The DONX Framework & The Dimensional Closure Principle : A Counter-Cantorian Approach to Infinite Sets via Eℕ

1. Introduction

The nature and hierarchy of infinite sets have fascinated mathematicians since Georg Cantor’s foundational work in the late 19th century. Cantor’s revolutionary theory introduced the concept of different sizes, or cardinalities, of infinity - notably distinguishing the countably infinite set of natural numbers, denoted ℕ, from the uncountably infinite set of real numbers, ℝ. This distinction rests on the fundamental insight that while ℕ can be put into a one-to-one correspondence with any of its infinite subsets, no such correspondence can exist between ℕ and ℝ. Cantor’s diagonalization argument formally proves the uncountability of ℝ, establishing a strict hierarchy of infinities with ℝ being strictly “larger” than ℕ.

However, classical set theory inherently assumes that every element of ℕ is a finite natural number, possessing a finite representation - for example, a finite string of digits in base 10. This assumption aligns well with our standard, intuitive understanding of counting numbers, yet it imposes a structural limitation that excludes objects with infinite digit expansions as members of ℕ. For instance, while real numbers admit infinite decimal expansions (e.g., the Champernowne constant), attempting to reinterpret these infinite digit sequences as natural numbers leads to conceptual contradictions within the standard framework.

Thus in classical set theory, a fundamental distinction is drawn between the cardinality of an infinite set and the magnitude of its individual elements. For example, the set of natural numbers, ℕ, has an infinite cardinality, denoted by ℵ₀, yet every member of this set is, by definition, a finite integer. This however creates a conceptual disjuncture / tension, when considering intuitive geometric processes.

For instance, consider a sequence of right-angled isosceles triangles where the base length b and height h are both elements of ℕ, and $b=h$. As we imagine this sequence extending indefinitely ($b\to \infty$), our geometric intuition suggests that the dimensions of the triangle are growing to an infinite size. However, standard set theory formally restricts the domain of b and h to finite values, accommodating only an infinitely large collection of finite triangles, not a single triangle with infinitely large dimensions. This structural constraint forces a separation between the infinite cardinality of the set of all such triangles and the inherent finiteness of any individual member's dimensions.

[Disjuncture of the Triangle -

Consider a sequence of right-angled isosceles triangles, each defined by an increasing integer valued base length : b

b ∈ ℕ

In each triangle in the sequence, due to the isosceles property and right-angle constraint, the height h satisfies:

h = b

As the base length increases without bounds (i.e., b→∞), so too does the corresponding height (visualise a ramp).

But as per definition of ℕ, real numbers cannot contain elements with infinite values...]

The Classical Set Theory resolves this disjuncture by strictly separating the infinite cardinality of the set ℕ from the finiteness of its individual elements, but this creates a tension between the conceptual intuition of infinite extensions and the formal structure that disallows infinite elements.

In this paper, we propose a novel approach called the Dimensional Closure Principle (DCP) to address this tension by extending the classical natural numbers into a new structure, denoted Eℕ (Extended Naturals).  

This modification allows Extended Natural Numbers to include infinite length elements, conceptualized as Dimensionally Closed Infinities of Degree 1. Moreover, we reinterpret real numbers ℝ as inhabiting a higher dimensional infinite Set (Degree 2), analogous to Infinite Arrays or Branching Structures, thus introducing a Dimensional Hierarchy of infinite sets.

The Dimensional Closure Principle asserts that infinite sets can be categorized and understood via their Dimensional Degree - a measure of their Structural Complexity in terms of infinite extent along one or more Dimensions. Under this principle, the classical countable infinity of ℕ corresponds to a Sub Set of Eℕ, which inturn is a one-dimensional infinity, while ℝ corresponds to a two-dimensional infinity, a product of two infinite linear structures.

This Framework proposes an alternative to the traditional Cantorian viewpoint on countability and cardinality by allowing infinite elements within Eℕ (Extended Natural Numbers). We show that, within this dimensional perspective, some classical results, such as Cantor’s Diagonalization, are reinterpreted or reframed. This offers fresh insights into the relationship between countable and uncountable infinities and introduces new conceptual tools for dealing with infinite structures.

The paper proceeds as follows: 

Section 2 reviews classical set theory notions relevant to our discussion, including Cantor’s cardinalities and ordinals. 

Section 3 introduces the Dimensional Closure Principle formally, defining dimensional degrees and the structure of Eℕ. 

Section 4 explores open versus closed Set representations via Venn diagrams to visualize dimensional closure.

Section 5 elaborates the Dimensional Hierarchy and the corresponding infinite sets, including a two-dimensional Hilbert Hotel analogy.

By situating infinite sets within a dimensional hierarchy and relaxing the finiteness constraint on elements, the Dimensional Closure Principle offers a novel mathematical landscape - one where infinity itself is stratified not merely by cardinality but also by dimensional complexity, opening pathways to reconcile longstanding paradoxes and enriching our understanding of the infinites.

2. Background: Classical Set Theory and Cantorian Hierarchy

Modern set theory, rooted in the work of Georg Cantor, provides the formal framework for distinguishing between different "sizes" of infinity. Cantor introduced two key notions: cardinality, which compares the sizes of sets based on bijective mappings, and ordinality, which captures the position or ordering of elements within well-ordered sets.

2.1 Countability and the Cardinality of ℕ

A set is said to be countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers ℕ = {0, 1, 2, 3, ...}. This means that the elements can be listed in a Sequence, even if the Sequence is infinitely long. The Set of even numbers, the Set of rationals ℚ, and any finite alphabet strings are all countably infinite.

Cantor showed that ℕ has a cardinality denoted by ℵ₀ (aleph-null), the smallest infinity in the hierarchy of transfinite numbers. Importantly, ℵ₀ describes not the size of each number, but the size of the Set - it contains an infinite number of finite elements.

2.2 The Uncountability of ℝ

In stark contrast to ℕ, Cantor proved that the set of real numbers ℝ is uncountable. Using his famous diagonalization argument, he demonstrated that no list of real numbers (between 0 and 1) can capture all possible infinite decimal expansions. By constructing a new real number that differs in the nth decimal place from the nth number in any such list, he showed that there's always at least one real number missing from any attempted enumeration. Hence, the real numbers have a greater cardinality than the naturals, denoted by 𝔠 (the cardinality of the continuum), with ℵ₀ < 𝔠.

2.3 Ordinals and ω

While cardinality compares size, ordinals capture order. The first infinite ordinal is ω, which represents the order type of the natural numbers. It is not a number in the usual arithmetic sense but a formal way to describe an infinite sequence.

Ordinal arithmetic behaves differently from finite arithmetic. For instance:

ω + 1 ≠ 1 + ω

ω + 1 is strictly greater than ω,

but 1 + ω = ω.

This reflects that adding something "after" an infinite sequence creates a new structure, while adding something "before" does not.

2.4 Cantor’s Diagonal Argument Revisited

The diagonal argument is central to Cantor’s case for the uncountability of ℝ. It assumes that any real number between 0 and 1 can be expressed as an infinite sequence (e.g., a binary or decimal expansion). Listing these sequences, Cantor constructed a new sequence that differs in the nth digit from the nth sequence in the list. Because this new number is not in the original list, he concluded that no complete enumeration is possible - hence, ℝ is uncountable.

Implicit in this argument is the assumption that each index n ∈ ℕ corresponds to a finite number, which can only access a finite length prefix of any infinite sequence. This assumption is central to the distinction between ℵ₀ and 𝔠.

2.5 Closed vs. Open Set Models

Standard Set Theory often models sets with closed boundaries — particularly in visualizations like Venn diagrams, where each set is a closed region with well-defined membership. In practice, this suits most mathematical applications, but it can impose unintended limitations when representing open-ended, infinite structures.

In this context, it becomes meaningful to explore Open sets or Dimensionally open and closed models of infinity. Thus allowing for Sets with boundaries open / closed, and sets may-not / may admit members with infinite size / representation. This opens the door to alternate definitions of Sets such as Eℕ and Eℝ that account for infinite length (infinite size) elements rather than only infinite size collections of finite elements.

3. The Dimensional Closure Principle & Dimensional Order Numbers Extended  (DONX) Framework.

3.1 Overview of the Dimensional Closure Principle

The Dimensional Closure Principle (DCP) asserts that if a Set is allowed to include the full, infinite extension of its generating dimension, then all internal constructs that depend on that dimension must also admit infinite length elements as valid members. Put simply:

A Set that spans an entire Dimension must be “closed” under the full structural expression of that Dimension - including elements that are infinite, along that Dimension.

[Mind your step... A Set that is Dimensionally Closed, will be represented via on Open Venn Diagram, as a requisite to hold the unbounded.]

In contrast to Cantor's framework, where the natural numbers (ℕ) are countably infinite (aka. the Set can have infinite elements) but all individual elements are finite (in size), the Extended Natural Numbers (Eℕ)  proposed here allow for inclusion of elements whose internal structure can span infinite size, just as real numbers are allowed to have infinite decimal expansions.

3.2 Extending ℕ, ℝ to Eℕ, Eℝ , DONX, DTN

To represent numbers in base 10:

Let D = {0,1,2,3,4,5,6,7,8,9} be the Set of Standard Decimal Digits.

Definition 1: Extended Natural (Eℕ) Numbers

An Extended Natural Number (an element of Eℕ) can be a finite or infinite sequence of digits of the form:

EN = (...i₃ i₂ i₁ i₀)

	∞
EN =	∑	in · bn
	n = 0

EN ∈ Eℕ

i_n ∈ D 

b : base of counting system

The sequence continues indefinitely to the left (rightmost least significant digit). For  representing finite numbers, all leading digits beyond the most significant bit can be set to zero.

Definition :

For base 10 numbers :

Eℕ = { (…d₃ d₂ d₁ d₀) ∣ d_i ∈ {0,1,…,9} }

The structure Eℕ is NOT merely a set of infinitely many finite values (as in Cantor’s ℕ), but an Infinite Set that can contain infinitely large values as well. 

