A hidden assumption

[doronshadmi]

Be glade, you are wrong.

Repeating 'catch phrases" like "step-by-step" makes me wonder about your creativity and therefore your ability to come up with anything original.

Parallel\step-by-step complementation.

Not just step-by-step or just parallel.

 

[jsfisher]

By the way do you still content that the a set of union of X any Y that are cut out of set X is still connected to set X?

To claim that X or Y that cut out of set X are still considered as its members, is nothing but a contradiction.

Shell we clap hands for the impressive success of the traditional education method on your "reasonability"?

"Cutting out" and other forms of bodily injury were your analogies, not mine.

If I recall correctly, you were babbling about if X = { E, F }, we'd need to "cut out" E from the set X to consider it separately...or something like that.

If it is important, we can find the original post, then you can tell us better what you meant. No, wait, you can't, can you?

 

[jsfisher]

I see we have arrived at corndog territory only with math instead of language.

Will we soon see the universal transform that can change one expression into any other?

Well, George's ET rules truly were universal, but at least George laid them all out for us (with only the occasional secret rule he'd apply when convenient). Doronshadmi just babbles gibberish.

It's sort of like the clarity of Christophera supplemented with the intelligence of George coupled with the charisma of DavidJayJordan. I call this the C\GRS\DJJ complementary triangularization.

 

[jsfisher]

Here's the prelude to the severed body part post.

If E and F are the members of X, then the union of the members of X is (by definition) E ∪ F, which is the set { A, B, C, D }.

Your mistake is very simple E and F are members of X only in this case which is X={E,F}.

If you take them out of X={E,F} then they are no longer considered as the members of X but they are the two independent sets E={A,B} or F={A,C,D}

Now let us look again at http://en.wikipedia.org/wiki/Union_(set_theory) :

If AA and BB are sets, then the union of AA and BB is the set that contains all elements of AA and all elements of BB, but no other elements.

My case is X=AA=BB, so AA U BB = X U X

My statement "the union of all members of X is set X" = XUX.

So {E,F}U{E,F}={E,F}="the union of all the members of set X=set X".


In other words you are wrong because {A,B}U{A,C,D}= the union of the all members of the all members of set X={A,B,C,D}≠XUX=the union of the all members of set X=X

You know what makes this so especially ironic? The starting point for much of this nonsense was that the "set concept" was some sort of hidden assumption. Yet, we see that doronshadmi is incapable of distinguishing between A and {A}.

The ironic thing jsfisher is that you think that set E and set F can be unioned out of X={E,F} and still are concidered as X members.

I have bolded and colored two sections in the original to show where doronshadmi expresses his belief that E U F and X = { E, F } are mutually exclusive constructs.

 

[tsig]

Parallel\step-by-step complementation.

Not just step-by-step or just parallel.

You mean parallel lines never meet?

Each word has meaning it is in the combination of them that fails.

 

[tsig]

Well, George's ET rules truly were universal, but at least George laid them all out for us (with only the occasional secret rule he'd apply when convenient). Doronshadmi just babbles gibberish.

It's sort of like the clarity of Christophera supplemented with the intelligence of George coupled with the charisma of DavidJayJordan. I call this the C\GRS\DJJ complementary triangularization.

Hope this doesn't turn out so bad.

Having read the thread, the Hidden Assumption is that he is smarter than all mathematicians living or dead if we will just let him change a few definitions.

 

[INRM]

Wait...

You're saying there isn't an infinite number of numbers?


INRM

 

[doronshadmi]

You mean parallel lines never meet?

Each word has meaning it is in the combination of them that fails.

No, I mean superposition of identities\distinct identities complementation.

 

[doronshadmi]

Wait...

You're saying there isn't an infinite number of numbers?


INRM

There are infinite number of numbers but their magnitude (measured by cardinality) does not get the magnitude of symmetry itself, which its magnitude is beyond any given cardinal.

Symmetry itself is a non-local atom, where anything that is based on the multitude (and cardinality is based on the multitude) is the result of non-locality\locality complementation, which its magnitude is less than the magnitude of non-locality at its self state as a non-local atom.

 

[doronshadmi]

"Cutting out" and other forms of bodily injury were your analogies, not mine.

If I recall correctly, you were babbling about if X = { E, F }, we'd need to "cut out" E from the set X to consider it separately...or something like that.

If it is important, we can find the original post, then you can tell us better what you meant. No, wait, you can't, can you?

