Let's Branch Into Infinity

Introduction:

Let’s start with the Champernowne constant, which is defined as:

C = 0.12345678910111213141516…

(Note : This number is constructed by concatenating all natural numbers after the decimal point.)

Next let's remove the decimal point:

You get: 12345678910111213141516…

In standard mathematics, Natural Numbers are defined to have finite length. That is, every natural number is made of finitely many digits.

So the number we just created:

1234567891011121314… continuing forever

is not a Natural Number in the usual sense - because it has infinitely many digits.

In fact, this object isn't even a Real Number anymore without a decimal point - it's more like a formal sequence, and it doesn't belong to ℕ or ℝ unless you define a new space to contain such objects (e.g., sequences, formal strings, or base-10 expansions with infinite length).

Key points:

1. Champernowne constant is a Real Number, because it has a well-defined infinite decimal expansion.

2. When you remove the decimal point, you are left with an infinite digit sequence, which cannot be treated as a Natural Number under standard definitions.

What Are Natural Numbers in Standard Mathematics?

In standard set theory, the Natural Numbers are defined inductively:

0∈ℕ

If  n∈ℕ, then n+1∈ℕ

This means:

Every natural number is built in finite steps from zero.

So the set 

ℕ={0,1,2,3,…} only includes numbers that can be reached in finitely many additions.

There are infinitely many of them, but each individual number is finite.

What Does “Finite Number” Mean Here?

A number is finite if:

It has a finite number of digits in base 10,

Or can be constructed with a finite description (like 7 or 10¹⁰⁰).

A number like:

123456789101112131415…(all natural numbers concatenated forever) is not finite - it has infinitely many digits. Therefore, it is not a member of ℕ.

Georg Cantor was operating within a framework where:

The Natural Numbers were what we now call ℕ - the finite, countable ordinals.

He explicitly distinguished between countable sets (those that can be placed in 1-to-1 correspondence with ℕ) and uncountable ones (like ℝ).

In his famous 1891 paper on the diagonal argument, he assumed the listing of Reals (between 0 and 1) was indexed by Natural Numbers - and the contradiction arises because this finite indexing cannot capture all Real Numbers, some of which require infinite precision.

If We Allowed “Infinite Natural Numbers”?

That would change everything.

If we define a new system where we allow “infinite Natural Numbers” (like the concatenated Champernowne number), then we’re leaving the domain of standard ℕ, and moving into:

Ordinal numbers (e.g., ω, ω+1, etc.)

But Cantor's theorems - especially about countability - all assume standard ℕ with only finite members.

Next take a look at the following :

Statement 1: The set ℕ (Natural Numbers) is countably infinite - meaning it has infinitely many elements.

Statement 2: Each element of ℕ is a finite natural number.

Wouldn't above statement 1 and 2, lead to a logical contradiction?

For example consider a right angled isosceles  triangle. Let the height of the triangle represent the Natural Number's index. And let the base of the triangle represent the size of the said Natural Number. Thus as Natural Number Set can have infinite elements, the size of the numbers should increase upto infinity as well (aka invalidating statement 2 above?)

So is it a contradiction?

No, not in formal set theory.

It seems paradoxical, but it’s not - because "infinite" applies to the set ℕ, not to its elements (via definition of ℕ).

But mathematics does allow infinite sets whose elements are themselves infinite in size or value - just not in ℕ.

So in modern set theory, there absolutely exist infinite sets whose elements are themselves infinite in size, length, or value.

But ℕ is not one of those - by definition.

(We repeated the above a few times, as it's fundamental to grasp this point...)

That’s why the (Right Angled Isosceles) Triangle Analogy hits a philosophical paradox in ℕ : the numbers (index) grow forever, yet the numbers are never individually allowed to be infinite (in size) - via definition of ℕ -

To address such Sets, one option would be to use Ordinal numbers, the such of ω. But these numbers are not the most intuitive and requires in-depth perusal...

THUS...

To address the above "Triangle Paradox" (aka. The widths of a triangles whose heights grows to infinity cannot be represented in ℕ)... This article presents a Dimensional  Ordinal Number System with Infinities of Degree 1, 2, 3...

[Note : This article  is a continuation / addendum to the following :

https://projectsofacademicinterests.blogspot.com/2025/03/a-new-perspective-on-infinity-via-2d.html ]

Dimensional Ordinal Numbers (DONs)

Ordinal Numbers of First Dimension and Ordinal Infinity of Degree 1.

This is the case of a 1D Hilbert Hotel, where we can have infinite rooms and the ID of the Rooms (the Room Numbers) can also extend to infinity.

Another example: The Set of triangle's base widths, whose height increases to infinity (in unit steps...) can be represented as Ordinal Numbers / Infinity of Degree 1.