In this way, Eℕ reflects Dimensional Closure under its generating Dimension.

Definition 2: Dimensional Order Number Extended (DONX)

DONX = ([0…∞] … i₂ i₁ i₀ . d_-1 d_-2 … [0…∞])

	∞				∞
DONX =	∑	in · bn		+	∑	dn · b-n
	n = 0				n = 1

DONX ∈ Eℝ

Where:

• The digits …i₂ i₁ i₀ extend infinitely to the left (representing the integer part).
• The digits d_-1 d_-2 … extend infinitely to the right (representing the fractional part).
• And [0…∞] denotes infinite digits of zeros.
• b : base of counting system (b=10 : Decimal, b=2 : Binary, b=16 : Hex...)

Examples of DONXs (b=10) :

1: O₁ = ...0.55555...

2: O₂ = ...0.77[0…∞]

3: O₃ = ...0.12345678910111213141516...

4: O₄ = ...161514131211.10987654321...

5: O₅ = [0…∞]55555

Infinite length numbers (constructed via rules of ordinal addition, multiplication, exponentiation, etc.) are also admitted.

[Mind your step : DONX represents a number inside the DONX Framework. Thus DONX on its own, represents a number.]

Definition 3: Extended Real (Eℝ) Numbers

DONX ∈ Eℝ

Thus for base 10 numbers:

Eℝ = { (…d₃ d₂ d₁ d₀. df_-1 df_-2...) ∣ d_i, df-_i ∈ {0,1,…,9} }

Definition 4: Dimensional Transfinite Numbers (DTN) - Numbers with digits sequences of potentially infinite length in either direction of decimal point.

Under DONX Framework, a DTN is defined as follows:

For base 10:

DTN = [+-]? ([0-9]* [∞↻D]? [0-9]+)? (. [0-9]+ [∞↻D]? [0-9]* )?

DTN ≠ ∅

For decimal number representation:

D = {0,1,2,3,4,5,6,7,8,9}   

Pattern Description:

[+-]?         → Optional sign
(
  [0-9]*      → Finite digits to the left of the special construct
  [∞↻D]?      → Optional infinite segment of digits from D
  [0-9]+      → More finite digits to the left of the decimal point
)?            → Optional integer portion
(
  .           → Decimal point
  [0-9]+      → Finite digits to the right
  [∞↻D]?      → Optional infinite tail to the right
  [0-9]*      → Optional digits after the infinite segment
)?            → Optional fractional portion

The construct [∞↻D] represents a boxing or confinement of an algorithmic digits generator / engine.

Examples for [∞↻D] algorithmic constructs

1. A random number generator inside infinite loop...

2. A π digits generator...

3. A even / odd numbers generator inside infinite loop...

4. A Champernowne constant digits generator...

5. Mandelbrot Set [∞↻D=(zₙ₊₁ = zₙ² + c)]  

etc.

Examples of valid DTNs

0.0

55555

12345.6789

.595

3.1415926535897932384626433832795028841971693993751058209749445923078164...

0.3333333...

0.1234567891011121314151617...

...1110987654321

...55555.55555...

...1110987654321.1234567891011...

3.3 Dimensionality of Infinity

We now formalize a core premise:

Set	Dimensional Class	Description
ℕ	1D	Linear, sequential infinity (finite-length numbers in sequence)
Eℕ	1D	Structure with infinite-length positions admitted.
ℝ	2D	Branching or planar infinity (infinite decimal trees or function spaces)

From this lens, the difference between ℕ / Eℕ and ℝ is not purely cardinal, but dimensional. (This will be further illustrated under 2D Hilbert Hotel). Where ℕ / Eℕ is 1-Dimensional and ℝ is 2-Dimensional - a proper comparison must respect their generative geometries. Thus, rather than viewing ℝ as a larger "bag of numbers", we see it as spanning an additional axis of complexity.

3.4 Dimensional Order Numbers Extended (DONX)

DONX explicitly defines and accommodates  "unbounded" numbers. It provides the mathematical "space" for them to exist in their full, infinite form.

DONXs are not some Abstract Set, but a direct counterpart to real numbers, where infinite sequences of digits represent valid, "unbounded" numbers.

Observe that: ℝ ⊂ Eℝ and ℕ ⊂ Eℕ

[Note : Under ZFC, Eℝ and Eℕ would be categorised as classes and not sets]

Extended Numbers of First Dimension (Degree-1 Ordinality)

In the first instance, a Degree-1 Ordinality corresponds to the a canonical Hilbert Hotel with a single infinite corridor - each room indexed by a Extended Natural Number (Eℕ). Under this extended Set, room numbers are permitted to take on infinite length identifiers (e.g., sequences of infinite digits), effectively reaching what we term as DONX Ordinal of Degree 1.

Examples of Degree-1 Ordinality:

• Example 1: A right angled triangle with a h = b, where the height : h and base : b increases discretely in unit steps, can  extend in value into a DONX Ordinal of Degree 1.
• Example 2: The Indexes of books on a linear bookshelf, where the shelf extends infinitely, also illustrates a DONX Ordinal of Degree 1.

In each of these cases, the underlying structure is one-dimensional, and the set of indices or positions can be captured fully by the DONX Framework Set Eℕ - based Extended Number Line of First Dimension, permitting infinite extensions but confined to a single linear axis.

Extended Numbers of Second Dimension (Degree-2 Ordinality)

By extending this framework to Two Dimensions, we arrive at Degree-2 Ordinality, which corresponds to the case of a 2D Hilbert Hotel. Here, we imagine an infinite corridor (along x-axis), where the boundary between each adjacent room,  opens into a second infinite corridor (along y-axis). This yields a grid or matrix of rooms, where both axes admit indexing by Eℕ elements.

Let the room at position (i, j) have coordinates in the form ri,j, with i, j ∈ Eℕ, including infinite indices, but unlike traditional representations, we do not collapse the 2D structure into 1D (as in the zigzag encoding of Cantor’s pairing function) - but we explicitly recognize and preserve the dimensionality of indexing.

Illustration of Degree-2 Ordinality:

• Room addresses take the form i.j, such as:
  • 0.0, 0.1, 0.2, … 0.9, 0.10, 0.11…
  • 1.0, 1.1, 1.2, … 1.9, 1.10, 1.11…
  • 2.0, …
  • :
• Thus room numbers can extend upto the unbounded ...

Thus, we define:

DONX Ordinal of Degree 2: The total ordering structure associated with a 2D grid indexed by two orthogonal axes each extending over to the unbounded, forming the DONX Ordinality of Second Degree.

This dimensional perspective allows us to represent complex hierarchical systems or nested enumerations where ordinal depth increases with each added axis.

This Framework / Layered Approach introduces a Hierarchy of Infinities, each corresponding to Higher-Dimensional enumeration schemes. While all these infinities remain unbounded and non-terminating, their cardinal structure, indexing capability, and dimensional expressiveness differ significantly. Thus, it is not contradictory to observe that:

“All infinities are unreachable, and yet, some infinities are strictly larger than others.”

Such distinctions become crucial when modeling complex information systems, multi-level enumerations in logic, and transfinite structures in computability and set theory.

3.5 Implications for Cantor’s Diagonal Argument

Cantor’s diagonal argument critically relies on the assumption that ℕ can only index elements of finite magnitudes. By extending ℕ to Eℕ, we enable indices that themselves possess infinite structure/ magnitude. 

This reframes the enumeration of real numbers not as an impossible task, but as a dimensional mismatch: attempting to list a 2D structure using a 1D traversal.

In a framework based on Eℕ, one could posit an indexed tree whose branches are navigated not just by finite steps, but by infinitely expressive indices. From this, the diagonal argument no longer excludes completeness - it simply demonstrates that 1D-finite indexing cannot exhaust a 2D-infinite structure.

3.6 Open Venn Sets and Infinite Membership under DONX Framework.

The visual implication of this approach can be represented via open-ended Venn sets - sets that grow continuously without boundary in one or more dimensions. This contrasts with closed set theory diagrams, where all elements are assumed to be bounded and finitely representable.

For instance, Eℕ can be visualized as an infinite line that not only extends infinitely but can also contain points of / upto infinite magnitudes.  (Visualise a ramp that extends to infinity.)

By contrast, ℕ (in the Cantorian sense) is an infinite list, where each position needs to be a finite object / magnitude.

In this revised Ontology / Framework:

Eℕ is the proper domain for indexing ℝ.

Infinite decimal expansions are not fundamentally unlistable, but require infinite-indexed structures to enumerate.

ℕ just crumbles at the limits of ω. But Eℕ explicitly allows for numbers that have an infinite string of digits, transcending the "finiteness" of classical natural numbers. (The Arithmetic for these infinite numbers exists, even if demanding infinite resources in space time and energy -Refer Addendum A-)

Mapping Between ℝ_[0,1) and Eℕ (Bijection)

For any real number x ∈ ℝ_[0,1), let its decimal expansion be x = 0.d₀ d₁ d₂ d₃ …. We define the corresponding Eℕ element f(x) as the infinite sequence of (...i₃ i₂ i₁ i₀).

[ Definition :

Under DONX Framework: Numbers the such of 0.2 are represented as 0.2000… (or 0.2[0…∞]) and not as 0.1999999999... 

Each real number in ℝ_[0,1) now has a unique (non-ambiguous) infinite decimal expansion, which directly corresponds to a unique Eℕ sequence.]

From Eℕ to ℝ_[0,1):

For any Eℕ element o = (...i₃ i₂ i₁ i₀), define g(o) as the real number 0.d₀ d₁ d₂ d₃ … such that  i_n= d_n  (n ∈ Eℕ).

