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 International Skeptics Forum Continuation Deeper than primes - Continuation 2

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 27th September 2017, 07:43 PM #2761 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 20,852 Originally Posted by doronshadmi Please observe 111...111. or 111111. The first is infinite... No, the first is nonsense as an infinite sequence. Once again, you are trying to present an infinite sequence that terminates. You really need to stop doing that. Infinite sequences do not terminate. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 27th September 2017, 08:02 PM #2762 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Originally Posted by jsfisher Infinite sequences do not terminate. 111...111. is not terminated. The details are given in http://www.internationalskeptics.com...postcount=2760. __________________ 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 )
 27th September 2017, 11:24 PM #2763 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Originally Posted by jsfisher Your own special view of Mathematics once again gets you the wrong answer. And once again, I will remind you that you don't get to redefine Mathematics to accommodate your misunderstandings. You binary tree has countably infinite levels. None of them correspond to an infinite level. These facts do not change just because you may disagree. 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. __________________ 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 ) Last edited by doronshadmi; 27th September 2017 at 11:32 PM.
 28th September 2017, 03:00 AM #2764 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 In http://www.internationalskeptics.com...postcount=2733 I was talking against ℵ0 as a fixed size, which means that by the fixed size argument, for example, ℵ0 = ℵ0 + 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). __________________ 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 ) Last edited by doronshadmi; 28th September 2017 at 03:05 AM.
 28th September 2017, 03:52 AM #2765 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Since the last discussion is based on the differences between Complete and Whole, please look at these links, for example: https://www.quora.com/What-is-the-di...e-and-complete https://english.stackexchange.com/qu...hole-vs-entire 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). __________________ 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 ) Last edited by doronshadmi; 28th September 2017 at 04:22 AM.
 28th September 2017, 04:57 AM #2766 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 20,852 Originally Posted by doronshadmi Originally Posted by jsfisher Infinite sequences do not terminate. 111...111. is not terminated. You clearly meant 111...111. to represent an infinite sequence, one that comes to an abrupt end just before that binary point. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 28th September 2017, 05:29 AM #2767 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Originally Posted by jsfisher You clearly meant 111...111. to represent an infinite sequence, one that comes to an abrupt end just before that binary point. 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. __________________ 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 ) Last edited by doronshadmi; 28th September 2017 at 05:47 AM.
 28th September 2017, 06:48 AM #2768 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 20,852 Originally Posted by doronshadmi 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. If at any point you have any Mathematics to present and discuss, let me know. Meanwhile, enjoy your fantasy play with words and malformed ideas. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 28th September 2017, 07:05 AM #2769 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Originally Posted by jsfisher If at any point you have any Mathematics to present and discuss, let me know. Meanwhile, enjoy your fantasy play with words and malformed ideas. Thank you jsfisher for the discussion, it enabled me to express my non-standard notions by further details that are also taken as a whole. I wish you the best. __________________ 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 ) Last edited by doronshadmi; 28th September 2017 at 07:06 AM.
 28th September 2017, 02:44 PM #2770 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 ℵ0 domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ0 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 ℵ0 domain, defines the ℵ1 domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ1 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 ℵ1 domain, defines the ℵ2 domain along the tree, where each cardinal number that is mapped with a given level in that domain, must be represented by at least ℵ2 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. __________________ 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 ) Last edited by doronshadmi; 28th September 2017 at 02:58 PM.
 29th September 2017, 07:04 AM #2771 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 12 places. The second finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number that is represented by at least 102 places. The third finite level (which includes bits) of The Infinite Binary, is covered by a cardinal number that is represented by at least 1002 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 ℵ0 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 ℵ0, is represented by at least ℵ1 places. By this observation ℵ1 > 2ℵ0 since ℵ1 is greater than any cardinal number that is involved with ℵ0 (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 ℵ1, is represented by at least ℵ2 places. ... __________________ 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 ) Last edited by doronshadmi; 29th September 2017 at 08:13 AM.
