A New Perspective on Infinity via a 2D Hilbert Hotel

Challenging Cantor: A New Perspective on Infinity via a 2D Hilbert Hotel

Synopsis Part I:

Georg Cantor's diagonal argument is widely used in Set Theory, to demonstrate the uncountability of real numbers. However, a critical examination reveals potential flaws in its logic. This article presents a compelling alternative perspective, using a "2D Hilbert Hotel" analogy to visualize the relationship between natural and real numbers, offering a more intuitive perspective of infinite cardinalities.

[Def : Any set X that has the same cardinality as the set of the Natural numbers, or | X | = | ℕ | = ℵ₀ is defined as a countably infinite set.]

In-Depth Detail:

Introduction:

Georg Cantor's diagonal argument is widely used in literature as a proof of the uncountability of real numbers, establishing that the set of real numbers is a "larger" infinity than the set of natural numbers. 

(Note: This diagonal argument was a simplified approach to this validation, which Greg Cantor introduced later.)

This article presents a critical analysis of the diagonal argument and introduces a novel approach using a "2D Hilbert Hotel" analogy to provide a more intuitive and robust understanding of infinite cardinalities.

The Diagonal Argument Under Scrutiny:

Cantor's diagonal argument relies on a proof by contradiction, assuming that real numbers can be listed and then constructing a number that is demonstrably not in that list. However, this argument faces several challenges:

The Problem of "Listability": The assumption that an infinite set of real numbers can be completely listed is inherently problematic. Even if a list is assumed, the constructed "diagonal number" will still exist within the infinite set, leading to questions about the validity of the contradiction.

For example when the diagonal process is applied to "countable" sets like Natural numbers, it fails to produce a number outside the set, raising concerns about its consistency and applicability.

[Side Note:

The following will highlight the issue with Cantor's Diagonal Argument.

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 ∞

(In above list, we have highlighted the diagonal letters in each line...)

Now the argument is that : by flipping / changing the A or B one finds along the diagonal, we could generate a new line that does not appear anywhere in List I.

But the issue with this logic is that, we have a list that is infinitely long. Thus this infinitely long list, should have all the permutations / combinations permissible with A & B. 

Another more logical / simpler way to look at this, whould be to visualize A & B as binary numbers. Thus if we rearrange List I : with 0 & 1 (instead of the A & B) and then re-arrange the list in ascending  "left-right flipped" binary order...

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 ∞

Thus by flipping / changing any Diagonal Values, we will never get a new number that is not in the original set.

End of Side Note ]

The nature of an infinite list: The idea of a complete infinite list is a paradox in itself (as further discussed in Part II)

The 2D Hilbert Hotel Analogy: A New Perspective:

To address these concerns, we propose a "2D Hilbert Hotel" analogy to visualize the relationship between natural and real numbers:

Natural Numbers (1D Hilbert Hotel): Natural numbers are represented as a single, infinitely long line of rooms, reflecting their discrete and sequential nature.

Real Numbers (2D Hilbert Hotel): Real numbers are represented as a grid, where each room (representing a natural number) has an infinitely long corridor next to it. Each corridor has infinite rooms, representing the infinite fractional part of a real number.

This analogy effectively demonstrates the "wider set" nature of real numbers, where each natural number is associated with an infinite number of real numbers. It provides a clear and intuitive understanding of why real numbers are uncountable, without relying on the contested assumptions of the diagonal argument.

Implications and Conclusion:

The 2D Hilbert Hotel analogy offers a compelling alternative to the diagonal argument, providing a more robust and intuitive way to understand the uncountability of real numbers. It challenges the traditional interpretation of countability and highlights the importance of visualizing infinite sets in a way that aligns with their inherent properties. 

This approach hopefully may open new avenues for exploring the complexities of infinite cardinalities and the foundations of set theory.

This article has shown that there is a logical argument against the validity of the Cantor Diagonal argument, and that another method, the 2D Hilbert Hotel, presents a more intuitive technique to demonstrate that real numbers are a larger set (infinity ?) than the natural numbers.

Synopsis Part II:

In this section, we shall look further into the countable and uncountable sets.