Bijection Confirmation: This establishes a perfect one-to-one correspondence (bijection) between the set of real numbers in ℝ_[0,1) and Eℕ. Therefore, within The DONX Framework, |ℝ_[0,1)| = |Eℕ|.

Since |ℝ_[0,1)| =  𝔠 (the cardinality of the continuum) 

|Eℕ| = 𝔠

4. Open vs. Closed Venn Diagram Representations under DONX Framework.

4.1 Introduction

Traditional set theory often employs closed Venn diagrams - finite, bounded loops - to represent the relationships between sets. These representations work well for finite and even countably infinite structures (like ℕ), but they struggle to convey the nature of true mathematical infinity, especially when the elements of a set themselves may be infinitely large or infinitely structured.

In this section, we propose an alternative visualization framework:

Open Venn diagrams, where boundaries are incomplete or conceptually “open-ended.”

This better reflects the character of sets that include infinite-length sequences, transfinite ordinals, or dimensionally closed domains like Eℕ.

4.2 Closed Venn Diagrams in Classical Set Theory

In classical logic and set theory, a closed Venn diagram uses enclosed, finite loops to indicate the containment or intersection of sets:

ℕ fits easily into this model, as each natural number is a discrete, finite object.

Even ℚ (the rationals) can be represented in this model, because each element is still a finite expression (ratio of integers).

However, when it comes to representing ℝ, things begin to break down:

Real numbers between 0 and 1 include infinitely long decimal expansions.

The “boundary” of the set ℝ cannot be meaningfully enclosed in a finite loop.

Cantor's approach manages this by working abstractly, using cardinality rather than visual representation - but it still leaves us with no geometric or diagrammatic intuition for how such "larger" sets relate to ℕ.

4.3 The Case for Open Venn Diagrams

We propose a complementary mode of representation:

Open Venn diagrams, where some boundaries are explicitly unclosed, to reflect that the set they represent:

Cannot be finitely circumscribed,

Admits infinite-length elements, or

Grows without bound in its dimensional space.

For instance:

The set Eℕ is like ℕ but allows infinite-magnitude elements.

The set {0,1}^ℕ, the set of all infinite binary sequences (i.e. real numbers in binary form), extends beyond the grasp of any finite container.

Open diagrams allow us to conceptually place these sets within a visual framework where closure is not assumed by default, but rather earned through Dimensional Sufficiency.

Thus...

⊚ – Standard closed Venn circle (e.g. set of planets in solar system)

⊂∞ – Open-ended Venn region (e.g. DONX Framework sets the such of Eℕ, Eℝ)

Point to ponder : Cantorian sets like ℕ, ℚ, are allowed to contain infinite number of elements. Thus under DONX Framework  should they be categorised as Open-ended sets?

4.4 Dimensional Closure and Diagram Topology

In the Dimensional Closure Principle (DCP), a set is considered “closed” if and only if it contains all elements made possible by the Dimensional rules of its own construction. 

[Note : Dimensionally Closed Sets are represented by Open Venn Diagrams]

Thus in the case of Eℕ and Eℝ their Venn Diagrams would be open.

This brings us to a core insight:

ℝ is thus uncountable relative to ℕ, because ℕ is dimensionally not closed with respect to ℝ's structure.

[Note : Under Eℕ, the same ℝ[0,1) appears dimensionally closed and hence "countably infinite" -in the Cantonian parlance-.

Under DONX Framework / parlance, Eℕ, Eℝ are unbounded and thus uncountable.

4.5 Summary

Traditional closed Venn diagrams are insufficient to capture sets containing infinite-length elements or transfinite structures.

Open Venn diagrams better reflect the Dimensional  Closure (capacity to hold unbounded) for example of Eℕ.

The Dimensional Closure Principle provide a consistent criterion for determining when a set is truly closed - not merely in size, but in dimensional sufficiency.

This leads to a new geometric metaphor for understanding countability and cardinality through the lens of diagram topology and dimensional form.

In the following chapter, we will build on this visual logic to formally reinterpret / reexamine Cantor’s diagonal argument, with respect to Eℕ and investigate how Dimensional Closure affects ℝ’s uncountability (under Eℕ indexing).

5. Revisiting Diagonalization and Countability

A DCP-Based Re-examination of Cantor’s Classic Argument

A cornerstone of Cantorian set theory's claim that the real numbers are uncountable lies in his famous diagonalization argument. This elegant proof demonstrates that, for any purported complete listing of real numbers between 0 and 1 (represented as infinite binary or decimal expansions), a new number can always be constructed by altering the diagonal entries - thereby escaping the list. This is taken to imply that the real numbers cannot be placed into one-to-one correspondence with the natural numbers, and hence that their cardinality strictly exceeds that of ℕ.

Under the Dimensional Closure Principle (DCP) and the introduction of the extended ordinal domain Eℕ, we recontextualize this conclusion - not by challenging the validity of the Diagonal Argument under ZFC- but via extension of ℕ onto Eℕ. 

Specifically, we argue that the diagonalization strategy implicitly relies on the restriction that ℕ can only index finite positions. By lifting this restriction and including infinite-length ordinal indices in our indexing class (Eℕ), we permit ourselves to fully inhabit the dimensional space required by the real numbers.

5.1 Diagonalization in the Cantorian Framework

Let us recall the setup of Cantor’s diagonal argument:

Assume a complete enumeration of real numbers between 0 and 1 is given:

r₀ = 0.d₀₀ d₀₁ d₀₂ d₀₃ d₀₄ ...

r₁ = 0.d₁₀ d₁₁ d₁₂ d₁₃ d₁₄ ...

r₂ = 0.d₂₀ d₂₁ d₂₂ d₂₃ d₂₄ ...

⋮

Then, construct a new number:

r′ = 0.d₀′ d₁′ d₂′ d₃′...

where d_n′ ≠ d_nn. This ensures that r′ differs from each listed number r_n in at least the n^th digit - thus not part of the list.

The proof crucially assumes that each index n ∈ ℕ,  and that digits are selected only at finite positions.

This limitation is precisely where DCP proposes a revision.

5.2 Diagonalization under Dimensional Closure

In the DCP framework, we admit Eℕ - a closed extension of ℕ that includes both finite and transfinite indices. The digit positions of real numbers are not limited to finite-length sequences but can extend into the transfinite. Thus, instead of indexing sequences with ℕ, we allow indexing by Eℕ.

Under this system:

The set of all infinite binary sequences becomes indexable by Eℕ.

Any new sequence produced by diagonalization remains within the Closure of the Dimension already enumerated.

Therefore, diagonalization does not escape the indexing system, because the system already includes all possible infinite-length sequences by construction.

This undermines the force of the diagonalization argument, not by breaking its logic, but by showing that it fails to prove incompleteness under Eℕ indexing. The constructed "diagonal" element is simply another element already present in the Dimensional Closure of the Space.

The following two Cases further illustrates this point :

Case 1-

Consider List I:

0: A A A A A B B B A A A A A A B A A ...

1: A B A A B A A A A B A A A A A B A ...

2: A A A A B A A A A A A B B A B A A ...

3: A A B A B B A A B A A A B A A A A ...

4: A A B A B A B A A A A A A A B A A ...

5: ...

upto ∞ (unbounded...)

(In above list, the diagonal letters have been highlighted...)

Now under Cantonian Framework : via flipping A / B along the diagonal, we should generate a new line that does not appear anywhere in List I.

But under Dimensional Order Numbers Extended (DONX) Framework, List I is Dimensionally Closed. Thus List I, should contain all permutations / combinations permissible with A & B. Thus flipping any elements will not produce any new item.

Case 2-

Consider List II having the binary numbers in sequence. (List II:  List of binary numbers in ascending order "left-right flipped" ...)

The resulting list SHOULD contain all binary numbers starting from zero upto infinity.

List II:

0: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

1: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

2: 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

3: 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

4: 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

5: ...

:

upto ∞ (unbounded...)

Now flipping the Diagonal Values: the new number 1 1 1 1 1... would appear in the Dimensionally Closed infinite List II.

[For example observe the binary representation of following numbers:

1    : 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

3    : 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

7    : 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

15  : 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

31  : 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 ...

63  : 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 ...

127: 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 ...

etc.]

The Set of all infinite binary expansions (2^ℵ0) has cardinality 𝔠 (the cardinality of the continuum). Also the Set of all infinite decimal expansions (10^ℵ0) has cardinality  𝔠 (the cardinality of the continuum)...

By defining Eℕ as the set of all infinite sequences of digits, we have |Eℕ|= 10^ℵ0 = 𝔠.

Thus, the claim |ℝ|=|Eℕ| is now inherent in the definition of Eℕ itself, at least for ℝ_[0,1) 

5.3 Countability Reinterpreted

Cantor defines a set as countable if its elements can be placed in bijection with ℕ. 

We are not challenging Cantor’s diagonal argument under ZFC. (Refer Addendum B) Instead, we propose a novel enumeration model based on Extended Natural Numbers (Eℕ) and dimensional indexing, under which ℝ becomes listable.

In our framework, we redefine countability dimensionally:

A set is countable if it is indexable by standard ℕ (Cantonian).

A set is Eℕ-countable (or 1D-Countable) if its elements can be exhaustively enumerated by extended indices, including infinite-length representations.

Thus under DONX Framework, the real numbers between 0 and 1 are Eℕ-countable. This reinterpretation aligns with the Dimensional classification.

Consider the following scenario :

Extended Real Numbers as Decimal Trees and Linear Lists

We consider two distinct yet interrelated numerical structures: one representing the set of Extended Natural Numbers (denoted Eℕ), and the other representing the fractional part of an Extended Real Number  (denoted Eℝ), commonly associated with decimal or fractional expansions.

Each of these sets can be constructed both as a Decimal Tree and as a Linear List, allowing us to define a precise location for any given number in both representations. 

For instance, the Extended Natural Number 52 can be uniquely traced in the corresponding Eℕ Tree structure or directly indexed in its Linear List. Similarly, the real number 0.52, belonging to the decimal portion of Eℝ, has an unambiguous placement within its respective Tree and List.

