not matter what you say or state, the non-relation *actually* is part of your resultant set.
Click to expand...

Being a part of some set is possible only if it is its member.


realpaladin said:
In your examples this is blatantly visible as 2 →

You *show* that the righthand set *is* different.

No matter of wordwrangling will change that.
Click to expand...

The righthand of 2 → does not exist, or in other words, there is no righthand (there is a function that does not return any value, and this is exactly the meaning of 2 →).
 
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EDIT:

realpaladin said:
Like saying your car has three wheels because you define the left-front as a rapuctor.
Click to expand...

No, its like saying that a part of my car, named left-front, does not return any value that is a part of the car.

As a result there are more sides than wheels.
 
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doronshadmi said:
EDIT:

It is some room's name without any relation. The room's name is indeed a part of the set, but the non-relation (the lack of output) is not part of this set.
Click to expand...
That is not possible. Even a 'non-relation' is a relation; namely the relation that defines the absence of a relation.



doronshadmi said:
Being a part of some set is possible only if it is its member.
Click to expand...
Correct. So the 'lack of output' *must* be part of the right-hand set OR you redefine the > operator to mean that the left-hand set is of a separate domain from the righthand set.

These are your definitions. Either the relation changes the domain (where the right-hand side acquires the 'lack of output') or the sets were never equal to begin with.


doronshadmi said:
The righthand of 2 → does not exist, or in other words, there is no righthand (there is a function that does not return any value, and this is exactly the meaning of 2 →).
Click to expand...
And the only logical conclusion from that is that the relation modifies the domains of the set.
 
doronshadmi said:
Thank you for your reduction (which is statement or its negation).

Yet, it does not change the fact of statement #1 OR statement #2
Click to expand...

Not a fact. Your 'or' would have us choosing which if and only if statement to apply. Choose unwisely and you'd conclude that, perhaps, |A| ^= |B| and completely miss the stronger conclusion that |A| > |B|.

and it does not change the validity of http://www.internationalskeptics.com/forums/showpost.php?p=9757927&postcount=3135 and the generalization of #1 or #2 (as defined in http://www.internationalskeptics.com/forums/showpost.php?p=9756363&postcount=3116) into "x in X has at most one y in Y", as follows:

x in X has at most one y in Y (where the term at most enables also doron-functions that have an input but not any output, as seen in definition #2 (A>B).
Click to expand...

Nope, doesn't change the validity at all. My prior objection stands unaddressed.
 
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doronshadmi said:
:bigclap
Click to expand...

Using kindergarten tactics again?

Your whole premisse is demolished before your eyes and all you can do is applaud?

Have a little streak of Nero?

But essentially, this proves; nothing Doron Shadmi provides is new or even onthologically complete or even logically consistent.

Therefore: Doron Shadmi's treatises are naught but an excercise in linguistics.
 
doronshadmi said:
Thank you for your reduction (which is statement or its negation).
Click to expand...

I missed a subtlety. Let's explore it:

Doron, a simple question:
Let's say for some pair of sets, A and B, Statement #1 is true. What can you conclude about the relationship between the doron-cardinalities of sets A and B?

(By the way, if your answer is |A| = |B|, then you are overlooking something rather important.)
 
jsfisher said:
I missed a subtlety. Let's explore it:

Doron, a simple question:
Let's say for some pair of sets, A and B, Statement #1 is true. What can you conclude about the relationship between the doron-cardinalities of sets A and B?

(By the way, if your answer is |A| = |B|, then you are overlooking something rather important.)
Click to expand...

Let's do it like that.

There are only 3 options: |A|=|B| OR |A|>|B| OR |A|<|B|

Also there are only 3 definitions for each case above, as follows:


The definition for option |A|=|B| is:

Given two sets, A and B, the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).


The definition for option |A|>|B| is:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).


The definition for option |A|<|B| is:

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one b in B that does not have any a in A).

I can clearly conclude about the relationship between the doron-cardinalities of sets A and B, only if they satisfy one and only one of the 3 definitions above.

So, after all, definition #3 is relevant.
 
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doronshadmi said:
Let's do it like that.

There are only 3 options: |A|=|B| OR |A|>|B| OR |A|<|B|

Also there are only 3 definitions for each case above, as follows:


The definition for option |A|=|B| is:

Given two sets, A and B, the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).


The definition for option |A|>|B| is:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).
Click to expand...

According to that "definition", it is impossible for any two sets not to have the same cardinality. Needless to say, that's just about as epic a fail can get.

The definition for option |A|<|B| is:

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one b in B that does not have any a in A).
Click to expand...

Same as above. Pulled a doron again there.

I can clearly conclude about the relationship between the doron-cardinalities of sets A and B, only if they satisfy one and only one of the 3 definitions above.

So, after all, definition #3 is relevant.
Click to expand...

It is only relevant in showing how clueless you are.
 
doronshadmi said:
[...]
The definition for option |A|>|B| is:
Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).

[...]
Click to expand...
The bolded conditions are directly contradictory. In other words, no set has greater cardinality than any other set.
 
doronshadmi said:
Let's do it like that.

There are only 3 options: |A|=|B| OR |A|>|B| OR |A|<|B|
...<snip>...
Click to expand...

You never answered the question.

For a given pair of sets, A and B, Statement #1, this statement, is true:
the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).​

What, if anything, can you conclude about the relative cardinality of the two sets knowing that Statement #1 is true?
 
BenjaminTR said:
The bolded conditions are directly contradictory. In other words, no set has greater cardinality than any other set.
Click to expand...

It was mentioned before, back where there were only two conditions AND'ed together. His response was to add the third condition.
 
