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27th September 2017, 07:43 PM  #2761 
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27th September 2017, 08:02 PM  #2762 
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111...111. is not terminated.
The details are given in http://www.internationalskeptics.com...postcount=2760. 
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27th September 2017, 11:24 PM  #2763 
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jsfisher, by deducing in terms of wholeness instead of in terms of completeness (as already given in http://www.internationalskeptics.com...postcount=2753) one redefines Mathematics.
If you disagree with this redefinition, then please demonstrate its inconsistency in terms of wholeness. In order to do it both your visual_spatial AND verbal_symbolic brain skills must be activated, in order to demonstrate its inconsistency in terms of wholeness. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

28th September 2017, 03:00 AM  #2764 
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In http://www.internationalskeptics.com...postcount=2733 I was talking against ℵ_{0} as a fixed size, which means that by the fixed size argument, for example, ℵ_{0} = ℵ_{0} + 1.
In http://www.internationalskeptics.com...postcount=2740 I show why this argument does not hold, in case that The Infinite Binary Tree is taken as a whole (which is not the same as complete). 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

28th September 2017, 03:52 AM  #2765 
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Since the last discussion is based on the differences between Complete and Whole, please look at these links, for example:
https://www.quora.com/Whatisthedi...eandcomplete https://english.stackexchange.com/qu...holevsentire http://wikidiff.com/wholeness/completeness If the set of natural numbers is taken as a whole, it does not mean that new natural numbers cannot be added to it, since being a whole is being both variant AND invariant without getting into contradiction (as shown in http://www.internationalskeptics.com...postcount=2740). 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

28th September 2017, 04:57 AM  #2766 
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28th September 2017, 05:29 AM  #2767 
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EDIT:
111...111. is a unique path of bits that permanently growing from within as a whole, so the radix point (which is not any one of the bits) is the level of The Infinite Binary Tree that defines the unique path of The Infinite Binary Tree as its infinite whole organ. 
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28th September 2017, 06:48 AM  #2768 
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28th September 2017, 07:05 AM  #2769 
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28th September 2017, 02:44 PM  #2770 
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Ok, let's continue.
The first level along The Infinite Binary Tree, which is mapped with the smallest cardinal number that is greater than any finite cardinal number, defines the ℵ_{0} domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ_{0} bits. The first level along The Infinite Binary Tree, which is mapped with the smallest cardinal number that is greater than any infinite cardinal number of ℵ_{0} domain, defines the ℵ_{1} domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ_{1} bits. The first level along The Infinite Binary Tree, which is mapped with the smallest cardinal number that is greater than any infinite cardinal number of ℵ_{1} domain, defines the ℵ_{2} domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ_{2} bits. etc. ... .<== radix point So, 111...111. is used (without loss of generality) in order to represent infinite cardinal numbers that are mapped with any infinite level in The Infinite Binary Tree. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

