ISF Logo   IS Forum
Forum Index Register Members List Events Mark Forums Read Help

Go Back   International Skeptics Forum » General Topics » Religion and Philosophy
 


Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today.
Reply
Old 27th September 2017, 07:43 PM   #2761
jsfisher
ETcorngods survivor
Moderator
 
jsfisher's Avatar
 
Join Date: Dec 2005
Posts: 20,964
Originally Posted by doronshadmi View Post
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
jsfisher is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 27th September 2017, 08:02 PM   #2762
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by jsfisher View Post
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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 27th September 2017, 11:24 PM   #2763
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by jsfisher View Post
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 03:00 AM   #2764
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 03:52 AM   #2765
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 04:57 AM   #2766
jsfisher
ETcorngods survivor
Moderator
 
jsfisher's Avatar
 
Join Date: Dec 2005
Posts: 20,964
Originally Posted by doronshadmi View Post
Originally Posted by jsfisher View Post
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
jsfisher is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 05:29 AM   #2767
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by jsfisher View Post
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 06:48 AM   #2768
jsfisher
ETcorngods survivor
Moderator
 
jsfisher's Avatar
 
Join Date: Dec 2005
Posts: 20,964
Originally Posted by doronshadmi View Post
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
jsfisher is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 07:05 AM   #2769
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by jsfisher View Post
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 28th September 2017, 02:44 PM   #2770
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 29th September 2017, 07:04 AM   #2771
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 > 20 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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 29th September 2017, 11:00 AM   #2772
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 > 20 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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 29th September 2017, 05:30 PM   #2773
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 places (it means that 0 places can represent any cardinal number from 0 up to 20 ).
...

The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 1 places (it means that 1 places can represent any cardinal number from 0 up to 21).

By this observation 1 > 20

...

The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 2 places (it means that 2 places can represent any cardinal number from 0 up to 22).

...

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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 30th September 2017, 01:13 AM   #2774
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 30th September 2017, 06:56 AM   #2775
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 places (it means that 0 places can represent any cardinal number from 0 up to 20 ).
...

The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 1 places (it means that 1 places can represent any cardinal number from 0 up to 21).

By this observation 1 > 20

...

The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 2 places (it means that 2 places can represent any cardinal number from 0 up to 22).

...

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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 30th September 2017, 09:59 AM   #2776
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 bits in order to represent 20 cardinal numbers.

The second infinite level of The Infinite Binary, needs cardinal number with 1 bits in order to represent 21 cardinal numbers.

By this observation 1 > 20

The third infinite level of The Infinite Binary, needs cardinal number with 2 bits in order to represent 22 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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 1st October 2017, 11:14 AM   #2777
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 places in order to represent 20 cardinal numbers.

The second infinite level of The Infinite Binary, needs cardinal number with 1 places in order to represent 21 cardinal numbers, such that 1 = 20

The third infinite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 22 cardinal numbers, such that 2 < 21

The forth infinite level of The Infinite Binary, needs cardinal number with 3 places in order to represent 23 cardinal numbers, such that 3 < 22


...

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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 1st October 2017, 01:22 PM   #2778
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 1st October 2017, 01:38 PM   #2779
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 20

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 0 places in order to represent 20 cardinal numbers from 0 to 20-1

The second infinite level of The Infinite Binary, needs cardinal number with 1 places in order to represent 21 cardinal numbers from 0 to 21-1

The third infinite level of The Infinite Binary, needs cardinal number with 2 places in order to represent 22 cardinal numbers from 0 to 22-1

...

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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 2nd October 2017, 10:45 AM   #2780
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 3rd October 2017, 09:37 AM   #2781
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th October 2017, 03:45 AM   #2782
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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:

0 + 2n = 0,
1 + 20 = 1,
2 + 21 = 2,
3 + 22 = 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 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 0 places (it means that 0 places can represent up to 20 cardinal numbers).
...

The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 1 places (it means that 1 places can represent up to 21 cardinal numbers).

By this observation 1 > 20, since 1 is a limit-cardinal of anything that is involved with 0 .

...

The third infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 2 places (it means that 2 places can represent can represent up to 22 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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 4th October 2017, 01:24 PM   #2783
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 places (it means that 0 places can represent up to 20 cardinal numbers).
...

The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by 1 places (it means that 1 places can represent up to 21 cardinal numbers).

By this observation 1 > 20, since 1 is a limit-cardinal of anything that is involved with 0, exactly as 0 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 2 places (it means that 2 places can represent can represent up to 22 cardinal numbers).

...

etc.

