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#1 This is Cool » On sets and power sets » 2005-03-26 01:11:16
- DoronShadmi
- Replies: 5
On P(Z*) > Z*
General:
We say that B > A if A not= B but there is an injection between A and subset of B .
Theorem: |P(Z*)| > |Z*|
Proof:
Step 1: (injection between set A and subset of set B ):
Let us define injection between each member of Z* to each member of P(Z*), which includes a singleton as its content.
We define f(z) where f(z)={z}, for example: 0 <--> {0} , 1 <--> {1} , 2 <--> {2}, and so on.
At this stage we can say that |P(Z*)| >= |Z*| .
Step 2:
Suppose there exists a bijection between P(Z*) and Z* .
We will show that this assumption leads us to contradiction.
There exists member S in P(Z*) which is consists of ALL members of Z* that are not themselves included in the P(Z*) members which they are mapped to, for example:
0 <--> {0,1} , 1 <--> {10,11,12} , 2 <-->{5,6} , 3 <--> {3,4,5} , 4 <--> {8,9},
In this example S ={1,2,4, }.
|Z*|=aleph0 , |P(Z*)|=2^aleph0
Bucause {} does not exist in Z*, but {} exists in P(Z*), then S has at least one member of Z*, for example:
z <--> {}
In this example S ={z}.
Now, let us say that there exist some member in Z* (let us call it t ) which is mapped with S (t <--> S ).
In this case we can ask: is t in S or t not in S ?
Options:
1) t in S , but by S definition t cannot be in S .
2) t not in S , in this case by S definition, t must be in S , but by (1) t cant be in S .
And so on, and so on.
As we can see, both options lead us to logical contradiction.
Therefore, there cannot be a bijection between P(Z*) and Z* and we can conclude that |P(Z*)| > |Z*| .
Q.E.D
A new point of view of this proof:
By using the empty set (with the Von Neumann Hierarchy), we can construct the set of Z* numbers {0,1,2,3,...}:
0 = |{ }| (notation = {})
1 = |{{ }}| (notation = {0})
2 = |{{ },{{ }}}| (notation = {0,1})
3 = |{{ },{{ }},{{ },{{ }}}}| (notation = {0,1,2})
...
So, through this point of view the atom is {} and the Z* numbers are structural|quantitative combinations of {}.
We have here an Hierarchy of infinitely many complexity levels, starting from {}.
If {} does not exist then we have no system to research, therefore we can learn
that in the base of this number system, there is the idea of dependency, which can be translated to the idea of the power of existence.
Let us look at Cantor's proof from this point of view.
As we all know, any set includes only unique members or no members at all.
Now, let us examine options 1 and 2 again.
Option 1:
All members which included in S , are different from each other.
Any member of Z* can be mapped with some member of P(Z*), once and only once.
Therefore t is different from each member in S, therefore t MUST BE INCLUDED in S.
(S exists iff t is out of the scope of Cantor's definition. it means that the word ALL is omitted from S definition).
Therefore S cannot be defined, and we cannot check our assumption in step 2 of Cantor's proof.
Option 2:
If we want to keep S as an existing member, we MUST NOT INCLUDE t in S .
(S exists iff t is out of the scope of Cantor's definition. it means that the word ALL is omitted from S definition).
In this case we also see that S cannot be defined, and again we cannot check our assumption in step 2 of Cantor's proof.
Therefore we cannot ask nor answer to anything that is connected to member S, because S definition cannot exist, without any connection to the map result between Z* and P(Z*).
The general idea behind this point of view is the power of existence of member t and member S .
More you simple less you depended, therefore more exist.
I'll write S definition again, in an informal way:
There exists member S in P(Z*) which is consists of ALL members of Z* that are not themselves included in the P(Z*) members which they are mapped to, for example:
0 <--> {0,1} , 1 <--> {10,11,12} , 2 <-->{5,6} , 3 <--> {3,4,5} , 4 <--> {8,9},
In this example S ={1,2,4, }.
Now, the power of existence of Z* members is bigger than the power of existence of P(Z*) members.
Member t, which is some arbitrary member of Z* (and mapped with S), is simpler than S, which is a member of P(Z*) .
Member S existence depends on objects like t , therefore we have to check S by t as we did here, and not t by S as Cantor did.
Conclusion: Cantor did not prove that |P(Z*)| > |Z*| because the definition S={z in Z* such that z is not in f(z)} cannot exist because P(Z*) cannot exist without Z*.
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{{}} can exist only if {} is nested in another {}.
In other words, there is a hierarchy of dependency of composed sets in non-composed sets, and this is exactly the deep meaning of the word hierarchy in Von Neumann Hierarchy, which its unique nested structure is based on 3 ZF axioms:
ZF pair axiom, ZF sum axiom and ZF infinity axiom.
Pair and sum axioms are its recursive side and infinity axiom is its inductive side.
In other words, there is no deep thing here but some arbitrary way to construct nested sets, by using the empty set as their building block, for example:
If → means implies then {} → {{}} → {{},{{}}} → {{},{{}},{{},{{}}}} etc. where the existence of the more composed element depends on the existence of the less composed element.