29th September 2017, 07:04 AM  #2771 
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Cardinality and positional notation
If one orderly marks each level of The Infinite Binary by cardinal numbers that are represented by bits, then the amount of places that is needed in order to represent each cardinal number is, at least, equal to the finite or infinite level number.
The first finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number that is represented by at least 1_{2} places. The second finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number that is represented by at least 10_{2} places. The third finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number that is represented by at least 100_{2} places. ... The first infinite level (which includes bits) of The Infinite Binary, which is marked by the smallest cardinal number that is greater than any finite cardinal number, is represented by at least ℵ_{0} places. ... The first infinite level (which includes bits) of The Infinite Binary, which is marked by the smallest cardinal number that is greater than any infinite cardinal number in domain ℵ_{0}, is represented by at least ℵ_{1} places. By this observation ℵ_{1} > 2^{ℵ0} since ℵ_{1} is greater than any cardinal number that is involved with ℵ_{0} (so CH is solved). ... The first infinite level (which includes bits) of The Infinite Binary, which is marked by the smallest cardinal number that is greater than any infinite cardinal number in domain ℵ_{1}, is represented by at least ℵ_{2} places. ... 
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29th September 2017, 11:00 AM  #2772 
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Cardinality and positional notation  The corrected post
Let's correct the previous post.
It has to be written as follows: If one orderly marks each level of The Infinite Binary by cardinal numbers that are represented by bits, then the amount of places that is needed in order to represent each cardinal number is, at least, equal to the finite (> 0) or infinite level's place in the The Infinite Binary, as follows: The first finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number (represented by bits) that is represented by at least 1_{2} places. The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by at least 10_{2} places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by at least 100_{2} places. ... The first infinite level of The Infinite Binary (which is marked by the smallest infinite cardinal number that is greater than any finite cardinal number) is covered by an infinite cardinal that is represented by at least ℵ_{0} places. ... The first infinite level of The Infinite Binary (which is marked by the smallest infinite cardinal number that is greater than any infinite cardinal number in domain ℵ_{0}) is covered by an infinite cardinal that is represented by at least ℵ_{1} places. By this observation ℵ_{1} > 2^{ℵ0} since ℵ_{1} is greater than any infinite cardinal number (represented by bits) that is involved with ℵ_{0} (so CH is solved). ... The first infinite level (which includes bits) of The Infinite Binary (which is marked by the smallest infinite cardinal number that is greater than any infinite cardinal number in domain ℵ_{1}) is covered by an infinite cardinal that is represented by at least ℵ_{2} places. ... 
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29th September 2017, 05:30 PM  #2773 
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Some CH observation
Ok, I do not need anymore the radix point in order to show the following:
By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, is covered by a cardinal number (represented by bits) that is represented by 1_{2} places. The term “covered by” means that 2^{1} numbers (represented by bits) can be represented by 1_{2} places. Generally, the number of places is determined by number ^{x} , which is used as the power value of any expression of the form 2^{x} . The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 10^{2} places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 100^{2} places. ... The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{0} places (it means that ℵ_{0} places can represent any cardinal number from 0 up to 2^{ℵ0} ). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{1} places (it means that ℵ_{1} places can represent any cardinal number from 0 up to 2^{ℵ1}). By this observation ℵ_{1} > 2^{ℵ0} ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{2} places (it means that ℵ_{2} places can represent any cardinal number from 0 up to 2^{ℵ2}). ... etc. 
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30th September 2017, 01:13 AM  #2774 
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Some GCH observation
Since the observation in the previous post holds for any base (finite or infinite) GCH is solved.
So The Infinite Binary Tree is some case without loss of generality. 
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30th September 2017, 06:56 AM  #2775 
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GCH (important typo correction)
By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree.
By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, is covered by a cardinal number (represented by bits) that is represented by 1 places. The term “covered by” means that 2^{1} numbers (represented by bits) can be represented by 1 places. Generally, the number of places is determined by number ^{x} , which is used as the power value of any expression of the form 2^{x} . The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 2 places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 3 places. ... The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{0} places (it means that ℵ_{0} places can represent any cardinal number from 0 up to 2^{ℵ0} ). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{1} places (it means that ℵ_{1} places can represent any cardinal number from 0 up to 2^{ℵ1}). By this observation ℵ_{1} > 2^{ℵ0} ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{2} places (it means that ℵ_{2} places can represent any cardinal number from 0 up to 2^{ℵ2}). ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved. So The Infinite Binary Tree is some case without loss of generality. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

30th September 2017, 09:59 AM  #2776 
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Some GCH observation, without the nonsense that I wrote before
Some GCH observation, without the nonsense that I wrote before.
By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, needs cardinal number with 1 bit in order to represent 2^{1} cardinal numbers. The second finite level of The Infinite Binary, needs cardinal number with 2 bits in order to represent 2^{2} cardinal numbers. The third finite level of The Infinite Binary, needs cardinal number with 3 bits in order to represent 2^{3} cardinal numbers. ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ_{0} bits in order to represent 2^{ℵ0} cardinal numbers. The second infinite level of The Infinite Binary, needs cardinal number with ℵ_{1} bits in order to represent 2^{ℵ1} cardinal numbers. By this observation ℵ_{1} > 2^{ℵ0} The third infinite level of The Infinite Binary, needs cardinal number with ℵ_{2} bits in order to represent 2^{ℵ2} cardinal numbers. ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved. So The Infinite Binary Tree is some case without loss of generality. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