Some examples:

0 + 2n = 0,
1 + 20 = 1,
2 + 21 = 2,
3 + 22 = 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.
__________________
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.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 11th October 2017, 01:42 PM   #2784
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 0 places (0 places are needed in order to represent the 20 members of the power set of S0, where each one of its unique members has 0 bits).
...

The second infinite level of The Infinite Binary is covered by an infinite strong limit cardinal that is represented by 1 places (1 places are needed in order to represent the 21 members of the power set of S1, where each one of its unique members has 1 bits).

By this observation 1 > 20, since 1 is a strong limit cardinal of anything that is involved with 0, exactly as 0 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 2 places (2 places are needed in order to represent the 22 members of the power set of S2, where each one of its unique members has 2 bits).

...

etc.

Some examples:

0 + 2n = 0,
1 + 20 = 1,
2 + 21 = 2,
3 + 22 = 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 )

Last edited by doronshadmi; 11th October 2017 at 02:05 PM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 12th October 2017, 04:29 AM   #2785
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 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 {0 ,1, 2, 3, 4, ...} are inaccessible to 0).

...

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 not-one-of-many observation, which naturally can be used as the substance of any one-of-many observation (not-one-of-many observation AND one-of-many 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 not-one-of-many observation AND one-of-many 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 verbal-symbolic 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 not-one-of-many observation AND one-of-many observation.

The minimal amount of one-of-many 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 one-of-many 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 not-one-of-many thing) where this knowledge is represented by the the outer "{" and "}" (which is not any one-of-many thing between them).

The visual_spatial knowledge of not-one-of-many thing (represented here by one edge along the Möbius strip), which is also represented by the outer "{" and "}", actually enables to gather one-of-many 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 step-by-step (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 non-entropic 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 Bose-Einstein statistics (even function) and Fermi-Dirac statistics (odd function) and the mathematical concepts of contradiction (expressed as sine-like wave) and tautology (expressed as cosine-like 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 non-composed endless straight line, which is essentially not the sum of its vibrations simply because being non-composed endless straight line is independent of its sine-like/cosine-like vibrations, yet they are depend on it.

This asymmetry of dependency defines a non-entropic realm since no amount of vibrations is the non-composed line in itself.

Being aware of the non-composed 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 non-composed 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 cross-universes' 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 non-composed endless straight line and a point not on that line, there are endless larger non-composed circles that are smaller than that line, and there are endless smaller non-composed circles that are larger than that point.

Yet pi is a proportion among the endless larger and the endless smaller non-composed 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 Bose-Einstein statistics (even function) and Fermi-Dirac 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 non-composed 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 )

Last edited by doronshadmi; 12th October 2017 at 05:54 AM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 12th October 2017, 07:34 AM   #2786
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 12th October 2017, 02:33 PM   #2787
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
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/think-it-up...h-performance/

[3] http://journals.plos.org/plosone/art...type=printable

[4] https://www.omicsgroup.org/journals/....php?aid=80807

[5] http://bernhard-hommel.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 )

Last edited by doronshadmi; 12th October 2017 at 03:08 PM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 16th October 2017, 11:00 PM   #2788
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
The usefulness of order and complementarity among two-valued logical oprators

Please observe the example of the following 16 two-valued logical operators on propositions p and q.

As can be seen they are ordered from contradiction to tautology or backwards.

Moreover, each two-valued logical operator has a complement.

Boolean algebra is a generalization of Power set algebra, but as can be seen in Wikipedia the order and complemntarity (as seen in the example of two-valued logical operators) are not defined.

Are there formal mathematical researches that are focusing on the order and complementarity among uncountable two-valued logical operators (please provide concrete examples)?
__________________
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; 16th October 2017 at 11:04 PM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 03:54 AM   #2789
Hevneren
Scholar
 
Join Date: Jul 2007
Posts: 115
Originally Posted by doronshadmi View Post

Yet pi is a proportion among the endless larger and the endless smaller non-composed 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.
Ok, so circles come in all sizes, but all of them have the same proportion between area and squared radius, namely pi. That's fifth grade curriculum.
Hevneren is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 04:28 AM   #2790
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by Hevneren View Post
Ok, so circles come in all sizes, but all of them have the same proportion between area and squared radius, namely pi. That's fifth grade curriculum.
There are no "all of them" if endless smaller or endless larger circles are considered, therefore it is definitely not fifth grade curriculum.