E={A,B}

F={A,C,D}

X={E,F}

There is nothing in E or F that automatically and unconditionally makes them the members of some set (it is X in this particular example) unless we explicitly say for each one of them that it is a member of a certain set (it is X in this particular example).

Now let us examine again the definition of the union:

http://en.wikipedia.org/wiki/Union_(set_theory) :

If AA and BB are sets, then the union of AA and BB is the set that contains all elements (members) of AA and all elements of BB, but no other elements.

If E or F are members of X then they must be examined as the members of X and not as E or F (that there is nothing that automatically and unconditionally makes them the members of X).

E_as_a_member_of_X is not E

F_as_a_member_ of_X is not F

E or F are not automatically and unconditionally members of any set, unless they are explicitly defined as its members.

One can ask:

So how (for example) the set of natural numbers exists if some natural number is not automatically and unconditionally its member.

My answer:

No researchable element exists because it is a member of some set, but because it is the result of invariance\variance complementation.

A set is a researchable element and its existence as a researchable element is not necessarily depend on being a member of a set.

When the notion of the Researchable\Non-researchable is understood, the concept of Set is changed by a paradigm-shift.

You may say:

You did not define your terms.

My answer:

On the contrary, I explicitly defined them.

You cannot get them because they are already used by you as a hidden assumption, you know, these little outer "{" and "}" which are inseparable of any set (empty or not) and are not counted as its members.

 

[jsfisher]

...gibberish...
Now let us examine again the definition of the union:

http://en.wikipedia.org/wiki/Union_(set_theory) :

If AA and BB are sets, then the union of AA and BB is the set that contains all elements (members) of AA and all elements of BB, but no other elements.

If you are going to quote something from another source, at least get the quotation right. You fail cut & paste.

If E or F are members of X then they must be examined as the members of X and not as E or F (that there is nothing that automatically and unconditionally makes them the members of X).

E_as_a_member_of_X is not E

F_as_a_member_ of_X is not F

So, in doronshadmi's world 2 (a member of the set of even numbers), 2 (a member of the set of natural numbers), 2 (a member of the set of prime numbers), and 2 (an integer) are all different and distinct.

...gibberish...

 

[doronshadmi]

If you are going to quote something from another source, at least get the quotation right. You fail cut & paste.

This is the origibal:
If A and B are sets, then the union of A and B is the set that contains all elements of A and all elements of B, but no other elements.


I have changed A to AA and B to BB in order to avoid confusion with E member that are {A,B} and F members that are {A,C,D}:
If AA and BB are sets, then the union of AA and BB is the set that contains all elements (members) of AA and all elements of BB, but no other elements.

So, in doronshadmi's world 2 (a member of the set of even numbers), 2 (a member of the set of natural numbers), 2 (a member of the set of prime numbers), and 2 (an integer) are all different and distinct.

So in jsfisher's word X=2 does not exist if it is not at least a member of some set for example {X}.

 

[jsfisher]

So in jsfisher's word X=2 does not exist if it is not at least a member of some set for example {X}.

Nope, not what I wrote, implied, nor meant.

On the other hand, you said that 2, the member of the set of even numbers, would be different from 2, the number. You are completely wrong on this.

You are also completely wrong that a set is the union of its members.

 

[doronshadmi]

Nope, not what I wrote, implied, nor meant.

On the other hand, you said that 2, the member of the set of even numbers, would be different from 2, the number. You are completely wrong on this.

You are also completely wrong that a set is the union of its members.

Furthermore, the rule that collects the members into a one set is not considered (it is used as a hidden assumption) and as a result each member can be cut out of the set but still be considered as its member.

2 is first of all an independent element. Also it can be a member of some set, but then it is a pert of some set, and being a part must not be ignored.

I know that by get 2 as the same object whether it is a member of some set or not, you think that you are using mathematical generalization.

So let me tell you something about generalization in general.

Generalization in general is the ability to get the common without losing the details.

Actually, more you get the common more you get the details, where the common and the details complement each other in more complex (not complicated but complex, which are not the same) and interesting ways.

For example: by using invariance\variance complementation I am enable to understand the Multitude as a result of this complementation (something that is beyond the ability of the traditional point of view). Furthermore, I can distinguish between being a member and not being a member from finer resolution which is complemented by greater generality.

This is the organic paradigm of the mathematical science, which does not ignore whole\parts relations.

You are actually use the organic paradigm when you distinguish between {1,2,3,4,...},{2,4,6,8,...},{2,3,5,7,...}, etc.

 

[jsfisher]

Furthermore, the rule that collects the members into a one set is not considered (it is used as a hidden assumption) and as a result each member can be cut out of the set but still be considered as its member.