1st October 2017, 11:14 AM  #2777 
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My previous post is wrong.
It has to be as follows: By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, needs cardinal number with 1 place in order to represent 2^{1} cardinal numbers. The second finite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 2^{2} cardinal numbers, such that 2 = 2^{1} The third finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 2^{3} cardinal numbers, such that 3 < 2^{2} The forth finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 2^{3} cardinal numbers, such that 4 < 2^{3} ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ_{0} places in order to represent 2^{ℵ0} cardinal numbers. The second infinite level of The Infinite Binary, needs cardinal number with ℵ_{1} places in order to represent 2^{ℵ1} cardinal numbers, such that ℵ_{1} = 2^{ℵ0} The third infinite level of The Infinite Binary, needs cardinal number with ℵ_{2} places in order to represent 2^{ℵ2} cardinal numbers, such that ℵ_{2} < 2^{ℵ1} The forth infinite level of The Infinite Binary, needs cardinal number with ℵ_{3} places in order to represent 2^{ℵ3} cardinal numbers, such that ℵ_{3} < 2^{ℵ2} ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved. So The Infinite Binary Tree is some case without loss of generality. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

1st October 2017, 01:22 PM  #2778 
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Some correction of the previous post.
The forth finite level of The Infinite Binary, needs cardinal number with 4 places in order to represent 2^{4} cardinal numbers, such that 4 < 2^{3} 
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1st October 2017, 01:38 PM  #2779 
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I am not comfortable with my last 3 posts about CH or GCH, so at this stage all I can show is that The Binary Tree is not limited by 2^{ℵ0}
By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, needs cardinal number with 1 place in order to represent 2^{1} cardinal numbers from 0 to 2^{1}1 The second finite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 2^{2} cardinal numbers from 0 to 2^{2}1 The third finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 2^{3} cardinal numbers from 0 to 2^{3}1 ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ_{0} places in order to represent 2^{ℵ0} cardinal numbers from 0 to 2^{ℵ0}1 The second infinite level of The Infinite Binary, needs cardinal number with ℵ_{1} places in order to represent 2^{ℵ1} cardinal numbers from 0 to 2^{ℵ1}1 The third infinite level of The Infinite Binary, needs cardinal number with ℵ_{2} places in order to represent 2^{ℵ2} cardinal numbers from 0 to 2^{ℵ2}1 ... etc.  The observation above holds for any base, finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides. So The Infinite Binary Tree is some case without loss of generality. 
__________________
As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