Theory of Data Structures dictates: Any Tree with finite number of elements (without any recursive branching), has equivalent Linear Array. Under DONX Framework, above is axiomatically extended upto infinite number of elements (unbounded).

DONX Axiom :

Under DONX Framework data structures the such of Lists -L, Trees -T, etc. can contain upto infinite elements. (e.g. ei ∈ T | i ∈ Eℕ)

The following section further illustrates this association :

Consider a Decimal Tree (N Tree / N-ary Tree ; N=10). From root (at level 0), 10 branches extends to level 1. From level 1, 100 (10²) branches to level 2...

At level n : 10ⁿ branches...

Consider real number 0.25391

Location of above number in Decimal Tree follows :

0 at level 0 (root)

2 at level 1 branch 3

5 at level 2 branch 6

3 at level 3 branch 4

9 at level 4 branch 10

1 at level 5 branch 2

Thus : Linear Array equivalence of a Decimal Tree to the first digit:

0 1 2 3 4 5 6 7 8 9

Linear Array equivalence of a Decimal Tree to the first two digits:

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 ... (repeat upto 100 branches...)

It follows that a number such of 7.52, comprising both integer and fractional component, requires dual indexing: one List/ Tree for the integer part (7), and another List/ Tree for the fractional part (0.52). This bifurcation allows for elegant parallelism and comparative structure between the integer and fractional parts.

Decimal Tree Traversal and Linear Correspondence

A key observation emerges when traversing the Decimal Tree Structure of Eℝ using a breadth-first search (BFS) traversal rather than a depth-first approach. This ensures that all decimal expansions - finite or infinite - are encountered in a systematically enumerable fashion, bypassing the unresolvable recursion that depth-first descent into infinite branches would otherwise entail.

Let us define:

• List I-Eℝ: Linear List representation of the integer part of Eℝ.
• List D-Eℝ: Linear List representation of the fractional/ decimal part of Eℝ.

Each list proceeds by enumerating digits at each branching level:

Linear List representation of Integer part of Extended Real (Eℝ) Numbers : List I-Eℝ

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 ... (repeat upto 100 branches...) 0 1 2 3 4 5 6 7 8 9 ... (repeat upto  1000 branches...) ... (unbounded...)

Linear List representation of Decimal part of Extended Real (Eℝ) Numbers : List D-Eℝ

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 ... (repeat upto 100 branches...) 0 1 2 3 4 5 6 7 8 9 ... (repeat upto  1000 branches...) ... (unbounded...)

Upon comparison, it becomes evident that Lists I-Eℝ and D-Eℝ are isomorphic in structure. This implies a one-to-one correspondence (geometric / data structural bijection) between the digit sequences of integer and fractional components - provided the integer portion of the real number is held fixed.

Thus we have proven geometrically (via data structures), that |ℝ_[n,n+1)| = |Eℝ|

[Note : Linearising the Decimal Tree into a List is not mandatory / essential. A BFS of the two Decimal Trees, yields the same isomorphic one-to-one correspondence.

But for a comparison of equivalence, it is requisite to traverse both trees in the Breath First Order. Thus for example, traversing the Integer portion in Breath First Order and the Decimal portion in a Depth First Order, invalidates the impartiality / equality of the investigative analysis.

This would be equivalent to providing one rowing team with proper ores and the other team with table spoons... Not a level playing field...

Further illustrating the necessity to do a BSF on both structures.]

5.4 Consequence for Cardinality in Higher Dimensions

This symmetry reveals a deeper structural insight: the decimal portion of any Real Number, represented via ℝ / Eℝ, possesses cardinality identical to that of the Extended Natural Number system Eℕ. That is, for a fixed integer component, the infinite permutations of the decimal tail map one-to-one with the infinite Extended Natural Numbers.

Under the analogy of the 2D Hilbert Hotel, this observation implies that the cardinality along any / each corridor along y-axis is equivalent to the cardinality along the corridor along x-axis - each capable of expressing infinite entries and structurally mirroring the other.

Bijection Confirmation: This establishes a perfect one-to-one correspondence (bijection) between the set of real numbers in ℝ_[0,1) and Eℕ. 

Therefore, within the DONX Framework, |ℝ_[0,1)| = |Eℕ| = 𝔠

Question : If |ℝ_[0,1)| =  𝔠 (the cardinality of the continuum), what would be |ℝ|?

Is is still 𝔠, or...

[We note that within the DONX Framework, we can generalise :|ℝ_[n,n+1)| = |Eℕ| = 𝔠 , where n ∈ Eℕ]

Next point...

NOTE : Under the standard framework of Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC), the line segment [0,1], the unit area (square) [0,1]×[0,1], and the unit volume (cube) [0,1]×[0,1]×[0,1] all have the same number of points and thus the same cardinality 𝔠.

(Under Cantonian framework 𝔠 . 𝔠 = 𝔠)

This brings up to the Atomicity Disjuncture:

If [0,1], [0,1]×[0,1] and [0,1]×[0,1]×[0,1] have the same number of points (cardinality 𝔠), this implies that a length of 5 cm, an area 5×5 cm2 and volume 5×5×5 cm3 have the same number of atoms (points) under ZFC.

The disconnect arises because standard ZFC set theory's definition of cardinality is purely and extremely abstract. It's only concerned with whether elements can be perfectly matched via pairing one-to-one (via set theory definitions), regardless of:

Distance or topology: It doesn't care about "closeness" of points.

Dimensionality: It treats all points as atomic entities without any inherent dimensional properties.

Resolution to Atomicity Disjuncture under DONX Framework.

[Definition:

Define a measuring unit of sufficient precision for the task at hand (e.g. 1 micron).

DONX linear cardinality : The number of unit 1-D points on a line segment.

DONX surface cardinality : The number of unit 2-D surface points on a surface area.

DONX volumetric cardinality : The number of unit 3-D volumetric points in a geometric volume.]

Take a 2D iscocelese right angled triangle (of the Disjuncture of the Triangle). Place the base length l, along the x axis of the 2D Hilbert Hotel. 

Now when we traverse along x / base ( b ∈ Eℕ )→∞, height ( h ∈ Eℕ ) →∞.

Now the above construct forms a 2D triangle (for reference, lets name it as a Hilbert H. Triangle), having an Area (number of unit points) of 1/2 b.h

Let's assign 𝔠 to the above identity 1/2 b.h ( number of points, when b→∞) - Result 1.

Next let's extend above construct to include the rooms of y direction. Thus now, we obtain a 3D Ramp (Hilbert H. Ramp), of Volume of 1/2 b.h.y

The cardinality in the y direction was shown to be |Eℕ| = 𝔠.

Thus "DONX Volumetric Cardinality" (number of unit points) Of 3D Hilbert H. Ramp = 1/2 b.h.y = 𝔠 . 𝔠 = 𝔠²  - Result 2.

Comparison of Result 1 and Result 2 implies that "DONX surface cardinality" (number of unit points) of / in a 2D Triangle does not equate to a "DONX volumetric cardinality" (number of unit points) of / in a 3D Ramp.

Definition : Under DONX Framework

|Eℕ| = 𝔠[D1] - cardinality of Degree 1

 |ℝ| = 𝔠[D2] - cardinality of Degree 2

[Section 5.5 Provides a Definition of Pairing Bijection Functions and then proceeds to elaborate the limitations of these functions... 

Section 5.6 will present a Geometric pairing Technique better suited for DONX Framework.]

5.5 Definition: Pairing Bijection Function

In the Dimensional Order Numbers Extended (DONX) Framework, which posits a foundational geometric and topological structure to number systems and their cardinalities, a function ℱ: A → B is considered a Valid Pairing Bijection Function if and only if it satisfies the following conditions:

1. Bijectivity (Standard Set-Theoretic):
   • ℱ is injective (one-to-one): For every a₁, a₂ ∈ A, if ℱ(a₁) = ℱ(a₂), then a₁ = a₂.
   • ℱ is surjective (onto): For every b ∈ B, there exists one a ∈ A such that ℱ(a) = b.
   • (Combined: ℱ establishes a one-to-one correspondence between the elements of A and B.)
2. Structural Preservation Requisite (DON-Specific):
   1. For Equinumerosity of Unbounded Sets (Dimensional Scale Preservation):
      If both the domain A and the codomain B are unbounded sets (e.g., Eℕ, ℝ, Eℕ × Eℕ), then ℱ must be a linear transformation of order 1.
      • This means ℱ must satisfy additivity (ℱ(x + y) = ℱ(x) + ℱ(y)) and homogeneity of degree 1 (ℱ(αx) = αℱ(x)), acting on appropriate vector spaces (or modules) defined over DON.
      • Justification: This condition is essential for establishing that two unbounded sets possess the same DONX-cardinality (e.g., 𝔠_[D1]), as it ensures that the function preserves the "dimensional scale" and does not distort or collapse the geometric extent of these spaces. Functions like x², e^x, tan(x), and polynomial / quadratic pairing functions are excluded here, along with any other non-linear functions.
   2. Exclusion of Squashing / Stretching Functions (No Dimensional Collapse / Expansion for Equinumerosity):
      A function ℱ cannot be a Valid Pairing Bijection Function if it maps an unbounded set to a bounded set, or vice-versa.
      • This explicitly excludes functions like the sigmoid function (mapping ℝ to (0,1)) and the tangent function (mapping (−π/2, π/2) to ℝ). While such functions may be bijections in standard set theory, they inherently "squashes" infinite geometric extent into finite space, or "stretch" finite space into infinite extent.
      • Justification: Such transformations are deemed to violate the preservation of fundamental geometric dimension and scale, and therefore cannot be used to assert that the domain and codomain share the same DONX-cardinality.