 29th September 2017, 11:00 AM #2772 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 12 places. The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by at least 102 places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by at least 1002 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 ℵ0 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 ℵ0) is covered by an infinite cardinal that is represented by at least ℵ1 places. By this observation ℵ1 > 2ℵ0 since ℵ1 is greater than any infinite cardinal number (represented by bits) that is involved with ℵ0 (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 ℵ1) is covered by an infinite cardinal that is represented by at least ℵ2 places. ... __________________ 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 ) Last edited by doronshadmi; 29th September 2017 at 12:52 PM.
 29th September 2017, 05:30 PM #2773 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 12 places. The term “covered by” means that 21 numbers (represented by bits) can be represented by 12 places. Generally, the number of places is determined by number x , which is used as the power value of any expression of the form 2x . The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 102 places. The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 1002 places. ... The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ0 places (it means that ℵ0 places can represent any cardinal number from 0 up to 2ℵ0 ). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ1 places (it means that ℵ1 places can represent any cardinal number from 0 up to 2ℵ1). By this observation ℵ1 > 2ℵ0 ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ2 places (it means that ℵ2 places can represent any cardinal number from 0 up to 2ℵ2). ... etc. __________________ 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 ) Last edited by doronshadmi; 29th September 2017 at 05:33 PM.
 30th September 2017, 01:13 AM #2774 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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. __________________ 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, 06:56 AM #2775 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 21 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 2x . 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 ℵ0 places (it means that ℵ0 places can represent any cardinal number from 0 up to 2ℵ0 ). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ1 places (it means that ℵ1 places can represent any cardinal number from 0 up to 2ℵ1). By this observation ℵ1 > 2ℵ0 ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ2 places (it means that ℵ2 places can represent any cardinal number from 0 up to 2ℵ2). ... 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. __________________ 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 ) Last edited by doronshadmi; 30th September 2017 at 08:30 AM.
 30th September 2017, 09:59 AM #2776 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 21 cardinal numbers. The second finite level of The Infinite Binary, needs cardinal number with 2 bits in order to represent 22 cardinal numbers. The third finite level of The Infinite Binary, needs cardinal number with 3 bits in order to represent 23 cardinal numbers. ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ0 bits in order to represent 2ℵ0 cardinal numbers. The second infinite level of The Infinite Binary, needs cardinal number with ℵ1 bits in order to represent 2ℵ1 cardinal numbers. By this observation ℵ1 > 2ℵ0 The third infinite level of The Infinite Binary, needs cardinal number with ℵ2 bits in order to represent 2ℵ2 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. __________________ 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 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 21 cardinal numbers. The second finite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 22 cardinal numbers, such that 2 = 21 The third finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 23 cardinal numbers, such that 3 < 22 The forth finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 23 cardinal numbers, such that 4 < 23 ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ0 places in order to represent 2ℵ0 cardinal numbers. The second infinite level of The Infinite Binary, needs cardinal number with ℵ1 places in order to represent 2ℵ1 cardinal numbers, such that ℵ1 = 2ℵ0 The third infinite level of The Infinite Binary, needs cardinal number with ℵ2 places in order to represent 2ℵ2 cardinal numbers, such that ℵ2 < 2ℵ1 The forth infinite level of The Infinite Binary, needs cardinal number with ℵ3 places in order to represent 2ℵ3 cardinal numbers, such that ℵ3 < 2ℵ2 ... 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. __________________ 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 ) Last edited by doronshadmi; 1st October 2017 at 12:34 PM.
 1st October 2017, 01:22 PM #2778 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 Some correction of the previous post. The forth finite level of The Infinite Binary, needs cardinal number with 4 places in order to represent 24 cardinal numbers, such that 4 < 23 __________________ 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, 01:38 PM #2779 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 2ℵ0 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 21 cardinal numbers from 0 to 21-1 The second finite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 22 cardinal numbers from 0 to 22-1 The third finite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 23 cardinal numbers from 0 to 23-1 ... The first infinite level of The Infinite Binary, needs cardinal number with ℵ0 places in order to represent 2ℵ0 cardinal numbers from 0 to 2ℵ0-1 The second infinite level of The Infinite Binary, needs cardinal number with ℵ1 places in order to represent 2ℵ1 cardinal numbers from 0 to 2ℵ1-1 The third infinite level of The Infinite Binary, needs cardinal number with ℵ2 places in order to represent 2ℵ2 cardinal numbers from 0 to 2ℵ2-1 ... 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 ) Last edited by doronshadmi; 1st October 2017 at 02:20 PM.