2nd October 2017, 10:45 AM  #2780 
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Let x be a placeholder for any cardinal number, finite or infinite.
Because the sequences along The Binary Tree represent ordered cardinal numbers, any cardinal number of the form 2^{x}1 < 2^{x} exactly by the finite cardinal 1. So one learns at least four novel things about infinite cardinals: 1) By using the binary tree (without loss of generality) one directly proves that x < 2^{x}. 2) Finite cardinals > 0 can be added to and subtracted from infinite cardinals, such that the result is different from the considered infinite cardinal. 3) The bijection between an infinite set and its proper subset, is an illusion based on finite presentation of infinite sets. 4) The Axiom Of Infinity guarantees the accessibility of a given set of elements to the levels above it. 
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3rd October 2017, 09:37 AM  #2781 
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The axiom of infinity (in words):
Quote:
Let's use The Infinite Binary Tree (without loss of generality) as follows: The idea is to define x>0 of 2^{x} as a placeholder for any given level with more than one node, such that the finite cardinal 1 is used as a successor in order to represent 2^{x} cardinal numbers, by unique sequences with x bits for each one of them. By doing so x can be any cardinal number, finite or infinite, and we get a binary tree which is not bounded by any x. 
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4th October 2017, 03:45 AM  #2782 
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The last 3 posts represent totally ordered cardinal numbers, such that cardinal number 1 is their successor, no matter what is the cardinal number of the placeholder, that is used to represent them.
 Such observation is not accepted by mathematicians that define infinite cardinal numbers in terms of limitcardinals, which means that given an infinite cardinal, no addition operation with some cardinal taken from any level below it, is used as a successor of this infinite cardinal, for example: ℵ_{0} + 2^{n} = ℵ_{0}, ℵ_{1} + 2^{ℵ0} = ℵ_{1}, ℵ_{2} + 2^{ℵ1} = ℵ_{2}, ℵ_{3} + 2^{ℵ2} = ℵ_{3}, ... etc. In this case the following holds: By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, is covered by a cardinal number (represented by bits) that is represented by 1 places. The term “covered by” means that 2^{1} numbers (represented by bits) can be represented by 1 places. Generally, the number of places is determined by number ^{x} , which is used as the power value of any expression of the form 2^{x} . The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 2 places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 3 places. ... The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{0} places (it means that ℵ_{0} places can represent up to 2^{ℵ0} cardinal numbers). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{1} places (it means that ℵ_{1} places can represent up to 2^{ℵ1} cardinal numbers). By this observation ℵ_{1} > 2^{ℵ0}, since ℵ_{1} is a limitcardinal of anything that is involved with ℵ_{0} . ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{2} places (it means that ℵ_{2} places can represent can represent up to 2^{ℵ2} cardinal numbers). ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved. So The Infinite Binary Tree is some case without loss of generality. 
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As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

4th October 2017, 01:24 PM  #2783 
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Strong limit cardinals, GCH and infinite trees
This post is an improved version of the previous post.