Interpretation and Implications within DONX:

• This definition rigorously enforces that only bijections that preserve both one-to-one correspondence and fundamental geometric properties (linearity for unbounded sets, and the preservation of boundedness / unboundedness) can establish DONX-cardinality.
• 
  
• The core distinctions between dimensional cardinalities (e.g., 𝔠_[D1], 𝔠_[D2], 𝔠_[D3]) are thus directly maintained because no Valid Pairing Bijection Function (as defined above) can exist between sets of different geometric dimensions (e.g., a 1D line and a 2D plane) or between sets of fundamentally different boundedness characteristics (e.g., an infinite line and a finite interval).
• 
  
• This provides the necessary mathematical machinery to support the geometric intuition that a line segment, an area, and a volume (even if "filled" with points) represent fundamentally different "sizes" in a dimensionally coherent universe.
• 
  
• This definition asserts that for two infinite sets to possess the "same size" or DONX-cardinality (denoted by 𝔠[Dₙ]), their elements must be capable of being paired via a function that preserves their intrinsic dimensional and geometric relationships.
• 
  
• Functions such as polynomial, exponential, trigonometric (e.g., tan), or other non-linear functions (e.g., the standard Cantor pairing function P(x, y)) are explicitly disqualified from being Valid Pairing Bijection Functions, even if they satisfy standard bijectivity in ZFC. This is because their non-linear nature is deemed to distort or collapse the inherent dimensional structure, violating the "topological sanity" paramount to the DONX framework.
• 
  
• The non-existence of a Valid Pairing Bijection Function between sets representing distinct geometric dimensions (e.g., a 1-dimensional Eℕ-line and a 2-dimensional Eℕ-plane) serves as the formal justification for their distinct DONX-cardinalities (e.g., 𝔠_[D1] ≠ 𝔠_[D2]). This fundamentally deviates from the standard cardinal arithmetic where κ · κ = κ for infinite cardinals κ.

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Notes :

The following would not qualify as valid pairing functions under DONX framework.

F(x) = tan(π(x - 1/2))

P(x, y) = ½(x + y)(x + y + 1) + y

Limitations of Pairing Functions under DONX:

The introduction of linear pairing functions  does resolve a few shortcomings of non linear bijections. But still they do not fully address the such of properly resolving, squashing / stretching... Take the function y = mx+c. For m=1, we obtain linear bijection without spacial distortions. But for m ~ 0 and m→∞, we yet again observe squashing / stretching.

Thus the Utilisation of Pairing Functions for cardinality testing purposes, is deprecated / not used under the DONX Framework. 

DONX framework instead utilises a Geometric Pairing Algorithm.

5.6 Definition : DONX Geometric Pairing via Inductive Extrapolation.

Example 1:

Consider the mapping of ℝ[0,1) onto Eℕ. To prove / check as to whether ℝ[0,1) has the same cardinality of Eℕ, Employ following geometric algorithm :

1. Consider the Decimal Trees or Linear Lists of items of ℝ[0,1) and Eℕ.

2. Traverse the two Trees / two Lists in suitable incremental steps. For this exercise, the incremental step, is set to the Level / Depth of Decimal Tree.

3. At the end of each step, List the items so far traversed in linear order.

4. Use the 

Inductive proof that |IR(n)| = |IEN(n)| for all n ∈ ℚ

We're proving by induction that the number of elements in the finite approximation IR(n) equals the number of elements in the finite approximation IEN(n). In other words:

P(n) : |IR(n)| = |IEN(n)|

Lets assign step = 10n, where n is the depth of the decimal tree.

1. Step 1: Base case (prove P(1))

   For n = 1:

   • IR(101) consists of 1-digit decimal strings (0.0, 0.1, …, 0.9). So |IR(10¹)| = 10.
   • IEN(101) consists of 1-digit numbers (0, 1, …, 9). So |IEN(10¹)| = 10.

   Since 10 = 10, we have |IR(10¹)| = |IEN(10¹)|. Thus P(1) is true.

2. Step 2: Inductive hypothesis (assume P(m))

   Assume for some arbitrary m ≥ 1 that:

   Inductive hypothesis: |IR(10^m)| = |IEN(10^m)|

3. Step 3: Inductive step (prove P(m+1) assuming P(m))

   Consider the next stage m+1:

   • For IR(m+1): it contains all elements of IR(m) plus the new (m+1)-digit decimal strings. The number of new strings is 10^m+1, so : (1)

     |IR(m+1)| = |IR(m)| + 10^m+1

   And same for |IEN(m+1)| = ... And Using the inductive hypothesis |IR(m)| = |IEN(m)|, substituting into the expression for IEN(m+1) : (2)

   |IEN(m+1)| = |IR(m)| + 10^m+1

   Comparing RHS of (1) and (2) we get:

   |IR(m+1)| = |IEN(m+1)|

   Hence P(m+1) is true assuming P(m).

4. Step 4: Conclusion

   Since the base case P(1) holds and the inductive implication P(m) ⇒ P(m+1) holds for arbitrary m, by the Principle of Mathematical Induction we conclude that P(n) is true for every natural number n ≥ 1. That is:

   For all n ∈ ℚ, |IR(n)| = |IEN(n)|.

Remark: This is the finite (natural-number) inductive portion of the DONX Transfinite Inductive Extrapolation Principle - The algorithm first operates within ℚ in a finite stage-by-stage induction process.

Next we extrapolate above inducted proof into DONX Transfinite Domain, via axiomatic extrapolation (Refer Addendum A -)

[For a derivation of above under the Axioms of ZFC and Cantonian Framework, refer Addendum B-]

Example 2:

Compare the Cardinality of all Even Extended Natural numbers and Extended Natural Numbers under the axioms of DONX Framework.

Let assign step (s) = 100 (scan items in groups of 100).

Let EEN(s.n) represent the Set of Even EN upto index s.n.

Let EN(s.n) represent the Set of EN upto index s.n.

Next let's apply the DONX Geometric Pairing algorithm to sets EEN(s.n) and EN(s.n)

Prove that |EEN(s.n)| = ½|EN(s.n)| for all n ∈ NE under the DONX Framework.

Inductive prove that |EEN(s.n)| = ½ |EN(s.n)| for all n ∈ ℚ

P(n) : |EEN(100.n)| = ½|EN(100.n)|

1. Step 1: Base case (prove P(1))

   For n = 1:

   • EEN(100.1) consists of even decimal numbers between [0 - 100) (0, 2, …, 98). So |EEN(100.1)| = 50.
   • EN(100.1) consists of  numbers (0, 1, …, 99). So |EN(100.1)| = 100.

   Since 50 = ½.100, we have |EEN(100.1)| = ½ |EN(100.1)|. Thus P(1) is true.

2. Step 2: Inductive hypothesis (assume P(m))

   Assume for some arbitrary m ≥ 1 that:

   Inductive hypothesis: |EEN(100.m)| = ½ |EN(100.m)|

3. Step 3: Inductive step (prove P(m+1) assuming P(m))

1. |EEN(100(m+1))| = |EEN(100m)| + 50
2. |EN(100(m+1))| = |EN(100m)| + 100
3. From the inductive hypothesis, we know |EEN(100m)| = (1/2)·|EN(100m)|.
4. Substitute and simplify:
   |EEN(100(m+1))| = (1/2)·|EN(100m)| + 50
5. We also know that |EN(100m)| = |EN(100(m+1))| − 100.
   Substituting this in:
   |EEN(100(m+1))| = (1/2)·(|EN(100(m+1))| − 100) + 50
   |EEN(100(m+1))| = (1/2)·|EN(100(m+1))| − 50 + 50
   |EEN(100(m+1))| = (1/2)·|EN(100(m+1))|

4. Step 4: Conclusion

   Since the base case P(1) holds and the inductive implication P(m) ⇒ P(m+1) holds for arbitrary m, by the Principle of Mathematical Induction we conclude that P(n) is true for every Natural number n ≥ 1. That is:

   For all n ∈ ℕ , |EEN(n)| = ½ |EN(n)|.

Next axiomatically we extend above result for all n ∈ Eℕ, under the DONX Framework.

Note: Under DONX Framework the cardinalities the such of  𝔠_[D1],  𝔠_[D2] = 𝔠_[D1]² ... are defined entities. And the association of a Test Set against any of the above defined cardinalities is achieved by Geometric Pairing via Induction.

(Under DONX Framework

|Eℕ| = 𝔠[D1] - cardinality of Degree 1

 |ℝ| = 𝔠[D2] - cardinality of Degree 2)

Thus as|EEN(n)| = ½ |EN(n)|

Under DONX Framework:

|EEN|= ½ 𝔠[D1] 

To further illustrate the above result consider two spacecrafts traveling at velocity v and 2v. Each unit distance traveled, an entry is made into the log file (DONX Set) of each craft. After infinite time, both crafts would have had travelled an infinite distance.

Question : After infinite time, which craft's log file would have ½ of the number of entries as the other?

Definition : Under DONX Framework, Cardinalities can be of rational multiples of the defined cardinals of 𝔠[D1], 𝔠[D2] etc.

Definition : Under DONX Framework,  Mathematical Induction is valid upto the unbounded. If a pattern holds for n ∈ ℕ or ℚ : Under the DONX Transfinite Inductive Extrapolation Axiom, this deduction / result is valid upto the unbounded. 

Note : The above definitions and results are clear deviations from the axioms and deductions of ZFC and Cantonian Framework.

5.7 Consequences for Computability and Foundations

If real numbers are Eℕ-countable, then a wide class of infinite binary strings - often considered uncomputable due to their “unlistability” - could, under this system, be treated as enumerable within a richer dimensional framework. This has far-reaching implications:

Computability Theory: The Church–Turing thesis and Turing degrees may need reformulation to accommodate transfinite input / output spaces.

Information Theory: Infinite information patterns need not be considered fundamentally unreachable; they may be encoded in dimensionally extended systems.

