Cont: Deeper than primes - Continuation 2

[jsfisher]

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.

 

[doronshadmi]

Infinite sequences do not terminate.

111...111. is not terminated.

The details are given in http://www.internationalskeptics.com/forums/showpost.php?p=12010739&postcount=2760.

 

[doronshadmi]

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/forums/showpost.php?p=12009858&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.

 

[doronshadmi]

In http://www.internationalskeptics.com/forums/showpost.php?p=11860589&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/forums/showpost.php?p=12008521&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).

 

[doronshadmi]

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-difference-between-entire-whole-and-complete

https://english.stackexchange.com/questions/9615/whole-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/forums/showpost.php?p=12008521&postcount=2740).

 

[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.

 

[doronshadmi]

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.

 

[jsfisher]

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.

 

[doronshadmi]

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.

 

[doronshadmi]

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.

 

[doronshadmi]

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 > 2^ℵ₀ 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.

...

 

[doronshadmi]

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 > 2^ℵ₀ 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.

...

 

[doronshadmi]

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 2^x .

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 2^ℵ₀ ).
...

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 2^ℵ₁).

By this observation ℵ₁ > 2^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 any cardinal number from 0 up to 2^ℵ₂).

...

etc.

 

[doronshadmi]

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.

 

[doronshadmi]

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 2^x .

The second finite level of The Infinite Binary, is covered by a cardinal number that is represented by 2 places.

The third finite level of The Infinite Binary, is covered by a cardinal number that is represented by 3 places.
...

The first infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ₀ places (it means that ℵ₀ places can represent any cardinal number from 0 up to 2^ℵ₀ ).
...

The second infinite level of The Infinite Binary is covered by an infinite cardinal that is represented by ℵ₁ places (it means that ℵ₁ places can represent any cardinal number from 0 up to 2^ℵ₁).

By this observation ℵ₁ > 2^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 any cardinal number from 0 up to 2^ℵ₂).

...

etc.

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

Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved.

So The Infinite Binary Tree is some case without loss of generality.

 

[doronshadmi]

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 2^ℵ₀ cardinal numbers.

The second infinite level of The Infinite Binary, needs cardinal number with ℵ₁ bits in order to represent 2^ℵ₁ cardinal numbers.

By this observation ℵ₁ > 2^ℵ₀
The third infinite level of The Infinite Binary, needs cardinal number with ℵ₂ bits in order to represent 2^ℵ₂ cardinal numbers.

...

etc.

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

Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved.

So The Infinite Binary Tree is some case without loss of generality.

 

[doronshadmi]

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 2^ℵ₀ cardinal numbers.

The second infinite level of The Infinite Binary, needs cardinal number with ℵ₁ places in order to represent 2^ℵ₁ cardinal numbers, such that ℵ₁ = 2^ℵ₀
The third infinite level of The Infinite Binary, needs cardinal number with ℵ₂ places in order to represent 2^ℵ₂ cardinal numbers, such that ℵ₂ < 2^ℵ₁
The forth infinite level of The Infinite Binary, needs cardinal number with ℵ₃ places in order to represent 2^ℵ₃ cardinal numbers, such that ℵ₃ < 2^ℵ₂

...

etc.

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

Since the observation above holds for any base (finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides) GCH is solved.

So The Infinite Binary Tree is some case without loss of generality.

 

[doronshadmi]

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

 

[doronshadmi]

I am not comfortable with my last 3 posts about CH or GCH, so at this stage all I can show is that The Binary Tree is not limited by 2^ℵ₀
By carefully observe this diagram I realized that every sequence of bits in its left side, has a complement in its right side and vise versa, such that no matter how many bits are involved, the complement property is invariant, which guarantees the uniqueness of each sequence along the tree.

By carefully observe these notions I have found the following:

The first finite level (the one that includes two bits) of The Infinite Binary, needs cardinal number with 1 place in order to represent 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 2^ℵ₀ cardinal numbers from 0 to 2^ℵ₀-1

The second infinite level of The Infinite Binary, needs cardinal number with ℵ₁ places in order to represent 2^ℵ₁ cardinal numbers from 0 to 2^ℵ₁-1

The third infinite level of The Infinite Binary, needs cardinal number with ℵ₂ places in order to represent 2^ℵ₂ cardinal numbers from 0 to 2^ℵ₂-1

...

etc.

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

The observation above holds for any base, finite or infinite, where the invariant complementary property is taken as an average between trees' left and right sides.

So The Infinite Binary Tree is some case without loss of generality.

 

[doronshadmi]

Let x be a placeholder for any cardinal number, finite or infinite.

Because the sequences along The Binary Tree represent ordered cardinal numbers, any cardinal number of the form 2^x-1 < 2^x exactly by the finite cardinal 1.

So one learns at least four novel things about infinite cardinals:

1) By using the binary tree (without loss of generality) one directly proves that x < 2^x.

2) Finite cardinals > 0 can be added to and subtracted from infinite cardinals, such that the result is different from the considered infinite cardinal.

3) The bijection between an infinite set and its proper subset, is an illusion based on finite presentation of infinite sets.

4) The Axiom Of Infinity guarantees the accessibility of a given set of elements to the levels above it.