It is important to expose the lack of traditional mathematicians to understand Set also in terms of visual_spatial brain's skills.
By not being restricted only to Geometry, visual_spatial brain's skills enable to understand that the outer "{" and "}" of a given set are not taken in terms of membership (in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
This post has 3 parts, but first let us use also visual_spatial brain's skills in order to minimally express the fundamental notion of Ploychotomy, which is the dichotomy of NO
thing and YES
thing complement opposites, as follows:
The definition (and the minimal needed symbolic expression) of the dichotomy of complement opposites:
NO
thing (not notated by any symbol) is that has no predecessor.
YES
thing (notated by the outer "{" and "}" symbols) is that has no successor.
According to these definitions (and the minimal needed symbolic expression), the empty set (notated as {}) is the minimal expression of NO
thing and YES
thing complement opposites, where:
1) NO
thing (not notated by any symbol) is below membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
2) YES
thing (notated by the outer "{" and "}") is above membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
(The complementarity of NO
thing and YES
thing is derived from
Unity (
thing) among them, and it is discussed in part 3 of this post).
By following the notions as described above, the empty set {} expression is understood as follows:
The outer "{" "}" represent YES
thing, no symbols between the outer "{" "}" represent NO
thing, where both NO
thing and YES
thing are not understood in terms of membership ("belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
By dealing also with the concept of non-empty set (according to the notions above) the universe of members is between YES
thing and NO
thing, where NO
thing and YES
thing are not understood in terms of membership ("belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
An example: The considered universe, in the case of members 2 and {2}, is {2,{2}}, where the outer "{" "}" is above membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
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Part 1:
In this part we are using the novel notion of outer "{" and "}" (as described above) in order to understand the relations among sets and members, by translating the Barber story into sets and members.
First, here is the story as quoted from Wikipedia (
http://en.wikipedia.org/wiki/Barber_paradox ):
The barber shaves only those men in town who do not shave themselves.
All this seems perfectly logical, until we pose the paradoxical question:
Who shaves the barber?
This question results in a paradox because, according to the statement above, he can either be shaven by:
- himself, or
- the barber (which happens to be himself).
However, none of these possibilities are valid! This is because:
- If the barber does shave himself, then the barber (himself) must not shave himself.
- If the barber does not shave himself, then the barber (himself) must shave himself.
Let us translate this story into the concept of sets and members, by using the novel relation between these concepts.
"
The barber shaves" is equivalent to the outer "{" and "}" (YES
thing), where the outer "{" "}" (YES
thing) is above membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
"
only those men in town who do not shave themselves." is equivalent to the members, which are (below the outer "{" "}" (YES
thing), which is above membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic))
AND (above NO
thing (not notated by any symbol), which is below membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic)).
So "
the Berber shaves" can't be below the outer "{" "}" (YES
thing), which is above membership (it is not understood in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic)).
The "paradox" is artificially derived from the attempt to define "
The barber shaves" is terms of membership by missing the fact that it is above membership ("
The barber shaves" can't be defined in terms of "belong to" , "does not belong to" or partial belonging as done by Fuzzy logic).
By understanding the difference of being a set and being a member of a given set, Russell's paradox is naturally avoided, without any need of special axioms (as done, for example, by ZF(C)).
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Part 2:
In this part we are using the novel notion of different level of membership, such that no member is reducible into NO
thing or extensible into YES
thing, as follows:
1) NO
thing (not notated by any symbol) is below membership.
2) YES
thing (notated by the outer "{" and "}") is above membership.
3) Membership can have infinitely many levels, where each level is wider (or higher) than the previews levels.
4) No level of membership is reducible into NO
thing or extensible into YES
thing.
5) The smallest level of membership is level 0, and examples of level 0 are: {}, 2, 236.67, etc...
6) The next level of membership is level 1, and examples of level 1 are: {{}, 2, 236.67}, {{}}, {2}, {236.67}, etc...
7) The next level of membership is level 2 , and examples of level 2 are: {{{}, 2, 236.67}}, {{{}}}, {{2}}, {{236.67}}, etc...
8) There can be several different levels of membership in a given expression, for example: {}, {{}, 2, 236.67}, 2 ,{{{}}}, {{{{{236.67}}, 2}}} , etc ...
(1) to (8) are not understood in terms of the standard notion of Set (which does not distinguish between the difference of being a set and being a member of a given set) but they are easily understood in terms of the novel notion of Set, as shown in part 3 of this post.
Level 0 of membership can't be but in-vitro (since NO
thing is below membership).
The wider (or higher) levels of membership can be in-vivo w.r.t lower levels, or in-vitro w.r.t higher levels.
The terms in-vitro (the object is isolated from a wider environment, for example: 2) and in-vivo (the object is not isolated from a wider environment, for example: {2} or {2,{2}} etc...) are not restricted here only to biological systems.
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Part 3:
In this part we define the considered framework in terms of
Unity, as follows:
Let's use a cross-section of
Riemann sphere through its 0 and ∞ poles.
The concept of Set is closed under the polychotomy of YES
thing and NO
thing.
That is among polychotomy is
thing (known also as
Unity), as follows:
[qimg]http://farm8.staticflickr.com/7043/6840987626_c9c426828a_z.jpg[/qimg]
NO
thing is weaker than any tool that is used to measure it.
YES
thing is stronger than any tool that is used to measure it.
Unity (
thing) is among NO,SOME,EVERY,YES ploychotomy.
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By following the notions above the outer "{" "}" represent YES
thing, no symbols between the outer "{" "}" represent NO
thing, and between these extremes we have SOME
thing and EVERY
thing.
According to these notions the universe of members is between YES
thing and NO
thing, where NO
thing and YES
thing are not one of the members (members that are not at level 0 can have outer "{" "}", which are always below the outer "{" "}" of a given set).
(An example: The considered universe in the case of 2 and {2} is {2,{2}}, where the outer "{" "}" is above membership).