---------------------------------

Please air your view about http://www.internationalskeptics.com...postcount=2788.
__________________
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; 17th October 2017 at 04:30 AM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 05:29 AM   #2791
Hevneren
Scholar
 
Join Date: Jul 2007
Posts: 115
Originally Posted by doronshadmi View Post
There are no "all of them" if endless smaller or endless larger circles are considered, therefore it is definitely not fifth grade curriculum.
Let me rephrase it without "all": A circle's area divided by the square of its radius equals pi. Relating a fixed value to infinitely many things isn't such a big deal: I hereby relate the word "fish" to the rational numbers. That was easy.

Originally Posted by doronshadmi View Post
Not easy to understand. Why did you mix boolean expressions with implications, and what were you thinking? And what does "p not implies q" mean? Is it "p doesn't imply q" or "p implies not-q"?

Last edited by Hevneren; 17th October 2017 at 05:30 AM.
Hevneren is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 02:26 PM   #2792
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by Hevneren View Post
Let me rephrase it without "all"
In that case you have no argument.

Originally Posted by Hevneren View Post
Not easy to understand.
https://en.wikipedia.org/wiki/Material_nonimplication and still you do not reply to http://www.internationalskeptics.com...postcount=2788
__________________
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 )
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 03:07 PM   #2793
Hevneren
Scholar
 
Join Date: Jul 2007
Posts: 115
Originally Posted by doronshadmi View Post
In that case you have no argument.
My argument is that this is fifth grade stuff. The area of a circle equals pi times the radius squared. Regardless of circle size.


Originally Posted by doronshadmi View Post
I replied to it in my previous post, where I asked you for some clarification. You answered with another hyperlink to the post I had commented on, and a Wikipedia link that may or may not be relevant. No clarification.
Hevneren is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 03:12 PM   #2794
Hevneren
Scholar
 
Join Date: Jul 2007
Posts: 115
Originally Posted by doronshadmi View Post
I guess "p not implies q" means "NOT (p implies q)" then?
Hevneren is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 17th October 2017, 03:22 PM   #2795
Hevneren
Scholar
 
Join Date: Jul 2007
Posts: 115
Originally Posted by doronshadmi View Post
Given a non-composed endless straight line and a point not on that line, there are endless larger non-composed circles that are smaller than that line, and there are endless smaller non-composed circles that are larger than that point.
I want to learn this properly, so I hope it's OK that I ask a couple of questions:

Are there any composed endless straight lines, and what's the difference between them and the non-composed ones?

Are there any composed circles, and what's the difference between them and the non-composed ones?

What do you mean when you say that a circle is smaller than a line?
Hevneren is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 18th October 2017, 05:15 AM   #2796
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Originally Posted by Hevneren View Post
I want to learn this properly, so I hope it's OK that I ask a couple of questions:
Questions and answers are like two legs for any abstract or non-abstract research.

Originally Posted by Hevneren View Post
Are there any composed endless straight lines, and what's the difference between them and the non-composed ones?
In my model there is one and only one non-composed endless line that is vibrating upon infinitely many smaller or larger scales.

A composed line is defined by collection parts, where a non-compsed line does not have parts.

Originally Posted by Hevneren View Post
Are there any composed circles, and what's the difference between them and the non-composed ones?
The same as explained about lines.

Originally Posted by Hevneren View Post
What do you mean when you say that a circle is smaller than a line?
Any non-composed circle is smaller than the endless non-composed straight line.

------------------------

Still not even a single word from you about my question in http://www.internationalskeptics.com...postcount=2788.
__________________
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; 18th October 2017 at 05:22 AM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 19th October 2017, 02:27 AM   #2797
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Whole yet incomplete

Ok, here is my answer to http://www.internationalskeptics.com...postcount=2788

First some wikipedia quote:
Quote:
The boots-and-socks metaphor was given in 1919 by Russell 1993, pp. 125–127. He suggested that a millionaire might have ℵ0 pairs of boots and ℵ0 pairs of socks.

Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair.

Russell generally used the term "multiplicative axiom" for the axiom of choice. Referring to the ordering of a countably infinite set of pairs of objects, he wrote:

There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible.

Russell then suggests using the location of the centre of mass of each sock as a selector.

The axiom of incompleteness:

Let ... be the inaccessibility of sequence S to target T.

By using an infinite tree of two-valued logical operators, we construct the following infinite sequences (where each sequence is a distinguished two-valued logical operator) such that they are ordered and complements of each other (like pairs of boots), yet they are inaccessible to a given target, as follows:

...000
...001
...010

...

...101
...110
...111

As constructively seen, each infinite sequence has:

a) An immediate predecessor or successor (where ...000 has no predecessor and ...111 has no successor).

b) A complement in the other side of the considered tree of infinite two-valued logical operators.

c) ... defines the inaccessibility of each sequence to the tree's root.

d) ... defines the inaccessibility of each sequence to the tree's centre.