Philosophy of Mathematics: The rejection of "completed infinities" in finitist philosophies may be reconciled with infinite sets through the structure of dimensional closure, offering a constructive framework that retains rigor.

[To do: Analyse Implications of above... The boundary between Real world / Real domains and Theoretical unbounded domains... Computations into unbounded domains will inturn consummate unbounded space time resources? - Refer Addendum A-]

5.8 Evolution and History of Number Systems.

To do...

5.9 Comparative Analysis of DONX Framework against ZFC and Cantonian Framework.

To do...

[Refer Addendum B-]

5.10 What's Next... Refinement of the Framework...

If the framework is found to be standing on solid grounds (no significant non-fixable contradictions...)

To do...

[Refer Addendum C-]

5.11 Summary

The Dimensional Order Numbers Extended (DONX) Framework emerged from a core conceptual challenge in standard set theory: Issues the such of "Disjuncture of the Triangle" and the disconnect between intuitive geometric notions of size and dimension (e.g., a line feels smaller than a plane) and the abstract set-theoretic cardinality (where a line segment, an infinite line, and a plane all have the same "continuum" cardinality, 𝔠). This led to the Atomicity Disjuncture and other "squashing" or "stretching" scenarios.

To address these, the framework introduced Dimensional Order Numbers Extended (DONXs) and the Dimensional Closure Principle (DCP), thus setting the stage for a new approach to classifying infinities.

At the heart of the DONX Framework are the Dimensional Order Numbers Extended (DONXs), defined as bi-infinite sequences of digits extending indefinitely to both the left (integer part) and the right (fractional part).

This foundation allowed for the conceptualization of distinct DONX  Dimensional Cardinalities such as 𝔠_[D1] (for a 1-dimensional continuum like Eℕ) and 𝔠_[D2] (for a 2-dimensional continuum like Eℕ × Eℕ, Eℝ), asserting their fundamental inequality.

Initially, a "Valid Pairing Bijection Function" was explored, requiring linearity to preserve structural integrity. However, this approach revealed limitations, particularly in addressing situations like the "5cm = 10cm" problem (atomicity conjecture) or the equivalence of even and all natural numbers, as standard linear bijections (e.g., f(x) = 2x) would still imply equal cardinality, contradicting geometric intuition about "halving" a set's size.

This critical limitation led to the development of the Geometric Pairing via Induction, which became the definitive method for comparing cardinalities within the framework. This algorithm operates by comparing the fixed, proportional counts of elements within finite "levels" or "blocks" of sets. If this proportionality holds true for all finite steps, the algorithm then axiomatically extrapolates this same proportional relationship to the infinite DON-cardinality of the complete sets.

This definition fundamentally alters how "size" is determined for infinite sets, directly allowing for fractional and multiple cardinalities (e.g.|Even Eℕ| = ½ |Eℕ|), and intrinsically maintaining distinctions between dimensional infinities.

The DONX framework, therefore, establishes a new, geometrically inclined system for classifying and comparing the "sizes" of infinite sets, sidestepping certain complexities and counter-intuitive results of standard ZFC cardinality regarding the continuum.

Addendum A

Refining the Representation of DONX Numbers

The structure of a DONX number:

N = ([0…∞] … i₂ i₁ i₀ . d_-1 d_-2 … [0…∞])

Where:

• The digits … i₂ i₁ i₀ extend infinitely to the left (representing the integer part).
• The digits d_-1 d_-2 … extend infinitely to the right (representing the fractional part).
• And [0…∞] denote infinite strings of zeros on both ends.

Arithmetic with Spatial Complexity

Definition :

1. "Practical" or "Finite-Precision" Addition in DONX

To perform addition of two DONX numbers, say A = [… a₂ a₁ a₀ . a_-1 a_-2 …] and B = [… b₂ b₁ b₀ . b_-1 b_-2 …], with specific spatial complexities:

• Decimal Part Spatial Complexity (say, k digits):  Sum the digits a_i + b_i starting from the rightmost required digit (d_-k) and propagate carries to the left, up to d₀. All digits d_j for j < -k are truncated (ignored for the result).
• Integer Part Spatial Complexity (say, m digits): Sum the digits a_i + b_i starting from i₀ and propagate carries to the left, up to i_m-1. All digits i_j for j ≥ m are truncated (ignored for the result).

• Here we are not computing the "exact, infinite" sum.
• We're computing a finite-precision approximation of the sum.
• This means the result will be a DON number with a finite, specified number of non-zero digits (or non-zero leading/trailing digits, if one retain zeros for padding).

Implications of this approach:

• Finite Time & Space: By definition, a computation with fixed spatial complexity will take a finite amount of time and use a finite amount of memory. This directly addresses the computability concerns.
• Loss of Exactness: The trade-off is that this arithmetic is approximate, not exact. A+B truncated to 9 decimal places and 5 integer places will not necessarily be the same as A+B truncated to 10 decimal places and 6 integer places.
• Practicality: This is how numbers are handled in most engineering and scientific computations.

Summation Multiplication and Comparison for DONXs (Dimensional Order Numbers Extended)

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1. Case 1: Finite Integer + Infinite Fractional DONXs

Structure:

X = a₀.a_-1a_-2a_-3…   where a₀ ∈ ℤ, and a_-i ∈ {0,1,…,b-1}

Algorithm (Addition):

1. Fix precision n (digits after the decimal).
2. Truncate X and Y to n fractional digits:
   X_n = a₀.a_-1…a_-n,   Y_n = b₀.b_-1…b_-n
3. Compute Z_n = X_n + Y_n using standard arithmetic.
4. Check convergence:
   If |Z_n – Z_n+1| < ε (requisite precission), return Z_n.
   Else, increment n and repeat.

Example:
Add X = 1.111… and Y = 2.222…:
- At n=3: 1.111 + 2.222 = 3.333
- At n=4: 1.1111 + 2.2222 = 3.3333
- Limit: X + Y = 3.333… (repeating)

Key Properties:

• Well-defined for convergent series (e.g., repeating decimals).
• Divergence detected if |Z_n| → ∞.

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2. Case 2: Infinite Leftward DONXs

Structure:

W = … w₂w₁w₀.w_-1…   (with infinitely many nonzero digits to the left)

Arithmetic Rules:

1. Addition/Multiplication:
   Any operation involving an infinite-leftward DONX yields infinity (or another infinite-leftward DONX).
   Example:
   (…111.1) + (…222.2) = …333.3   (still infinite, digit-wise valid).
2. Comparison:
   Use lexicographical order (left-to-right).
   Example: …999.9 > …999.1

Key Properties:

• No convergence: All such numbers are "infinite" in magnitude.

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3. Implementation Notes

(A) Precision Handling via convergence

• For fractional DONXs, use adaptive precision (increase n until error < ε).
• Example pseudocode:

def add_dtn(X, Y, epsilon=1e-6, max_iter=1000):
    n = 1
    while n <= max_iter:
        Z_n = truncate(X, n) + truncate(Y, n)
        if abs(Z_n - truncate(X, n+1) - truncate(Y, n+1)) < epsilon:
            return Z_n
        n += 1
    raise DivergenceError("Sum does not converge to precision.")

(B) Infinite-Leftward Warnings

• Operations on infinite-leftward DONXs return symbolic infinity (or propagate infinite structure).

DONX Convergence

Consider the DONXs generated by Σ 1/(2r) and Σ k/(2r) where r = 1 → ∞ and k = constant. The partial sum Σ 1/(2r) converges toward 1, and partial sum Σ k/(2r) converges toward k. The addition of the two partial sums would converge toward (k+1). The subtraction of the two partial sums would converge toward either  ±(k-1), multiplication towards k and division towards either of k or 1/k.

Multiplication of DONX (numbers) via Convolution

The formula for convolution-based multiplication algorithm for Dimensional Order Numbers Extended (DONXs).

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1. The Convolution Formula

Given two DONXs:

A = (… a₂a₁a₀),   B = (… b₂b₁b₀)

Their product C = A × B is computed as:

C = (… c₂c₁c₀),   where   c_k = ( Σ_i+j=k a_ib_j + carry ) mod b_

b_ being the base.

Key Components:

1. Digit-wise Multiplication:
   For each digit position k in C, sum the products a_i × b_j where i + j = k.
   Example: c₃ = a₀b₃ + a₁b₂ + a₂b₁ + a₃b₀.
2. Modulo Operation (mod b):
   Ensures each digit c_k stays within the base b (e.g., b_ = 10 for decimal).
   Example: If b_ = 10 and a₂b₁ = 24, then 24 mod 10 = 4.
3. Carry Propagation:
   The integer division ⌊sum / b_⌋ becomes the carry to the next higher digit.
   Example: If Σ a_ib_j = 13 and b_ = 10, then c_k = 3 and carry = 1.

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2. Step-by-Step Example

Let’s multiply two DONXs in base b = 10:

A = (… 0 0 3 2),   B = (… 0 0 4 1)
(These represent finite numbers: A = 32, B = 41).

Step 1: Compute c_k for each digit position

k	Pairs (i,j)	Calculation	Sum + Carry	ck	New Carry
0	(0,0)	2 × 1 = 2	2 + 0 = 2	2	0
1	(0,1), (1,0)	2 × 4 + 3 × 1 = 11	11 + 0 = 11	1	1
2	(0,2), (1,1), (2,0)	2 × 0 + 3 × 4 + 0 × 1 = 12	12 + 1 = 13	3	1
3	(1,2), (2,1), (3,0)	3 × 0 + 0 × 4 + 0 × 1 = 0	0 + 1 = 1	1	0

Result:
C = (… 0 0 1 3 1 2), which is 1312, matching 32 × 41.

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3. Special Cases & Edge Conditions

(A) Infinite Leftward DONXs (e.g., …111)

• The convolution sum becomes an infinite series for each c_k.
• Solution: Truncate to a finite depth n for practical computation.