By carefully observe this diagram I realized (by exclude the root, which is not defined by any bit) that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree. By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, is covered by a cardinal number (represented by bits) that is represented by 1 places. The term “covered by” means that 2^{1} cardinal numbers (represented by bits) can be represented by 1 places. Generally, the number of places is determined by number ^{x} , which is used as the power value of any expression of the form 2^{x}, whether ^{x} is finite or infinite cardinal number. The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 2 places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 3 places. ... The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{0} places (it means that ℵ_{0} places can represent up to 2^{ℵ0} cardinal numbers). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{1} places (it means that ℵ_{1} places can represent up to 2^{ℵ1} cardinal numbers). By this observation ℵ_{1} > 2^{ℵ0}, since ℵ_{1} is a limitcardinal of anything that is involved with ℵ_{0}, exactly as ℵ_{0} is a strong limitcardinal of anything that is involved with n (where n is any finite cardinal number). ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{2} places (it means that ℵ_{2} places can represent can represent up to 2^{ℵ2} cardinal numbers). ... etc. Some examples: ℵ_{0} + 2^{n} = ℵ_{0}, ℵ_{1} + 2^{ℵ0} = ℵ_{1}, ℵ_{2} + 2^{ℵ1} = ℵ_{2}, ℵ_{3} + 2^{ℵ2} = ℵ_{3}, ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved. So The Infinite Binary Tree is some case without loss of generality. 
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11th October 2017, 01:42 PM  #2784 
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Strong limit cardinal numbers and infinite trees
By carefully observe this diagram I realized (by exclude the root, which is not defined by any bit) that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree.
By carefully observe these notions I have found the following: The first finite level (the one that includes two bits) of The Infinite Binary, is covered by a finite cardinal number that is represented by 1 place (1 place is needed in order to represent the members of {0, 1}). Generally, the number of places is determined by number ^{x} , which is used as the power value of any expression of the form 2^{x}, whether ^{x} is finite or infinite cardinal number. The second finite level of The Infinite Binary, is covered by a finite cardinal number that is represented by 2 places (2 places are needed in order to represent the members of {00, 01, 10, 11}). The third finite level of The Infinite Binary, is covered by a finite cardinal number that is represented by 3 places (3 places are needed in order to represent the members of {000, 001, 010, 011, 100, 101, 110, 111). ... The first infinite level of The Infinite Binary is covered by an infinite strong limit cardinal (please look at https://en.wikipedia.org/wiki/Limit_cardinal) that is represented by ℵ_{0} places (ℵ_{0} places are needed in order to represent the 2^{ℵ0} members of the power set of S_{0}, where each one of its unique members has ℵ_{0} bits). ... The second infinite level of The Infinite Binary is covered by an infinite strong limit cardinal that is represented by ℵ_{1} places (ℵ_{1} places are needed in order to represent the 2^{ℵ1} members of the power set of S_{1}, where each one of its unique members has ℵ_{1} bits). By this observation ℵ_{1} > 2^{ℵ0}, since ℵ_{1} is a strong limit cardinal of anything that is involved with ℵ_{0}, exactly as ℵ_{0} is a strong limitcardinal of anything that is involved with n (where n is any finite cardinal number). ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ_{2} places (ℵ_{2} places are needed in order to represent the 2^{ℵ2} members of the power set of S_{2}, where each one of its unique members has ℵ_{2} bits). ... etc. Some examples: ℵ_{0} + 2^{n} = ℵ_{0}, ℵ_{1} + 2^{ℵ0} = ℵ_{1}, ℵ_{2} + 2^{ℵ1} = ℵ_{2}, ℵ_{3} + 2^{ℵ2} = ℵ_{3}, ... etc.  Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) The Infinite Binary Tree is some case without loss of generality. 
__________________
As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