Another example : The Index of the books in a library bookshelf, that extends to infinity, can be expressed / represented by Ordinal Numbers of First Dimension / Infinity of Degree 1.

Thus under this definition Ordinal Natural Numbers can have a value that extends to  Ordinal Infinity of Degree 1 (unlike in ℕ - where the definition prohibits such)

Ordinal Numbers of Second Dimension and Ordinal Infinity of Degree 2.

This would be the case of a 2D Hilbert Hotel. Thus in this scenario, we have infinite rooms along a corridor along the x direction. And then, adjacent to each of the above rooms, there is a corridor into the y direction. And those corridors themselves extends to infinity.

Now observe that we have a matrix / array of rooms (having infinite rooms in both x and y directions).

Now in this scenario, the rooms of this 2D Hilbert Hotel can reach Ordinal Infinity of Degree 2 (as described below...)

Consider our 2D Hilbert Hotel...

In the corridor along x axis, we have infinite rooms, each having an Index (room number) in our Ordinal Natural (Oℕ) Number Set (aka. room number also can be infinite). Thus room numbers will be {0,1, 2, ....  ∞ (or infinite digits)}

And along each corridor that extends in the y direction again we have room numbers that can extend upto infinity aka. {0,1, 2, .... ∞ (or infinite digits)}

And along the corridor (in the y direction) between room 0 and room 1, the room number allocations could be : 0.0, 0.1, 0.2 ... 0.9, 0.10, 0.11, 0.12, ... 0.19, 0.20, 0.21, 0.22,... 0.29, ... (upto infinite digits)

Thus taking a cursive look /birdeye look at this 2D Hotel, we can easily Deduct / Observe that this Hotel must have ∞ * ∞ or ∞² rooms.

Definition:

Thus we define ∞ * ∞ or ∞² as an Ordinal Infinity of Degree 2.

Point to ponder : Reflect as to why all Infinities are unbounded / uncountable / unreachable and yet, some Infinities are larger than others.

(And next we can even extend this hotel to 3D and obtain Ordinal Infinity of Degree 3 etc...)

Definition:

Ordinal Natural Numbers (Oℕ)

0 ∈ Oℕ

If  I_n ∈ Oℕ , then (I_n+1 = I_n + 1) ∈ Oℕ

When n →∞ , I_n →∞

Having set this Foundation / Definition of Ordinal Numbers and Ordinal Infinity, let's next move to prove / deduct that under these definitions (under these terms), Real Numbers ℝ, falls into the set of Ordinal Numbers of Degree 2.

Real Numbers ℝ, falls into the Set of Ordinal Numbers of Degree 2.

Proof (Soft Proof...)

Lets consider a Decimal Tree (N Tree / N-ary Tree where N=10). Thus in this tree we have a root (at level 0) and 10 branches extending from root (to level 1). And then from each of these branches extends another 10 branches. Thus at level 2 we have a total of 100 branches (10²). And so on... 

And thus at level n we should have 10ⁿ branches...

Next let's take the Real number 0.25391

The above number can be represented / located in our Decimal Tree, as follows :

0 will be at level 0 (the root)

2 will be at level 1 branch 3

5 will be at level 2 branch 6

3 will be at level 3 branch 4

9 will be at level 4 branch 10

1 will be at level 5 branch 2

Likeso any Real Numbers can be represented in our Decimal Tree. To represent more digits, we only need to expand the levels / height of the tree.

(One can have one Decimal Tree for the digits, infront of the decimal point -integer part- and another tree for digits after the decimal point -fractional part-.)

[One may ask, what about the digits after the last digit of our number 0.25391?

Let's me answer that with a story : A curious young boy once asked : why doesn't our earth fall into the void of space? And this individual answered : son the earth is held on the back of a giant tortoise. The boy's curiosity searing into new heights, asks : but then... what is the tortoise standing on? The person maintaining his calm responds : son... it's tortoises all the way down...

Likeso... What's after the last digit? : it's zeros all the way down...]

Now where were we... Ah yes the Tree of Ordinal Numbers...

Now observe that in Georg Cantor's take on Real Numbers ℝ, he dwelt into them in a depth down or depth first approach of our decimal tree. And guess what... This tree has no known finite depth. Aka it can have infinite digits, thus the depth of this decimal tree itself is infinite. Thus attempting to reach the bottom of this tree is a futile exercise in a finite space-time. (Thus he concluded / deducted that Real Numbers are uncountable.)

THUS... Now that we know that, that parth entraps / binds one to bottomless "Fractal Geometries"...

Let's do something else, take a different approach...

How about we instead, traverse this tree breath wise (breath first search...)