(B) Fractional DONXs (e.g., 0.333…)

• Extend the convolution to negative indices (right of the decimal):

c_-k = Σ_i+j=-k a_ib_j mod b_

• Example: 0.333… × 0.666… converges to 0.222…

(C) Modulo Requirement

• Why modulo?
  Ensures digits stay in [0, b_ -1].
  Without it, digits could overflow (e.g., 7 × 8 = 56 in base 10 would need c_k = 6 and carry = 5).

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4. Algorithm Pseudocode

def multiply_donx(A, B, base=10, max_iter=1000):
    C = []
    carry = 0
    for k in range(-max_iter, max_iter + 1):  # Cover all digit positions
        total = carry
        for i in range(-max_iter, max_iter + 1):
            j = k - i
            if -max_iter <= j <= max_iter:
                total += A[i] * B[j]
        digit = total % base
        carry = total // base
        C.append(digit)
    return C

---

The convolution formula:

1. Multiplies digit pairs (a_i, b_j) where i + j = k.
2. Uses modulo to keep digits valid in base b_.
3. Propagates carries to higher digits.

This method works for both finite and infinite DONXs, provided convergence is handled carefully.

Operational Semantics for DTNs and DONXs

Operand Classes	Addition	Subtraction	Multiplication	Division
DONX only (no ∞/↻)	Standard DONX arithmetic	Standard DONX arithmetic	Standard DONX arithmetic	Standard DONX arithmetic
Left ∞/↻ only (infinite integer part)	Result = ∞ (asymptotic sum)	Depends on cardinalities: - Equal: 0 - DTN₁ > DTN₂: ∞ - DTN₁ < DTN₂: 0	Result = ∞ (asymptotic product)	Depends on cardinalities: - DTN₁ > DTN₂: ∞ - DTN₁ < DTN₂: 0 - Equal: finite? (depends on convergence)
Right ∞/↻ only (infinite fractional part)	Check convergence: - Converges: use converged value - Divergent: undefined / declare divergence	Check convergence: - Converges: finite result - Divergent: undefined	Check convergence: - Converges: finite product - Divergent: undefined	Check convergence: - Converges: finite quotient - Divergent: undefined
Both sides ∞/↻ (full DTN)	Combination of left & right rules; asymptotic left dominates	Combination of left & right; left dominates for infinite part	Combination; left dominates, multiply fractional if convergent	Combination; left dominates, divide fractional if convergent

DTN Classes (Dimensional Transfinite Numbers)

DTNs are classified based on where ∞/↻D appears:

• F — Finite/DONX: no transfinite marker on either side (ordinary DONX arithmetic).
• R∞ — Right-transfinite only: fractional tail is infinite (treat via limits/convergence).
• L∞ — Left-transfinite only: integer side is infinite (treat via growth/cardinality/asymptotics).
• LR∞ — Both sides transfinite: combine L∞ dominance (for growth) with R∞ convergence (for tails).

Philosophy: Left ∞ = magnitude/growth, Right ∞ = limit/convergence, None = inherit DONX.

Canonical Measures

• Left-growth rate γ(x) for L∞/LR∞: asymptotic digit generation rate.
• Right-limit for R∞/LR∞: arithmetic defined as limit of partial truncations, with chosen resolution.

Operations Table

Operation	Operands	Result Class	Rule / Intuition
Addition	F + F	F	Ordinary DONX arithmetic.
Addition	F + R∞	R∞	Limit of partial sums; converges to a real.
Addition	F + L∞	L∞	Dominated by left-infinite magnitude.
Addition	R∞ + R∞	R∞	Sum of two real limits.
Subtraction	L∞ − L∞	F, 0, or L∞	Use left-growth comparison γ(x)/γ(y) for scale; if equal, may collapse to finite residue.
Multiplication	L∞ · L∞	L∞	Left-infinite magnitude times left-infinite, stays L∞.
Division	L∞ ÷ L∞	F, 0, or ∞	Use growth ratio γ(x)/γ(y) to determine scale; 0, or ∞ if ratio is extreme, else F.

Oscillation & Convergence

Right-tail generators may oscillate. Operations are defined only if the sequence of partial results is Cauchy under decimal metric. Fixing k fractional digits bounds the subtraction error by 10-k.

Example: Compare 5.23453... and 5.22222 with first 5 fractional digits known:

• Difference at 5 digits: 0.01231
• Unknown tail adds at most 10^-5
• True difference ∈ [0.01231, 0.01232)

If only 4 digits fixed, range ∈ (0.0122, 0.0124). This matches the intended resolution rule.

Summary

• Class detection based on ∞ markers: F, R∞, L∞, LR∞
• Right arithmetic: limits of truncations with resolution
• Left arithmetic: growth comparison γ(x), limsup ratio for division
• Cross-class operations: left growth dominates, right tails ride along when finite scale occurs

Axiom: The DONX Transfinite Inductive Extrapolation Principle

Let S represent a mathematical structure or system defined within the DONX Framework (e.g., sets of DONX numbers, operations on DONX numbers, or relationships between them). We assume that S can be rigorously conceptualized through a canonical sequence of finite approximations or "levels," indexed by natural numbers n ∈ N.

For any well-defined Finite-Level Proposition P(n), which asserts a specific property or relationship concerning the finite approximation of S at level n, there exists a corresponding Transfinite Proposition P∞

The Transfinite Proposition P∞  is the direct extension or limit of P(n), applying the same property or relationship to the full, infinite (transfinite) form of S that is completed or derived from its finite approximations.

Assertion:

If the Finite-Level Proposition P(n) is true for every natural number n∈N (i.e., ∀n∈N, P(n) holds true), then it is axiomatically asserted that the corresponding Transfinite Proposition P∞ is true.

The Principle of Mathematical Induction (The Axiom):

"If a property P holds for n=0 (or n=1), AND if for every natural number k, the truth of P for k implies the truth of P for k+1, THEN P holds for all natural numbers n."

Next via axiom of DONX Transfinite Inductive Extrapolation, we axiomatically  extrapolate the deduction P, to hold for all n ∈ EN.

Distinguishing Between Ideal and Practical Arithmetic in Eℕ

1. Ideal (Exact) Eℕ Arithmetic:

• The mathematical definition of addition for two full, infinite-length DONX numbers.
• This operation conceptually exists, even if its "computation" takes infinite time.
• How it works (conceptually):
  • Fractional Part: Summing 0.d_-1 d_-2 … with 0.e_-1 e_-2 … results in 0.s_-1 s_-2 … by propagating carries leftward (from infinitely far right). This is standard for real number addition.
  • Integer Part: Summing … i₂ i₁ i₀ with … e₂ e₁ e₀ results in … s₂ s₁ s₀ by propagating carries rightward (from i₀). This is analogous to how p-adic numbers are added, and it results in a well-defined infinite sequence to the left.
• The result of this ideal addition is always another full DONX number. This maintains the closure and completeness of DONX.

2. Practical (Approximate) Eℕ Arithmetic (with Spatial Complexity):

• It's an algorithm for computing an approximation of the ideal sum.
• It's performed with a defined "spatial complexity" (precision/truncation limits for both integer and fractional parts).
• This produces a finite-representation DONX number within the chosen limits.

Resolves The Following:

• "ℕ crumbles at the limits of ω". Eℕ explicitly allows for numbers that have an infinite string of digits, transcending the "finiteness" of classical natural numbers.
• "Boundedness": The numbers themselves in Eℕ are still unbounded (they are infinite sequences). It's only the computation of their exact sum that forces you to choose a finite bound for practical purposes. This doesn't make the numbers themselves "bounded."

The "loss of exactness" is contingent on the nature of the DONX numbers itself:

1. For DONX numbers with finite, terminating representations:
   Like the example 5950.123456789. This is represented in DONX as [0…∞]5950.123456789[0…∞].
   If one adds two such numbers, and the chosen "spatial complexity" (precision) is sufficient to capture all their non-zero digits (i.e., beyond the last 9 and the first 5), then the result will be exact and terminate with [0…∞]. There is no loss of information for such numbers. This requires only a finite spatial complexity, and thus a finite time domain.

2. For truly unbounded DONX numbers (like π):
   Represented as [0…∞]3.1415926535...

3. Here, any finite spatial complexity will indeed result in an approximation. One will lose information beyond the chosen limit, and the result is not exact.
   
   

• DONX explicitly defines and accommodates these "unbounded" numbers. It provides the mathematical "space" for them to exist in their full, infinite extent.
• Exact arithmetic for these truly unbounded numbers is an idealization that requires unbounded time. It's a conceptual operation within the DONX framework, even if practically unachievable in finite space (ex. memory / storage) nor finite time.
• For practical purposes, finite spatial complexity provides necessary approximations. This bridges the gap between the theoretical exactness of DONX and the realities of computation.

• The two-tiered view of arithmetic:
  • Ideal Arithmetic: Exact, conceptually performed on the full infinite sequence, requires unbounded space-time for truly unbounded numbers. This is where DONX truly embraces "infinity and beyond."
  • Practical Arithmetic: Approximate, performed within a defined "spatial complexity" (finite precision), yields results in finite time. This is for the "finite chronological / time domain."

Addendum B

Consider the following right angled triangle. Observe that when 𝛂 → 0, we get / satisfy the axiom of ZFC (viz. the set of Natural numbers can extend upto infinity but their magnitude have to be finite.) Thus by appropriately selecting a suitably small value of 𝛂 we can obtain magnitudes of base (b), to satisfy the Natural Numbers of ZFC.

But the moment  𝛂 > 0, the magnitude of the Natural Numbers representing b, goes out of scope / bounds of Natural Numbers (when h→∞).

And the above clearly describes as to why Cantor's Diagonal Argument works under ZFC (No Natural Number is allowed to extend upto infinity in magnitude under axioms of ZFC).

The following example will further illustrated this point.

Assume a set of numbers, which is only allowed to extend upto 10 digits (aka. finite magnitude).