12th October 2017, 04:29 AM  #2785 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,641

As observed in http://www.internationalskeptics.com...postcount=2784 the cardinality of the places that are needed in order to represent each unique sequence (whether it has finite or infinite members) is equal to ^{x}, which is used as a placeholder of the power value of any expression of the form [base]^{x}, whether ^{x} is finite or infinite cardinal number, where [base] is a placeholder for any cardinal number > 1 .
Moreover, even if strong limit cardinals are defined as inaccessible to the members of the set of cardinals below them (for example: the members of {0, 1 ,2, 3, 4, 5, ...} are inaccessible to ℵ_{0}), the members of the set of these strong limit cardinals are inaccessible to a strong limit cardinal above them (for example: the members of {ℵ_{0} ,ℵ_{1}, ℵ_{2}, ℵ_{3}, ℵ_{4}, ...} are inaccessible to ℵ_{ℵ0}). ... etc. So, a collection of members with a common property can be taken as a whole, yet it is incomplete since it is inaccessible to a given limit above it, and this inaccessibility is essential to each one of the members with the common property in the given collection.  So the notion of inaccessibility as an essential property of collections, actually goes beyond the multiple (the domain of collections), and we do not need the transfinite universe in order to deal with this essential property, as follows: As I understand it, useful science is actually done by achieving results that can be tested and used beyond some particular observation. It means that the observation is notoneofmany observation, which naturally can be used as the substance of any oneofmany observation (notoneofmany observation AND oneofmany observation will be explained later by using the model of Möbius strip). I find Mathematics as one of the most powerful tools that enable to develop the natural linkage among notoneofmany observation AND oneofmany observation. In my opinion, in order to actually be such a tool, mathematicians can't partially use their own brain during mathematical activity. Unfortunately, for the past 250 years almost every mathematical work has to be translated into verbalsymbolic form in order to be accepted as a valid mathematical work. As I see it, in order to correct it, both visual_spatial AND verbal_symbolic brain skills have to be used during mathematical activity, where mathematical results are accepted as the association among both visual_spatial AND verbal_symbolic brain skills. For example, let's use both visual_spatial AND verbal_symbolic brain skills in order to define the natural linkage among notoneofmany observation AND oneofmany observation. The minimal amount of oneofmany things is two things. Question: What enables the knowledge that there are two things, in the first place? By using almost only verbal_symbolic brain skills, this question is not asked at the basis of fundamental mathematical concept like Set, where {5,pi} is some example (without loss of generality) of two things (the outer "{" and "}" is not mathematically defined). So, by Modern Mathematics (which is currntly mostly verbal_symbolic_only framework) the ability to gather at least two things is arbitrarily done. A simple visual_spatial AND verbal_symbolic tool, which naturally enables to answer to this question (by also define the outer "{" and "}" as an essential thing of fundamental mathematical concept like Set), is Möbius strip: By locally observe it, it has two edges (represented here by brown color), where each edge is oneofmany thing. This observation is actually possible because also global observation is done, which provides the knowledge that there is one edge along the Möbius strip (which is notoneofmany thing) where this knowledge is represented by the the outer "{" and "}" (which is not any oneofmany thing between them). The visual_spatial knowledge of notoneofmany thing (represented here by one edge along the Möbius strip), which is also represented by the outer "{" and "}", actually enables to gather oneofmany things like 5,pi into a set (for example: {5,pi}). So, fundamental mathematical concept like Set is not rigorously defined, unless both visual_spatial AND verbal_symbolic brain skills are used. The model of Möbius strip naturally demonstrates the linkage among being aware of one thing at once (the visual_spatial brain skill) as the substance of being aware of multiple things stepbystep (the verbal_symbolic brain skill). Moreover, this awareness (may be known also as Unity consciousness) is exactly the core of useful scientific work, which actually achieving results that can be tested and used beyond some particular observation. Furthermore, the observed, the observer and the tool of observation are actually one thing, which is not the sum of its expressions, and as a fundamental result we get nonentropic realm, which is essentially open to new expressions (infinitely many things can't be defined by a fixed limit, fixed cardinality, etc., as unfortunately done by the current main stream of Modern Mathematics). Generally, by this notion one's awareness is fully activated only by simultaneously using his\her visual_spatial AND verbal_symbolic brain skills. As I understand it ,without it, no actual scientific progression is possible. I do not think that mathematical work that is done by using only verbal_symbolic brain skills (where visual_spatial brain skills are used only as tools for demonstration, and not as fundamental entities of mathematical work) enables to actually gather the observed, the observer and the tool of observation into Unity consciousness. Moreover, in my opinion, the challenge is to develop natural Unity consciousness among us as human beings directly within our biological systems, instead of seeking after our external agents in terms of Artificial Intelligence. Uncertainty by this notion, is exactly the field of creativity for any brain that enables to define the linkage among Simplicity and Complexity in daily life (where the simple is not the trivial and the complex is not the complicated, such that the wrong pair is trivial_complicated and the right pair is simple_complex).  