 2nd October 2017, 10:45 AM #2780 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 2x-1 < 2x 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 < 2x. 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. __________________ 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 ) Last edited by doronshadmi; 2nd October 2017 at 12:32 PM.
 3rd October 2017, 09:37 AM #2781 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 The axiom of infinity (in words): Quote: there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set. Such an induction is limited to members with finite cardinality (i.e. the natural numbers). Let's use The Infinite Binary Tree (without loss of generality) as follows: The idea is to define x>0 of 2x 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 2x 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. __________________ 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 ) Last edited by doronshadmi; 3rd October 2017 at 09:44 AM.
 4th October 2017, 03:45 AM #2782 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 limit-cardinals, 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: ℵ0 + 2n = ℵ0, ℵ1 + 2ℵ0 = ℵ1, ℵ2 + 2ℵ1 = ℵ2, ℵ3 + 2ℵ2 = ℵ3, ... 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 21 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 2x . 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 ℵ0 places (it means that ℵ0 places can represent up to 2ℵ0 cardinal numbers). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ1 places (it means that ℵ1 places can represent up to 2ℵ1 cardinal numbers). By this observation ℵ1 > 2ℵ0, since ℵ1 is a limit-cardinal of anything that is involved with ℵ0 . ... The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ2 places (it means that ℵ2 places can represent can represent up to 2ℵ2 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. __________________ 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 ) Last edited by doronshadmi; 4th October 2017 at 04:37 AM.
 4th October 2017, 01:24 PM #2783 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 21 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 2x, 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 ℵ0 places (it means that ℵ0 places can represent up to 2ℵ0 cardinal numbers). ... The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ1 places (it means that ℵ1 places can represent up to 2ℵ1 cardinal numbers). By this observation ℵ1 > 2ℵ0, since ℵ1 is a limit-cardinal of anything that is involved with ℵ0, exactly as ℵ0 is a strong limit-cardinal 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 ℵ2 places (it means that ℵ2 places can represent can represent up to 2ℵ2 cardinal numbers). ... etc. Some examples: ℵ0 + 2n = ℵ0, ℵ1 + 2ℵ0 = ℵ1, ℵ2 + 2ℵ1 = ℵ2, ℵ3 + 2ℵ2 = ℵ3, ... 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. __________________ 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 ) Last edited by doronshadmi; 4th October 2017 at 02:31 PM.
 11th October 2017, 01:42 PM #2784 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 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 2x, 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 ℵ0 places (ℵ0 places are needed in order to represent the 2ℵ0 members of the power set of S0, where each one of its unique members has ℵ0 bits). ... The second infinite level of The Infinite Binary is covered by an infinite strong limit cardinal that is represented by ℵ1 places (ℵ1 places are needed in order to represent the 2ℵ1 members of the power set of S1, where each one of its unique members has ℵ1 bits). By this observation ℵ1 > 2ℵ0, since ℵ1 is a strong limit cardinal of anything that is involved with ℵ0, exactly as ℵ0 is a strong limit-cardinal 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 ℵ2 places (ℵ2 places are needed in order to represent the 2ℵ2 members of the power set of S2, where each one of its unique members has ℵ2 bits). ... etc. Some examples: ℵ0 + 2n = ℵ0, ℵ1 + 2ℵ0 = ℵ1, ℵ2 + 2ℵ1 = ℵ2, ℵ3 + 2ℵ2 = ℵ3, ... 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 ) Last edited by doronshadmi; 11th October 2017 at 02:05 PM.
 12th October 2017, 07:34 AM #2786 doronshadmi 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 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,641 __________________ 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 ) Last edited by doronshadmi; 12th October 2017 at 03:08 PM.

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