Collects? Cut out? You have a fixation for verbs for some reason.

The [membership] rule does not collect the members. The rule simple describes the conditions that must be satisfied for an object to be a member. No rule runs around collecting the members; no container exists holding the members. The expression "the set contains..." is a colloquialism, nothing more, for the abstract notions of set and its membership.

The set of all prime numbers includes 2 as a member simply because 2 is a prime number. And when 2 is under consideration, its membership in the set of prime numbers may be relevant or irrelevant; it all depends on what's being considered. And no act of "cutting out" is required to consider 2 separate from the set of prime numbers.

 

[doronshadmi]

Collects? Cut out? You have a fixation for verbs for some reason.

The [membership] rule does not collect the members. The rule simple describes the conditions that must be satisfied for an object to be a member. No rule runs around collecting the members; no container exists holding the members. The expression "the set contains..." is a colloquialism, nothing more, for the abstract notions of set and its membership.

The set of all prime numbers includes 2 as a member simply because 2 is a prime number. And when 2 is under consideration, its membership in the set of prime numbers may be relevant or irrelevant; it all depends on what's being considered. And no act of "cutting out" is required to consider 2 separate from the set of prime numbers.

This is my English problem (it is not my first lenguage) please understand any verb as a noun (as a condition, not as an action).

So 2 is a prime number, also 2 is an integer or even an even number, according to the membership that we choose, isn’t it?

Also it can be just 2 without any membership title.

Please define "consideration" (when you write "what's being considered" is it a noun or a verb?).

Also you write: "... its membership in the set of prime numbers may be relevant or irrelevant" , In order to conclude that you must choose, and choosing is may be "the mother of all actions".

Hay, I got it, Mathematics does not need mathematicians at all.

 

[jsfisher]

So 2 is a prime number, also 2 is an integer or even an even number, according to the membership that we choose, isn’t it?

It doesn't matter what we choose, 2 continues to be a prime number, an even number, an integer, and so on, all at the same time.

Please define "consideration" (when you write "what's being considered" is it a noun or a verb?).

The set of all prime numbers includes 2 as a member simply because 2 is a prime number. And when 2 is under consideration, its membership in the set of prime numbers may be relevant or irrelevant; it all depends on what's being considered. And no act of "cutting out" is required to consider 2 separate from the set of prime numbers.

As used above, the phrase, "under consideration", was meant as "being used or looked at for some reason." Similarly, "what's being considered" was meant to refer back to the reason as "why it is being used or looked at."

If, for example, we were studying the binary system (base-2 radix notation), the number 2 would be a central part of our study. (Two would be "under consideration.") Two has many, many properties, but its membership in the set of prime numbers would be of no relevance to the study. Still, 2 continues to be a member of the set of prime numbers, even though that property of 2 didn't come up in studying the binary system.

Also you write: "... its membership in the set of prime numbers may be relevant or irrelevant" , In order to conclude that you must choose, and choosing is may be "the mother of all actions".

You make it sound like it's an exclusive process. "Choose" isn't really, either. Some properties may be important while others unimportant. In the Sieve of Eratosthenes, for example, the numerical value of 2 and its primeness both come up.

 

[jsfisher]

..."Choose" isn't really, either...

"Choose" isn't really the right word, either.

Sigh.

 

[PixyMisa]

It doesn't matter what we choose, 2 continues to be a prime number, an even number, an integer, and so on, all at the same time.

Indeed. 2 is always cardinal, natural, integer, prime, real, complex, rational and even. It belongs to all of those sets simultaneously; it can't not belong to those sets. You can ignore the sets that are not relevant to a particular theorem, but that says nothing except that you are ignoring those sets because they are not relevant.

So 2 is a prime number, also 2 is an integer or even an even number, according to the membership that we choose, isn’t it?

No. 2 is invariant, so its membership in those sets is invariant. Choice has no meaning here.

 

[nathan]

If, for example, we were studying the binary system (base-2 radix notation), the number 2 would be a central part of our study. (Two would be "under consideration.") Two has many, many properties, but its membership in the set of prime numbers would be of no relevance to the study. Still, 2 continues to be a member of the set of prime numbers, even though that property of 2 didn't come up in studying the binary system.

ahem, I wouldn't want to derail this truly fascinating discussion in any way -- no really I wouldn't. But I feel compelled to point out that 2's primality is important when considering binary approximations of decimal approximations of real numbers (and decimal approximations of binary approximations of real numbers).

Carry on.