Also a non-nested collection is simpler than a nested collection, even if they have the same cardinal, for example:
|{{}}| is the same as |{{}}|
|{{},{}}| is simpler than |{{},{{}}}|
|{{},{},{}}| is simpler than |{{},{{}},{{},{{}}}}|
|{{},{},{},{}}| is simpler than |{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}|
etc.
Simplicity is one of the most powerful attitudes that can be used in order to understand fine concepts, and I used simplicity in order to research non-empty finite and non-finite collections.
The simplest way to define the unique cardinal of each n in N, is based on the quantity of the empty sets that can be found within each n.
For example:
If we examine the 1-1 mapping between each N member and each member of a proper subset of N (notated as S) of even members, we realize that set S is nothing but a particular way to pack the empty sets, which is based on a successor of exactly two empty sets.
Let a successor be the result of card(nth n)-card(nth-1 n) where (nth-1 n)<(nth n).
If nth-1=0 then n has no successor.
Actually we can define ordered non-finite sets of N members, which are different from each other by the sequences of their successors.
But we have to understand that no matter what combination we use, it does not change the 1-1 mapping between a collection of infinitely many empty sets, into itself, for example:
If we take the above example of vertical collections of N members, which are constructed by empty sets, and order them in a horizontal way, we clearly realize that Peano or ZF axioms ignore the empty sets themselves and use a 1-1 mapping between N members, which are actually constructed by packs of empty sets.
The Cantorean 1-1 mapping ignores the level of empty sets, for example:
By ignoring the most fundamental level of the empty sets, we can get the illusion that {2,4,6,8,...} is a proper subset of {1,2,3,4,...}.
When we ignore the packed level of both {2,4,6,8,...} and {1,2,3,4,...} and define 1-1 mapping between the empty sets that are included within each natural number, we realize that we actually deal with the same non-finite collection of empty sets.
Furthermore, from this basic level, we also realize that a non-finite collection has no identity map because its 1-1 mapping cannot be satisfied.
In order to understand why, please look at:
#2 This is Cool » Two different models of infinity » 2005-03-26 01:01:22
- DoronShadmi
- Replies: 8
Important: This topic is based on proofs without words ( http://mathworld.wolfram.com/ProofwithoutWords.html ).
A one rotation of the Archimedean Spiral is exactly 1/3 of the circles area ( http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Pappus/Bookiv/Pappus.iv.21-25/Pappus.iv.21_25.html#Prop.%2022 ):
If this area is made of infinitely many triangles (as can be seen in the picture below) , it cannot reach 1/3 exactly as 0.33333... cannot reach 1/3:
In order to understand better why 0.33333 < 1/3 please define a 1-1 mapping between each blue level of the multi-scaled Kochs fractal that is found below, and each member of the infinitely long addition 0.3 + 0.03 + 0.003 + 0.0003 + that is equivalent to 0.3333
( http://members.cox.net/fractalenc/fr6g6s.577m2.html )
In any arbitrary level that we choose, the outer boundary of this multi-fractal has sharp edges.
0.333 = 1/3 only if the outer boundary has no sharp edges.
Since this is not the case, then 0.333 < 1/3.
Actually, we can generalize this conclusion to any 0.xxx form and in this case 0.999 < 1 where 0.999 is a single path along a fractal that exists upon infinitely many different scales, where 1 is a smooth and non-composed element.
Now we can understand that a one rotation of the Archimedean Spiral is exactly 1/3 of the circles area only if we are no longer in a model of infinitely many elements, but in a model that is based on smooth and non-composed elements (and in this case the elements are a one rotation of a smooth and non-composed Archimedean Spiral and a one smooth and non-composed circle).
A model of infinitely many elements and a model of a non-composed element have a XOR connective between them.
Therefore the Cantorean aleph0 cannot be considered as the cardinal of N , because N is a collection of infinitely many elements that cannot be completed exactly as 0.9999... < 1.
In other words, by defining the Cantorean aleph0 as an exact cardinal of infinitely many elements, we are no longer in any relation with N, because N is based on a model of infinitely many elements and the Cantorean aleph0 cannot be but a non-composed and infinitely long element, which is too strong to be used as an input by any mathematical tool, and therefore it cannot be manipulated by the language of Mathematics.
Some words about Riemann's Ball:
By using Riemann's Ball we can clearly distinguish between potential infinity and actual infinity.
As we can see from the above example, no infinitely many objects (where an object = an intersection in this model) can reach actual infinity.
In our example we represent only Z* numbers, but between any two of them we can find rational and irrational numbers.
Riemann's limits are 0 and oo (or -oo), and all our number systems are limited to potential infinities, existing in the open intervals (0,oo) or (-oo,0).
When we reach actual infinity, then we have no information for any method that defines infinity by infinitely many objects.
Also oo cannot be defined as a point at infinity in this model, because no intersection (therefore no point) can be found when we reach oo.