So AC is not used, and unlike Russel's notion that is limited to ℵ0 bits, the inaccessibility a given S to a given T, is not limited by any particular infinite cardinality.

------------------

Please pay attention that an infinite tree of two-valued logical operators is taken as a whole from ...000 up to ...111, yet it is incomplete by its inaccessibility to its root an its centre.

Actually we get a constructive proof of Godell's incompleteness theorems, which directly shows that given infinite logical operators, there are always logical operators (truth values) that are inaccessible to a given target.
__________________
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; 19th October 2017 at 02:47 AM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Old 21st October 2017, 12:49 PM   #2798
doronshadmi
Penultimate Amazing
 
doronshadmi's Avatar
 
Join Date: Mar 2008
Posts: 12,647
Wholeness and Incompleteness, understand Mathematics in constructive terms

Ok, let's put the last posts in one typo free (I hope) comprehensive post.

---------------------------

I read Foundations of mathematics: an optimistic message.

I wish to share with you my view on the issue at hand.

Please observe the example of the following 16 two-valued logical operators on propositions p and q.

As can be seen they are ordered from contradiction to tautology or backwards.

Moreover, each two-valued logical operator has a complement.

Boolean algebra is a generalization of Power set algebra, but as can be seen in Wikipedia the order and complemntarity (as seen in the example of two-valued logical operators) are not defined (by generalization I mean that x (which is used as the power value of the form 2x) is a placeholder for any cardinal number, whether it is finite or infinite).

I did not find formal mathematical researches that use the order and complementarity among infinitely many two-valued logical operators (I think that it is important, since by doing so order and comlementarity enable to constructively distinguish between logical operators without the need of AC, even if uncountable logical operators are involved).

My work on the issue at hand (in case of infinite sequences) is as follows:

First some wikipedia quote:

Quote:
The boots-and-socks metaphor was given in 1919 by Russell 1993, pp. 125–127. He suggested that a millionaire might have 0 pairs of boots and 0 pairs of socks.

Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair.

Russell generally used the term "multiplicative axiom" for the axiom of choice. Referring to the ordering of a countably infinite set of pairs of objects, he wrote:

There is no difficulty in doing this with the boots. The pairs are given as forming an 0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible.

Russell then suggests using the location of the centre of mass of each sock as a selector.

-----------------------------------------


The axiom of incompleteness:

Let ... be the inaccessibility of sequence S to target T.

By using an infinite tree of two-valued logical operators, we construct the following infinite sequences (where each sequence is a distinguished two-valued logical operator) such that they are ordered and complements of each other (like pairs of boots) yet they are inaccessible to a given target, as follows:

...000
...001
...010

...

...101
...110
...111

As constructively seen, each infinite sequence has:

a) An immediate predecessor or successor (where ...000 has no predecessor and ...111 has no successor).

b) A complement in the other side of the considered tree of infinite two-valued logical operators.

c) ... defines the inaccessibility of each sequence to the tree's root.

d) ... defines the inaccessibility of each sequence to the tree's centre.

So AC is not used and, unlike Russel's notion that is limited to 0 bits, the inaccessibility of a given S to a given T, is not limited by any particular infinite cardinality.

------------------

Please pay attention that an infinite tree of two-valued logical operators is taken as a whole from ...000 up to ...111 and it has a root (notated as "Unity") yet it is incomplete by its inaccessibility to its root and its centre.

Actually we get a constructive proof of Godell's incompleteness theorems, which directly shows that given infinite logical operators, there are always logical operators (truth values) that are inaccessible to a given target, even if 2P bits (where P > 0) are involved.

------------------

My optimistic view of Mathematics development is its wholeness in spite of its incompleteness.

Furthermore, in my opinion, incompleteness is an essential signature of openness and endless creativity of Mathematics as a whole.
__________________
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; 21st October 2017 at 12:52 PM.
doronshadmi is offline   Quote this post in a PM   Nominate this post for this month's language award Copy a direct link to this post Reply With Quote Back to Top
Reply

International Skeptics Forum » General Topics » Religion and Philosophy

Bookmarks

Thread Tools

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Forum Jump


All times are GMT -7. The time now is 08:50 AM.
Powered by vBulletin. Copyright ©2000 - 2017, Jelsoft Enterprises Ltd.
© 2014, TribeTech AB. All Rights Reserved.
This forum began as part of the James Randi Education Foundation (JREF). However, the forum now exists as
an independent entity with no affiliation with or endorsement by the JREF, including the section in reference to "JREF" topics.

Disclaimer: Messages posted in the Forum are solely the opinion of their authors.