0000000000

1010000001

0110000000

1111110000

0000011111

1111011111

0000000111

1100110011

0000001111

1000000000

Now when we flip the diagonal values in the above finite set, the new number 1100101101, is not present in above finite set.

This is why Cantor's Diagonal Argument holds under the axioms of ZFC.

Mapping Between ℝ[0,1) and Eℕ (Bijection), via Cantonian bijection: 

For any Eℕ element o = (...i3 i2 i1 i0), define g(o) as the real number 0.d0 d1 d2 d3… such that i_n = d_n.

Bijection Confirmation: This establishes a perfect one-to-one correspondence (bijection) between the set of real numbers in ℝ[0,1) and Eℕ. Therefore, within The DONX Framework, |ℝ[0,1)| = |Eℕ|.

Since |ℝ[0,1)| = 𝔠 (the cardinality of the continuum) 

|Eℕ| = 𝔠

Formal Addendum B: From Geometric Analogy to Rigorous Proof

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1. Formalizing the Triangle Analogy

Claim:

For a right-angled triangle with base b, height h, and angle α:

• If α →0, b remains finite as h → ∞ (ZFC-compliant).
• If α >0, b → ∞ as h → ∞ (requires DONX’s infinite magnitudes).

Proof:

1. By definition, tan α = b / h.
2. As h → ∞:
   • Case α > 0: tan α > 0, so b = h · tan α → ∞.
     But in ZFC, b must be finite for any h ∈ ℕ. Contradiction!
     Resolution: ZFC implicitly assumes α → 0 as h → ∞ to keep b finite.
   • Case α > 0: tan α > 0, so b → ∞.
     DONX allows b ∈ Eℕ to be infinite.

Conclusion:
ZFC’s natural numbers are the α → 0 limit of the triangle, while DONX permits α to  increase beyond 0.

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2. Cantor’s Diagonal Argument in DONX

ZFC’s Version:

• Assume a list f: ℕ → [0,1) exists.
• Construct r ∈ [0,1) by flipping the n-th digit of f(n).
• r cannot be in the list because it differs from every f(n).

DONX’s Counter:

• Replace ℕ with Eℕ. Now f: Eℕ → [0,1).
• The diagonal number r is (... d₂ d₁ d₀), where dₙ is the flipped n-th digit of f(n).
• But r ∈ Eℕ by construction! Thus, r is in the "list" (which is now a complete Eℕ-indexed set).

Key Insight:
In ZFC, ℕ cannot index all infinite-digit sequences (uncountability).
In DONX, Eℕ can index them (|Eℕ| = 𝔠), so diagonalization fails.

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3. Rigorous Bijection Proof: ℝ[0,1) ↔ Eℕ

Definition:
Define g: Eℕ → [0,1) as:
g(... i₂ i₁ i₀) = 0.i₀ i₁ i₂ ...

Proof Steps:

1. Injectivity:
   If g(o₁) = g(o₂), then their digits satisfy i_k,1 = i_k,2 for all k, so o₁ = o₂.
   
2. Surjectivity:
   For any r = 0.d₀ d₁ d₂ ... ∈ [0,1), construct o = (... d₂ d₁ d₀) ∈ Eℕ. Then g(o) = r.
3. Cardinality:
   g is a bijection, so |Eℕ| = |ℝ[0,1)| = 𝔠.
   
   

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Addendum B (Formalized Version)

Theorem 1 (Triangle Constraint):
In ZFC, the set { b ∈ ℕ | b = h · tan α, h ∈ ℕ } is finite for any α > 0 as h → ∞.
Proof: ZFC’s axiom of infinity enforces b < ∞, requiring tan α = o(1/h).

Theorem 2 (Diagonalization in DONX):
Let f: Eℕ → [0,1) be any function. The diagonal number r constructed by flipping digits of f(n) is in Eℕ, so f cannot be surjective.
Proof: By construction, r ∈ Eℕ, but r ∉ image(f) would contradict Eℕ’s completeness.

Theorem 3 (Bijection):
The map g: Eℕ → [0,1) defined by digit reversal is a bijection.
Proof: As above.

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5. Implications for DONX

1. ZFC is a Subcase: When α → 0, DONX reduces to ZFC-like finiteness.
2. Dimensionality ≠ Cardinality:
   • Eℕ is 1D but has 𝔠 elements because its "infinite depth" compensates for lacking a second dimension.

Addendum C

To Do... Check the mathematical validity of the following "induction technique"... (Mathematical Induction ver.2)

First prove for base cases (k = 0, 1), then prove for k=m, m+1

-At end of step (k=1):

IR(1) = Items of ℝ[0,1) : 0 1 2 3 4 5 6 7 8 9

IEN(1) = Items of  Eℕ : 0 1 2 3 4 5 6 7 8 9

|IR(1)| = |IEN(1)|     -----(1)

-At end of step (k=2):

IR(2)  = Items of ℝ[0,1) : 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 [...all 100 branches]

IEN(2) = Items of  Eℕ : 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 [...all 100 branches]

|IR(2)| = |IEN(2)|     -----(2)

-At end of step (k=m):

Items of ℝ[0,1) : 0 1 2 3 4 5 6 7 8 9 [...upto all 10^m branches of level m]

Items of Eℕ: 0 1 2 3 4 5 6 7 8 9 [...upto all 10^m branches of level m]

|IR(m)| = |IEN(m)|     -----(3)

-At end of step (k=m+1):

Items of ℝ[0,1) : 0 1 2 3 4 5 6 7 8 9 [...upto all 10^m+1 branches of level m+1]

Items of Eℕ : 0 1 2 3 4 5 6 7 8 9 [...upto all 10^m+1 branches of level m+1]

|IR(m+1)| = |IEN(m+1)|     -----(4)

Thus (1) (2) (3) (4) when Applied to "Mathematical Induction ver.2" : yields ?

for all n ∈ Eℕ : |IR(n)| = |IEN(n)| 

Thus  |ℝ[0,1)|= |Eℕ|  (= 𝔠_[D1] : via def.)

Example 2:

Compare the Cardinality of all Even Extended Natural numbers and Extended Natural Numbers.

Let EENn represent the Set of Even EN upto index n.

Let ENn represent the Set of EN upto index n.

Let assign step = 100 (scan items in units of 100).

Next let's apply the DONX Geometric Pairing algorithm to sets EENn and ENn

-At end of step (k=1):

EEN1 [0,100) = {0,2,...,96,98}

EN1 [0,100) = {0,1,...,98,99}

| EEN1 | = 50

| EN1 | = 100

| EEON1 | = 1/2 | EN1 | ---(1)

-At end of step (k=2):

EEN2 [100,200) = {100,102,...,196,198}

EN2 [100,200) = {100,101,...,198,199}

| EEN2 | = 1/2 | EN2 | ---(2)

-At end of step (k=m):

EENm [(10*m),(10*m)+100) = {(10*m),(10*m)+2,...,(10*m)+96, (10*m)+98}

ENm [(10*m),(10*m)+100) = {(10*m),(10*m)+1,...,(10*m)+98, (10*m)+99}

| EENm| = 1/2|ENm | ---(3)

-At end of step (k=m+1):

EENm+1 [10*(m+1),10*(m+1)+100) = {10*(m+1),10*(m+1)+2,...,10*(m+1)+96, 10*(m+1)+98}

ENm+1 [10*(m+1),10*(m+1)+100) = {10*(m+1),10*(m+1)+1,...,10*(m+1)+98, 10*(m+1)+99}

| EENm+1 | = 1/2 | ENm+1 |  ---(4)

Thus (1) (2) (3) (4) when Applied to "Mathematical Induction ver.2" : yields ? 

for all n ∈ Eℕ : |EEN(n)| = 1/2 |EN(n)| 

Thus |EEℕ|= 1/2|Eℕ| 

|Eℕ| = 𝔠_[D1] : via definition

Thus : |EEℕ| = 1/2 𝔠_[D1]

Disjuncture of the Triangle (Revisited) :

Consider a right-angled isosceles triangle where the two equal sides constitute base length : b and height : h (visualise a ramp)

Now, let the base length b, be restricted to increments within the set of natural numbers, such that b ∈ ℕ = {1, 2, 3, …}. As the set of natural numbers is infinite, the base length b can increase without bounds (i.e. b → ∞). Consequently, the corresponding height h, also increases without bound, approaching infinity (h → ∞).

The set of all possible heights derived from this process { 1, 2, 3, … }, is unequivocally the Set of natural numbers, ℕ. However, by the standard definition of the natural numbers, every individual element within ℕ is a finite value. This presents a conceptual discontinuity: while the geometric construction demonstrates a quantity (height) capable of unbounded growth, approaching infinity, the formal mathematical set (ℕ) intended to contain these values does not include any elements representing actual infinity. 

This observation points to an inherent tension between unbounded processes and the finite nature of elements within the axioms of ZFC.

[This Journey began with a necessity to find a solution to the Disjuncture of the Triangle. Thus we end this article with a restatement of the point of origin...]

Author:

Ly De Sandaru

In a collaboration with: ChatGPT, Gemini, Grok, DeepSeek (AI Learning & Research Assistants)

[Note: For a detailed description of AI contributions kindly refer following:

ChatGPT:

https://lydesandarureviews.blogspot.com/2025/07/reviews-paths-of-refinements-of-infinity.html 

https://lydesandarureviews.blogspot.com/2025/08/don-naming-misnomer.html

Gemini:

https://lydesandarureviews.blogspot.com/2025/07/review-gemini-to-infinity-and-beyond.html 

https://lydesandarureviews.blogspot.com/2025/08/reviews-gemini-don-geometric-pairing.html

Grok:

https://lydesandarureviews.blogspot.com/2025/08/reviews-grok-on-donx-framework.html

DeepSeek:

https://lydesandarureviews.blogspot.com/2025/08/reviews-deepseek-on-donx-framework.html

]