Cont: Deeper than primes - Continuation 1/3*9

jsfisher said:
A and B are arbitrary sets (as was strongly implied), and |.| is a standard notation for cardinality (as was strongly implied).
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If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Cardinality (which is for a relative measure, i.e., a comparison) is:

Definition of Cardinality: The cardinality of A [SIZE=-2]<also notated as |A|>[/SIZE] <= the cardinality of B [SIZE=-2]<also notated as |B|>[/SIZE] iff there is an injection from A to B.

(Also you skipped on http://www.internationalskeptics.com/forums/showpost.php?p=13136028&postcount=18).
 
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doronshadmi said:
If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Cardinality (which is for a relative measure, i.e., a comparison) is:

Definition of Cardinality: The cardinality of A [SIZE=-2]<also notated as |A|>[/SIZE] <= the cardinality of B [SIZE=-2]<also notated as |B|>[/SIZE] iff there is an injection from A to B.
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No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
 
jsfisher said:
No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
Click to expand...

A relative measure is defined iff what is compared and how it is compared, are both or neither taken.

In this case what is compared is the cardinalities of sets A and B (which are |A| and |B|), where how they are compared is done by one-to-one function (also called injection, such that |A| <= |B|).

Now look at this:

If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Injection (which is for a relative measure, i.e., a comparison) is:

Definition of Injection : There is a one-to-one function, also called injection, from A to B iff The cardinality of A [SIZE=-2]<also notated as |A|>[/SIZE] <= the cardinality of B [SIZE=-2]<also notated as |B|>[/SIZE]

Definition of Cardinality: The cardinality of A [SIZE=-2]<also notated as |A|>[/SIZE] <= the cardinality of B [SIZE=-2]<also notated as |B|>[/SIZE] iff there is an injection from A to B.

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

(Also you skipped on http://www.internationalskeptics.com/forums/showpost.php?p=13136028&postcount=18).
 
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doronshadmi said:
If A and B are arbitrary sets and |.| is a standard notation for cardinality, then your definition of Injection (which is for a relative measure, i.e., a comparison) is....
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I did not define injection. I defined a relationship between the cardinalities of two sets.
 
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doronshadmi said:
I do not repeating my self in http://www.internationalskeptics.com/forums/showpost.php?p=13135979&postcount=15 since it is a correction of http://www.internationalskeptics.com/forums/showpost.php?p=13133503&postcount=4.
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Not much of a correction, and you didn't address the objections raised.

Nonetheless...
doronshadmi said:
Let _ be a placeholder for any given set.
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Or we could just follow convention and use a letter of the alphabet instead of a cryptic underbar.

Definition 1: Cardinality is the "size" of _ (notated as |_|), which is determined by the members of N, where 0 is a member of N.
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Determined how?
N? What is N? A set, I presume, but what set?
0? What is 0? Set theory has nothing like that. You'll need to define it.

Let < be less than.
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"Less than"? How is that defined? Set theory doesn't have a less-than concept on its own.

...and so on for the rest of the "corrected" post.


You have not moved any closer to a definition. Please keep trying, though.
 
jsfisher said:
Not much of a correction, and you didn't address the objections raised.

Nonetheless...


Or we could just follow convention and use a letter of the alphabet instead of a cryptic underbar.



Determined how?
N? What is N? A set, I presume, but what set?
0? What is 0? Set theory has nothing like that. You'll need to define it.



"Less than"? How is that defined? Set theory doesn't have a less-than concept on its own.

...and so on for the rest of the "corrected" post.


You have not moved any closer to a definition. Please keep trying, though.
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As long as you try to understand http://www.internationalskeptics.com/forums/showpost.php?p=13135979&postcount=15 in terms of ZF(C), you are surly missing it.
 
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jsfisher said:
No, it is not. The definition is for a specific relationship between cardinalities of sets. Your propensity for overstating the trivial and the bloody obvious notwithstanding, |A| <= |B| is what is being defined. Not |A| nor |B|.

That is the only way a relative measure can be defined, i.e., in terms of how measurements would compare.
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jsfisher, no relative measure is defined unless what is compared (and in this case it is the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)) and how it is compared (and in this case it is an injection from set A to set B), where the logical connective between what is compared and how it is compared is iff, otherwise |A| <= |B| can't be defined.

Any attempt to define X in terms of relative measure, can't avoid X as a part of the definition.

And this is exactly what I show in http://www.internationalskeptics.com/forums/showpost.php?p=13140061&postcount=23, which you get as gibberish as long as you are missing the truth table of iff where:

In case of relative measure

Code: 
p="what is measured" 
q="how it is muasured"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T

where relative measure can't be established unless p iff q.
 
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jsfisher said:
That post defines understanding at many levels and in so many ways.
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What is defined in http://www.internationalskeptics.com/forums/showpost.php?p=13135979&postcount=15 is simpler than your definitions that are done in terms of ZF(C) (your definition of Cardinality, which uses cardinality as a part of the definition).

For example:

1) I do not need iff in order to define Cardinality by the members of the set of natural numbers (notated as N), where natural numbers (including number 0) are naturally understood (no extra maneuvers are needed).

2) I provide symbols to the concepts "less than", "equal to", "greater than" (which I tend to replace by "more than") which are naturally understood (no extra maneuvers are needed).

3) I use iff in a vary simple way in case of relative measure, between two arbitrary sets A and B, where their cardinalities (what is compared) and the terms of how they are compared are simply and intuitively addressed, because of the simple use of iff.

4) By being simple and intuitive in definitions 1 and 2, I am able to very simply define finite (definition 3) and non-finite (definition 4) sets, without any need to add anything, which are not already given by definitions 1 to 4.

5) Definition 5 very simply addresses non-strict inequality as a range between strict inequality (< or >) and equality (=).