Also Theory of Data Structures dictates that any Tree (without any recursive branching), can be expressed in an equivalent Linear Array.

Thus the Linear Array of a Decimal Tree that can represent all the possible first digits is:

0 1 2 3 4 5 6 7 8 9

And Linear Array that is the equivalent of a Decimal Tree that can represent all the first two digits will NEXT have all the digits represented by the 100 branches of level 2 of our Tree...

Aka:

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 untill all the 100 branches are included...)

And next, the Linear Array that can represent all the number combinations upto 3 digits will be:

(Left as an optional exercise for the reader...)

Thus using this Tree-List Notation let's dig into another round of Ordinal Natural (Oℕ) and Ordinal Real (Oℝ) Numbers.

Let's consider Ordinal Natural (Oℕ) number 52 and Ordinal Real (Oℝ) number 0.52

Next let's built 2 sets of Decimal Trees and their equivalent Lists. 

Thus now we have two Lists and two Trees. One set representing Ordinal Natural (Oℕ) Numbers (also known as Positive Integers in English) and the other the Fractional part of Ordinal Real (Oℝ) Numbers (English : Decimal or Fractional).

Thus now we can locate any Ordinal Natural  Number (ex. 52), both in it's Tree as well as List.

Likeso we can also locate any Ordinal Real Number (ex. 0.52) in the respective Tree and List.

(Note: To Represent a Real Number the likes of 7.52 in this representation, we will require two Trees or two Lists. One for the Integer part and one for the Fractional / Decimal part)

Thus when one observes carefully a few very subtle and yet elegant patterns arises.

Any Ordinal Real (Oℝ) Number, can be located in the Decimal Tree, when one opts to do a breath first search of the digits (without getting lost into the recursive infinity of depth first search).

Next let's jot down the Linear List representation of Integer part of Ordinal Real (Oℝ) Numbers : List I-Oℝ

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 untill all the 100 branches of level 2 are included...) 0 1 2 3 4 5 6 7 8 9 ... (Repeat untill all the 1000 branches of level 3 are included...) ...

Next let's jot down the Linear List representation of Decimal / Fractional part of Ordinal Real (Oℝ) Numbers : List D-Oℝ

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 untill all the 100 branches of level 2 are included...) 0 1 2 3 4 5 6 7 8 9 ... (Repeat untill all the 1000 branches of level 3 are included...) ...

Observe that Lists I-Oℝ & D-Oℝ are identical.

Ie. There is a 1-1 or Linear Association between items in above two Lists.

Thus it becomes apparent that:

Provided the Integer part of a Real Number is kept constant, the Fractional part of the Number (with finite or infinite digits), has a 1-1 association with the finite or infinite digits of Ordinal Natural (Oℕ) Numbers (the Tree-List of Integers).

(In the 2D Hilbert Hotel analogy : The Ordinality along any corridor along the y axis is equal to the Ordinality along the corridor in the x axis.)

Thus analysing the above, the following  observations comes to light :

1 : Integers fall into the Set of Ordinal Natural (Oℕ) Numbers of Degree 1 (with an Ordinal Infinity of Degree 1)

2: The fractional part of any  Real Number ℝ (as well as fractional part of any Ordinal Real (Oℝ) Number) has a 1-1 association with the Set of Ordinal Natural (Oℕ) Numbers.

3: All Real numbers in ℝ, falls into the Set of Ordinal Real (Oℝ) Numbers of Degree 2. (Refer a 2D Hilbert Hotel...)

Rationale for point 3: A Real Number will have an Integer part and fractional part. When we construct a Tree-List for the Integer part, for each of those Integers we next get an associated Tree-List representing the decimal / fractional part.

In the 2D Hilbert Hotel analogy, we have a corridor extending in the y direction between each two rooms of the corridor along x direction. Thus in Total we have a product of ∞ elements in x direction and another ∞  in y direction. 

Likewise the Tree / List of Integers and Tree / List of Decimals / Fractions has ∞ elements...

Thus we can deduct that the Set of Real Numbers (made via the combination of Integers and Fractions) has ∞ * ∞ or ∞² elements.

For a further discussion of this topic, kindly refer :

https://projectsofacademicinterests.blogspot.com/2025/07/the-dimensional-closure-principle.html

[Point to ponder: When we allocate the room numbers of the 2D Hilbert Hotel (along corridors in y direction) or even when constructing Real Numbers from the Linear List, should we allow / consider numbers the such of n.00, n.000, ... , n.10, n.100, ... or should we omit them?

Hint / Answer: Wether we include or omit them, there is still a linear association, between the Set of the Fractional / Decimal part of a Real Number and the Set of Ordinal Natural (Oℕ) Numbers.]

Ly De Sandaru