According to latest scientific researches (for example: [1],[2],[3],[4],[5],6]) visual_spatial AND verbal_symbolic brain skills are activated during mathematical work. I wish to share with you some model (some analogy based on visual_spatial AND verbal_symbolic brain skills) that demonstrates the linkage between Logic and Physics. The attached schema (https://c1.staticflickr.com/4/3929/1...023ed9f0_b.jpg) illustrates the possible associations between BoseEinstein statistics (even function) and FermiDirac statistics (odd function) and the mathematical concepts of contradiction (expressed as sinelike wave) and tautology (expressed as cosinelike wave). By this model contradiction is the separator among universes, where each universe has its own tautology (its own natural constants and, so called, the laws of nature). The simplest form in this model is illustrated by noncomposed endless straight line, which is essentially not the sum of its vibrations simply because being noncomposed endless straight line is independent of its sinelike/cosinelike vibrations, yet they are depend on it. This asymmetry of dependency defines a nonentropic realm since no amount of vibrations is the noncomposed line in itself. Being aware of the noncomposed line during its vibrations, is actually Unity consciousness that is, in my opinion, the ultimate goal of life phenomena, that has no limits to its developed expressions, exactly because no amount of vibrations is the noncomposed line in itself. By not being limited by contradiction and tautology, life phenomena is beyond AND among the expressed universes, such that they become harmonious organs of an organism, which is ever developed exactly because no organ harms the developments of the other organs, since the development is naturally derived from crossuniverses' source, which is their cause. From this notion, Mathematics is a fulfilled tool only if it supports the development of Unity consciousness among the expressed universes, such that they become ever developed organs of one harmonious self aware realm. As I see it, Contemporary Mathematics that defines collections of infinitely many members by using fixed sizes like N, R , or defines the sum of infinitely many numbers, or multiplication of infinitely many (0,1] members by a single finite result, blocks the ever developed nature of multiplicity and the direct awareness of Unity as the cause of multiplicity, which can't express the full potential of Unity, since, unlike Unity, multiplicity is limited by contradiction and tautology. I wish to clarify something. Given a noncomposed endless straight line and a point not on that line, there are endless larger noncomposed circles that are smaller than that line, and there are endless smaller noncomposed circles that are larger than that point. Yet pi is a proportion among the endless larger and the endless smaller noncomposed circles. So a fixed value can be related to infinitely many things as long as it is not used to define their amount, their sum or any other fixed value that contradicts their property of being endless larger or endless smaller things. For more details, please search for Nicholas of Cusa. Here is an example of a mathematical work that is done by using visual_spatial AND verbal_symbolic brain skills: Please observe the diagram in http://farm5.staticflickr.com/4015/4...686e9e991b.jpg. X>0 AND X>2(a+b+c+d+...) are inseparable in the following diagram (where X is the constant positive value of the infinitely many orange figures in that diagram, whether they are straight (the first orange figure) or bent (the rest infinitely many orange figures, which their endpoints are projected upon the straight orange figure, and define 2(a+b+c+d+...) as an endless collection of added positive values that are < X, exactly because no infinitely many orange figures with constant value X>0 can have value 0)). By using mathematics that is done only by verbal_symbolic brain skills, X=2(a+b+c+d+...), which means that X>0 AND X>2(a+b+c+d+...) are separable in the considered diagram, which (in my opinion) it is the result of separating visual_spatial and verbal_symbolic brain skills during mathematical work. As I see it, Mathematics is actually fulfilled in case that (by using visual_spatial AND verbal_symbolic brain skills) it also goes beyond contradiction and tautology, and enables one to directly be aware of Unity as their cause (or in terms of Physics, it enables one to directly be aware of Unity as the cause of BoseEinstein statistics (even function) and FermiDirac statistics (odd function)). Direct awareness of Unity is not achieved at the realm of thoughts process, which is closed under multiplicity. As I get it, one enables to transcend multiplicity by being aware of finer vibrations' levels of thoughts' process without being restricted to their semantics (they are taken only in terms of different levels of vibrations)) which enables quantum leap into the noncomposed source of multiplicity and quantum leap back to multiplicity. More such bidirectional quantum leaps are exercised in one's brain, more one's brain enables the awareness of Unity as the source of multiplicity during daily life. In my opinion, Mathematical science has to deeply be involved in such exercises in order to really be fulfilled, where reasoning at the level of thoughts (which is restricted by contradiction and tautology, like any multiple phenomena) as currently done by Contemporary Mathematics, can't fulfill the actual abilities of this science among human beings. 
__________________
As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

12th October 2017, 07:34 AM  #2786 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,641

More details about the previous post are already given in http://www.internationalskeptics.com...postcount=2733 and http://www.internationalskeptics.com...postcount=2734.

__________________
As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

12th October 2017, 02:33 PM  #2787 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,641

Here are the references mentioned in http://www.internationalskeptics.com...postcount=2785:
[1] https://software.rc.fas.harvard.edu/...ehaene1999.pdf [2] https://www.youcubed.org/thinkitup...hperformance/ [3] http://journals.plos.org/plosone/art...type=printable [4] https://www.omicsgroup.org/journals/....php?aid=80807 [5] http://bernhardhommel.eu/Effectiven...%20prompts.pdf [6] http://www.mitpressjournals.org/doi/...9#.WMfarIVOIeF[/quote] 
__________________
As long as Comparison is impossible because of the imbalance of one's mind, new glasses will not help.  If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com ) 

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