More information of this subject can be found in:
http://www.geocities.com/complementarytheory/ed.pdf
http://www.geocities.com/complementarytheory/Successor.pdf
I am a Monadist.
In Monadic Mathematics there are two separated models of the non-finite:
a) A model that is based on the term "infinitely many ...".
b) A model that is based on the term "infinitely long (non-composed) ...".
The Cantorean universe is based only on (a) model.
Because of this reason Cantor did not understand that when he use an AND connective between totality (the term 'all') and a collection of infinitely many ... , he immediately find himself in (b) model.
Please read very carefully my Riemann's Ball argument , in order to understand the phase transition between (a) model and (b) model (and vise versa).
If you understand Riemann's Ball argument then you can clearly see that Aleph0 cannot be but a (b) model.
Since there is a XOR connective between (a) model and (b) model, there is no relation between Aleph0, which is a (b) model, and set N, which is an (a) model.
The foundations of Monadic Mathematics:
A scope is a marked zone where an abstract/non-abstract discussable entity can be examined.
An atom is a non-composed scope.
Examples: {} (= an empty scope), . (= a point), ._. (=a segment),
__ or .__ or __. (= an infinitely long entity).
An empty scope is a marked zone without any content.
An example: {}
A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .
An example: .
A segment is a non-composed and non-empty scope that has directions which are closed upon themselves, or has at least two reachable edges.
An example: O , .__.
Each segment can have a unique name, which is based on its ratio to some arbitrary segment, which its name is 0_1.
An infinitely long entity is a non-composed non-empty scope which is not closed on itself and has no more than one reachable edge.
An example: __ , .__ , __.
Non-atom (or notom) is a scope that includes at leat one scope as its content.
An example: {{}} ,{__} ,{ {},{...,{{}},{},{}}}, {... ,{{}} , . , ._.} etc.
A sub-scope is a scope that exists within another scope.
An Open notom (or Onotom) is a collection of sub-scopes that has no first sub-scope and not a last sub-scope, or a one and only one infinitely long entity with no edges.
An example: {... ,{},{},{}, ...} ,{__} ,{... ,{{}},{},{}, ...} etc.
A Half-Closed notom (or Hnotom) is a scope that includes a first sub-scope but not a last sub-scope, or a last sub-scope and not a first sub-scope.
Also a Hnotom can be based on a one infinitely long entity that has at least one reachable edge.
An example: {{},{},{}, }, {.__}, {__.} etc.
A Closed notom (or Cnotom) is a scope that includes a first sub-scope and a last sub-scope, and it does not include Hnotom or Onotom.
An example: {{},{},{}}, {{}}, {{}, {{},{{}}, ._.} etc.
A Nested-Level is a common environment for a finite or non-finite collection of sub-scopes.
If a notom includes identical sub-scopes ( __ , .__ or __. are excluded), then it is called a First-Order Collection (or FOC).
An example:
,{... ,{},{},{}} ,{... ._. , ._. , ._.} ,{... ,{},{}, ...} ,{... , ._. , ._. , ...}
{{},{}} ,{{{}},{{}},{{}}} ,{... , {.},{.},{.}} ,{{._.},{._.}} etc.
The name of an atom or a notom within some FOC is determined by its internal property and/or its place in the collection. From this definition it is understood that each atom or notom within a FOC, has more than one name.
Non-FOC (or NFOC) is a nested-level that does not include identical sub-scopes.
An example:
,{... , {} , . , {}} ,{... , {._.} , ._. , ._.} ,{... ,{.},{}, ...} ,{... , ._. ,{._.} , ...}
{{},{.}} ,{{{}} ,{} ,{{}}} ,{... ,{},{.},{.}} ,{{},{._.}} etc.
Any atom ( __ is excluded) or notom has a unique name only if it can be distinguished from the other atoms or notoms that share with it the same nested level.
Let redundancy be: more than one copy of the same entity can be found.
Let uncertainty be: more than a one unique name is related to an entity.
An edge and a point:
A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .
An example: .
An edge is an inseparable part of an atom that has a direction.
An example: ._. , .__ , __.
A more developed version of this framework (but with different names) can be found in:
http://www.geocities.com/complementarytheory/My-first-axioms.pdf
you seem to reject the notion of limits.
What is considered as a limit of some sequence that can be found upon infinitely many ordered scales, cannot be the limit of this sequence.
The reason is very simple, because if we examine the absolute value of the gap (the segment length) between any member in the sequence and the element that is considered as the limit of the sequence, we get the ratio 0_x/0 , where 0_x is the gap > 0 (which is a segment) and 0 is the gap between the limit to itself (which is a point).
From a point of view of a point (which is the hypothetic limit) each segment has the same length, and therefore nothing is converged to the point from the point's point of view, and the point cannot be considered as the limit of any segment.
The ratio 0_x/0 clearly gives us the notion that a point is not a limit of a segment (where a segment in this case is any gap > 0).
Instead of the limit concept, we can take any arbitrary segment and check the gaps (segments) relations of the sequence members, according to it (for example 0_x/0_s where 0_s is the arbitrary segment and 0_x is any member of the examined sequence).
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