BenjaminTR said:
The bolded conditions are directly contradictory. In other words, no set has greater cardinality than any other set.
Click to expand...
You are absolutely right.

Definitions #2 can't address Hilbert's Hotel case:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
…


Definitions #3 can't address Hilbert's Hotel case:

1 ← 1
← 2
2 ← 3
3 ← 4
4 ← 5
…

So we need finer logic in order to address these two cases, which does not use bijection as a part of definitions #2 or #3.

Let's see how it can be done by using further work on this interesting subject.
 
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No. The comment is about how you can't define things.

doronshadmi said:
The definition for option |A|>|B| is:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).


The definition for option |A|<|B| is:

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one b in B that does not have any a in A).
Click to expand...


Since I'm using a mobile device, you will have to pardon some of the formatting.

Doron-definition.

#2 |A|<|B|

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that [snipped AND statement] (for each b in B there is exactly one a in A) AND (there is at least one b in B that does not have any a in A).

Based on that you need to have a 1:1 ratio of elements in set A and B, and have more elements in B than A at the same time, you can never have a TRUE result.
 
|A|^=|B| is not clear since it is equivalent to |A|>|B| OR |A|<|B|

|A|^>|B| is not clear since it is equivalent to |A|=|B| OR |A|<|B|

|A|^<|B| is not clear since it is equivalent to |A|=|B| OR |A|>|B|


So, there are only 3 clear options: |A|=|B| OR |A|>|B| OR |A|<|B|


Also there are only 3 definitions for each case above, as follows:


The definition for option |A|=|B| is:

Given two sets, A and B, the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).


The definition for option |A|>|B| is:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (at least for each a in A there is at most one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).


The definition for option |A|<|B| is:

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (at least for each b in B there is at most one a in A) AND (there is at least one b in B that does not have any a in A).

I can clearly conclude about the relationship between the doron-cardinalities of sets A and B, only if they satisfy one and only one of the 3 definitions above.
 
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jsfisher said:
You never answered the question.

For a given pair of sets, A and B, Statement #1, this statement, is true:
the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).​

What, if anything, can you conclude about the relative cardinality of the two sets knowing that Statement #1 is true?
Click to expand...
In case of sets with bounded amount of elements, the answer is trivial, because we can count the elements of both sets, and if they satisfy Statement #1 we know that indeed A = B.

In case of sets with unbounded amount of elements, it depends on how they satisfy one and only one of the three definitions in http://www.internationalskeptics.com/forums/showpost.php?p=9760542&postcount=3153.
 
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BenjaminTR said:
The bolded conditions are directly contradictory. In other words, no set has greater cardinality than any other set.
Click to expand...

Well, that is basically what I tried to convey; if he insists on these definitions, then he redefines his operators.

Also, after the big hand clap, I never got a response :)
 
Little 10 Toes said:
No. The comment is about how you can't define things.




Since I'm using a mobile device, you will have to pardon some of the formatting.

Doron-definition.

#2 |A|<|B|

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that [snipped AND statement] (for each b in B there is exactly one a in A) AND (there is at least one b in B that does not have any a in A).

Based on that you need to have a 1:1 ratio of elements in set A and B, and have more elements in B than A at the same time, you can never have a TRUE result.
Click to expand...

See my discourse with him; he tries to weasel out of that by acting as if there is such a thing as 'non-relation' which magically increases the possibilities whilst at the same time keeping the cardinality of the set of possibilities equal...
 
There are simply no words, doron. Instead, I'm just going to highlight another gem of yours:

at least for each a in A

Here's the laughing dog that goes with it:
:dl:

Congrats.
 
Little 10 Toes said:
Based on that you need to have a 1:1 ratio of elements in set A and B, and have more elements in B than A at the same time, you can never have a TRUE result.
Click to expand...
It is not possible among sets with bounded amount of elements.

But we are talking here about sets with unbounded amount of elements.

The definition for |A|>|B| was fixed as follows:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is at most one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).

This definition provides the logical basis for the following case:

1 → 1
2 →
3 → 2
4 → 3
5 → 4
…

where (at least) one of the functions has an input but not any output.

Since both sides are actually one and only one thing, which is the set of all natural numbers (notated as N), and since there are also doron-functions that have an input but not any output, we are able to conclude that |N|>|N|.
 
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Another correction (I have change "at least for" to simply "for", in definitions #2 or #3).

|A|^=|B| is not clear since it is equivalent to |A|>|B| OR |A|<|B|

|A|^>|B| is not clear since it is equivalent to |A|=|B| OR |A|<|B|

|A|^<|B| is not clear since it is equivalent to |A|=|B| OR |A|>|B|


So, there are only 3 clear options: |A|=|B| OR |A|>|B| OR |A|<|B|


Also there are only 3 definitions for each case above, as follows:


The definition for option |A|=|B| is:

Given two sets, A and B, the doron-cardinality of the set A = the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is exactly one a in A).


The definition for option |A|>|B| is:

Given two sets, A and B, the doron-cardinality of the set A > the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is at most one b in B) AND (for each b in B there is exactly one a in A) AND (there is at least one a in A that does not have any b in B).


The definition for option |A|<|B| is:

Given two sets, A and B, the doron-cardinality of the set A < the doron-cardinality of set B if and only if there is doron-function from A to B such that (for each a in A there is exactly one b in B) AND (for each b in B there is at most one a in A) AND (there is at least one b in B that does not have any a in A).

I can clearly conclude about the relationship between the doron-cardinalities of sets A and B, only if they satisfy one and only one of the 3 definitions above.
 
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