You are invited to criticize what is written in this post and in http://www.internationalskeptics.com/forums/showpost.php?p=13135979&postcount=15 post.

Thank you.
 
doronshadmi said:
jsfisher, no relative measure is defined unless what is compared (and in this case it is the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)) and how it is compared (and in this case it is an injection from set A to set B), where the logical connective between what is compared and how it is compared is iff, otherwise |A| <= |B| can't be defined.
Click to expand...


You used a lot of words there. Did you have a point?

My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.

...and if you can't get along without supplemental information, no matter how obvious: ...where A and B are each sets, |.| is used to denote "cardinality of", and <= is the relationship that is the subject of the definition.
 
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jsfisher said:
My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.
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Let's see:

jsfisher said:
My definition (which is for a relative measure, i.e., a comparison) is:

|A| <= |B| [SIZE=-2]<the thing being defined>[/SIZE] if and only if [size=-2]<equivalent to saying "is defined as" or "means">[/size] there is an injection from A to B.
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So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?

jsfisher said:
Here is your first post about it:

For that matter, you need to be clear what you mean by "cardinality". No, wait, better still, I will provide you a definition of cardinality for the purposes of this discussion.

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)

Note that this definition requires only the introduction of mappings into the set theory
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Mapping between sets' cardinalities, so you are using cardinality as a part of your definition of Cardinality.

Moreover, you have said that you will provide me a definition of cardinality.

jsfisher said:
...and if you can't get along without supplemental information, no matter how obvious:
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Obvious?
 
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doronshadmi said:
Let's see:



So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?
Click to expand...


That' it? Petty attempts at semantic nitpicks?

Come back when you have something of substance. Meanwhile, Mathematics remains intact.
 
doronshadmi said:
Let's see:



So, your definition is for:

Relative measure.

Relative cardinalities.

Will you make up your mind?



Mapping between sets' cardinalities, so you are using cardinality as a part of your definition of Cardinality.

Moreover, you have said that you will provide me a definition of cardinality.



Obvious?
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Once again, you get things wrong. jsfisher is NOT "mapping between set's cardinality" so he is NOT using cardinality as a part of his definition. Moreover, is has provided you a definition of cardinality, multiple times.

Post #2: No. I have defined what the expression, |A| <= |B|, means. It means there exists an injection from A to B.
Post #4: Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (quoted by you!)
Post #14: Cardinality is a relative measure of "size" of sets where |A| <= |B| iff there exists an injection from A to B. (quoted by you, again)
Post #20: |A| <= |B| <the thing being defined> if and only if <equivalent to saying "is defined as" or "means"> there is an injection from A to B.
Post #21 is you misquoting (on purpose?) the definition given in post #20.
Post #23 is again you misquoting (on purpose?) the definition given in post #20.
Post #32: My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.
 
Little 10 Toes said:
jsfisher is NOT "mapping between set's cardinality" so he is NOT using cardinality as a part of his definition.
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By jsfisher's definition of Cardinality, |A| and |B| (which are actually the cardinalities of sets A and B) are inseparable parts of his definition, exactly as injection function from set A to set B is an inseparable part of his definition ,where this inseparability is based on IFF logical connective as as seen by its truth table, about the considered case, which is:

p="the cardinality of set A (notated as |A|) <= the cardinality of set B (notated as |B|))"

q= "there is injection from set A to set B (notated as <= between the cardinality of set A (notated as |A|) and the cardinality of set B (notated as |B|)"

Here is the iff truth table

Code: 
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T

and as logically seen, cardinality is an inseparable part of his definition of Cardinality, such that p and q are both true or both false.

Little 10 Toes said:
Moreover, is has provided you a definition of cardinality, multiple times.
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And in all these times, cardinality is an inseparable part of his definition of Cardinality.

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

Also please see my post about relative measure in http://www.internationalskeptics.com/forums/showpost.php?p=13142407&postcount=30
 
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jsfisher said:
My definition for relative cardinalities still stands: |A| <= |B| means/is defined as/if and only if there is an injection from A to B.

...and if you can't get along without supplemental information, no matter how obvious: ...where A and B are each sets, |.| is used to denote "cardinality of", and <= is the relationship that is the subject of the definition.
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Ok let's re-examine what you wrote as follows:

An injection from set A to set B is a one-to-one function from the members of set A to the members of set B, such that for every member of set A there is at most one matched member from set B, which provides at most two options by this relative measurement as follows:

(The "size" of set A is relatively less than the size of set B) OR (The "size" of set A is relatively the same as the "size" of set B).

Now comes the trick of "washing-machine of words to symbols" by replacing "the "size" of set A" with |A| symbol, and "the "size" of set B" with symbol |B|.

Also by this trick of "washing-machine of words to symbols" "relatively less than" is replaced by <, and "relatively the same as the "size" of" is replaced by symbol =.

And now after the "washing-machine of words to symbols" did its job, we can write:

(|A|<|B|) OR (|A|=|B|) (also written as |A|<=|B|) means (written as iff logical connective) that there is injection from set A to set B.

Now we take this "washing-machine of words to symbols" result and express it like this:

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)

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

jsfisher, no matter how your "washing-machine of words to symbols" works, you are using cardinality as a part of your definition of Cardinality.

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

A more straightforward framework of this subject can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=13142488&postcount=31, where any criticism is welcome.

jsfisher said:
That' it? Petty attempts at semantic nitpicks?

Come back when you have something of substance. Meanwhile, Mathematics remains intact.
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Mathematics remains intact according to traditional mathematicians, if they take collections and their relations in terms of Plato philosophy school of thought.

jsfisher, since you are using the term "intact" I believe that "set N is a complete whole" may be understood by you as "set N is intact".
 
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