That only increases the distance between you and anything related to math.
|{-1, 2, -3}| = 3 is not the same as abs({-1, 2, -3}) = {1, 2, 3}.
In other words, your claim has no basis.
Again, you demonstrated your misunderstanding of the standard notions, and did nothing in order to get the non-standard notions (you simply ignored the first part of
http://www.internationalskeptics.com/forums/showpost.php?p=7665090&postcount=16566.
My initial objection concerned your "non-standard expression" |{x}|. It automatically implies the absolute value of x in a single-member collection which is not a set.
Wrong. I explicitly defined x as a placeholder for collection of members, but since you ignore the first part of
http://www.internationalskeptics.com/forums/showpost.php?p=7665090&postcount=16566, you don't know that.
See, "cardinality" always refers to the size of a set -- it's not used with other types of collections.
Another demonstration of your ignorance of the first part of
http://www.internationalskeptics.com/forums/showpost.php?p=7665090&postcount=16566.
In this case you ignored this part:
doronshadmi said:
by non-standard notions Cardinality is extended beyond the concept of Collections
that leads toward the result that looks like a smiley: (=3
Vary "funny". So you ignored the rest of what is written between the brackets, which is:
doronshadmi said:
(=3, because the values and order of {-1, 2, -3} have no significance)
Shall we

for you?
That's why there exist collections called lists that allow handling vectors.
Since you are using the standard notions and notations, you must know that {-1, 2, -3} is a set and |{-1, 2, -3}| is the cardinality of that set.
I would chose completely different set of symbols not to drive those folks who practise math crazy.
I do not agree with your anti-evolutionist approach, which does not allow mutations of already agreed notions and/or notations.
This time please really read this:
doronshadmi said:
EDITED:
By non-standard notions Cardinality is extended beyond the concept of Collections, and my non-standard notions are expressed by non-standard notations (and mutations of already agreed notations) as follows:
x is a placeholder for collection of members.
My non-standard notation (and notion) {|x|} is equivalent to standard notation (and notion) |{x}|.
My non-standard notation (and notion) {||} is equivalent to standard notation (and notion) |{}|.
My non-standard notation (and notion) |{x}| = ∞ is not equivalent to standard notation (and notion) |{x}|.
My non-standard notation (and notion) |{}| = ∞ is not equivalent to standard notation (and notion) |{}| = 0.
Also the order of the members of a given collection that its cardinality > 1 AND < ∞ has no significance.
If one insists to ignore these extensions, then it is obviously chooses to not communicate about the extensions and their implications on the standard notions\notations of the concept of Collection.
In this case there is no use to continue the discussion with this person, because he\she already chose the limits of the discussion, where any extension of it is doomed to fail in terms of these limits.
Again, this is a philosophical forum, where fundamental notions (including already agreed notations) are re-examined and may be changed (even by mutations).
If one insists to stick to the standard notions\notations of a given subject, there is no use to discuss with him\her about re-examination of already agreed standard notions\notations.
So what is left is to clarify the difference between the non-standard notions\notations and the standard notions\notations of a given subject, and my last posts is an attempt to clarify the extensions of the non-standard notions\notations, by using (as much as possible) notations that may help the reader to make the extensions in his\her mind, which allow him\her to get the standard notions\notations from the new extended notions of the discussed subject (which is not free of mutations).
You still do not get that mutations of already agreed notions and/or notations are inseparable part of my non-standard approach of the mathematical science.
epix, your last post is a concrete example of how you do nothing in order to get the non-standard notions and notations (and mutations of already agreed notions\notations) which only increases the distance between you and anything related to real maths' re-search, where real maths' re-search is not limited to already agreed notions\notations and it is not free of mutations.
Again, according to my non-standard approach of the mathematical science, this science obeys the principles of Evolution, where mutations are one of the fundamental principles of Evolution.
In other words epix, you have no clue (yet) about my non-standard approach of the mathematical science, as follows:
In my opinion an anthropologist researches a given subject from within (by being involved with the researched subject) and from an external point of view (by not being involved with the researched subject), in order to get valuable and useful results.
I also think that we have to be aware of our verbal_symbolic
AND visual_spatial brain's skills if we wish to understand a given subject.
For example, by "Traditional" Mathematics (which is mostly expressed by verbal_symbolic skills) 0.111...
2 = 0.999...
10 =
1 where
1 is the considered mathematical object (the number itself) and 0.111...
2 or 0.999...
10 are some numerals (out of many representations) that represent number
1.
By using verbal_symbolic
AND visual_spatial skills as follows:
one may understand that no branch of that tree actually reaches any other branch of that tree "downward" , no matter how many levels that tree has (in other words, there is no homeomorphism between 0 dimensional space (notated by "0";"1" symbols) and 1 dimensional space (notated by "_____" spatial non-composed object)).
According to this framework
0.111...2 is a number of its own < number
1 by
0.000...12 where the "
...1" part of that number is the irreducibility of ___ 1 dimensional space into 0 dimensional space (known as a point).
By using verbal_symbolic
AND visual_spatial skills one enables to distinguish between non-local numbers like
0.111...2 or
0.000...12, and local numbers like
1 or
0.
Furthermore, no collection of, for example, 0 dimensional spaces or segments on 1 dimensional space has the power of the continuum of 1 dimensional space.
By understanding the power of the continuum in terms of spatial skills, one may understand that no collection of sub-objects of a given space (mathematical or physical) has the power of the continuum of that space, or in other words, any given collection of "hosted" sub-objects is incomplete with respect to the "host" space.
The terms "host"\"hosted" are used here in order to clarify that the the "host" and the "hosted" are defined but not made of each other.
The non-locality of
0.111...2 or
0.000...12 is "naturally vague" in terms of location, and one actually discovers/invents that the
Real-line has a non-empty collection of non-local numbers between 0 dimensional space and 1 dimensional space.
By generalization, given a "host" space, no collection of "hosted" spaces has the power of the "host" space.
Let us do a further step and look at the Mathematical Science by using the "host"\"hosted" view.
From this view, any mathematical theory is (hopefully) a consistent framework of unproved collection of decelerations.
Also form this view, the mathematical science is generally a collection of isolated (context-dependent) frameworks, where each framework has its own consistency.
From time to time it is discovered\invented that there are deeper connections between some context-dependent frameworks, but these discovered\invented connections are based on sporadic\random approach of these cross-contexts linkages.
It has to be stressed that the use of the word "branches" for these context-dependent frameworks is misleading, if there is no comprehensive framework of these context-dependent frameworks, which rigorously demonstrates the linkage between them, such that they can be considered as "branches of a one tree" or as "organs of a one organism".
By the current paradigm, which is generally based on isolated and context-dependent frameworks, any given professional mathematician (or group of professional mathematicians) is asked to invent\discover his\their context-dependent framework by avoiding any changes of already agreed context-dependent frameworks.
This current paradigm of the Mathematical science of isolated and context-dependent developments, can't agree with a paradigm of cross-contexts framework of this science.
In my opinion, the notion of Non-locality (the "host" aspect of "host"\"hosted" framework) is essential to cross-contexts approach and essentially forbidden by the paradigm of context-dependent approach.
The current community of mathematicians is mostly based on the paradigm of the context-dependent frameworks, and according to this paradigm any paradigm of cross-contexts framework, is automatically considered as non-mathematical.
The evolutionary approach of the mathematical science (which is cross-contexts
AND context-dependent framework) changes this paradigm.