One observes that to describe Natural numbers the term : Countably Infinite Sets, is used to distinguish them from the larger set of Real Numbers (the Uncountable Set).

But the fact remains that Natural Numbers themselves compose an infinite list/ set, which then brings up the point that the naming "Countable Set" to be a misnomer (semantically).

In-Depth Detail:

Thus to be more precise (semantically and mathematically), we shall use 1D and 2D Hilbert Hotel Sets (H.H Sets for short, to remove any possible naming ambiguity with Hilbert Spaces)

As discussed in Part I, Natural Numbers would thus fall into a 1D H.H Set, where as Real Numbers to a 2D H.H Set.

(In this context, the term Hilbert Hotel Set denotes a Set where : Each Dimension extending to infinity, which is a mandatory condition for the Hilbert Hotel's conceptual functionality.)

Thus via common logic we can reason that any entity that is countable will be in an Finite Set, where as any entity that is uncountable will be in an Infinite Set.

Thus for example to represent infinite  sets one can use a Venn diagram circle that extends to infinity on one side (like a tube, with one end open...)

To further clarify this point consider the following representation to denote a Real number ℝ.

 ℝ : (±)?  (ℕ)+ ([∞↻ℕ])? (. ([∞↻ℕ])?(ℕ)+)? 

What the above notation describes is a Real (ℝ) number: 

with optional (±)?

followed by one or more Natural (ℕ) number digits : (ℕ)+

followed the optional recursive Natural number digit generator ([∞↻ℕ])?

followed by an optional fractional component (. ([∞↻ℕ])? (ℕ)+ )? 

The above jumble becomes clearer when we consider a few real world examples...

Thus if we want to generate number 1050.59

We could use the following notation:

ℝ :  (ℕ)+ (. (ℕ)+)?

And for 7300

ℝ :  (ℕ)+

etc.

Thus in above, we have omited the optional recursive Natural number digit generator ([∞↻ℕ])?

Thus observe all Countable Sets can be generated / represented using the above notation.

Now in most all physical / practical / engineering / real world applications, the invocation of the ([∞↻ℕ)])? factor whould not be required.

But on the other hand, for the such of theoretical / fundamental mathematics, where one needs to conceptualise infinity / infinite numbers etc. one of the following could be used.

ℝ: (±)? (ℕ)+ ([∞↻ℕ])? (. ([∞↻ℕ])? (ℕ)+)? 

ℝ: (±)?  (ℕ)+ (. ([∞↻ℕ])? (ℕ)+)? 

ℝ: (±)?  (ℕ)+ ([∞↻ℕ])? (. (ℕ)+)? 

It would be left to the reader to conceptualise what the above three notations represents...

But one key take away point is that, once the [∞↻ℕ] operator is invoked what one gets is an infinite series of digits, aka an uncountable number of digits. Or we enter Infinite Realm(s).

Thus in above notation:

(±)? (ℕ)+ (. (ℕ)+)? would  represent / belongs to the Finite Domain / Realm.

Whereas invoking the following:

(±)? (ℕ)+ ([∞↻ℕ])? (. ([∞↻ℕ])? (ℕ)+)? causes above number to enter into the Infinite Domain / Realm.

One advantage of this notation is that we can conceptually lock / bind / delegate / confine the recursively iterating to infinity into the black box of  [∞↻ℕ] 

The following are a few more examples of Real (ℝ) numbers... 

ℝ : (±)?  (ℕ)+ ([∞↻ℕ] )? (. [∞↻ℕ] (ℕ)+ )?

       + 10 . [∞↻ℕ] 1

          54 . [∞↻ℕ] 2

       - 7700 . [∞↻ℕ] 5555

          2 . [∞↻ℕ] 7001001 

70007 [∞↻ℕ] . 0

Thus observe that all the digits highlighted above, resides in Infinite domain(s). Aka a domain / domains, we can only reach conceptually but never physically.

And yet for example, take the Real  number : + 10 . [∞↻ℕ] 505

Thus even though the digits 505 resides in an Infinite domain, the above number is definitely in the Finite domain (between 10.0 and 11.0). 

Thus can we say that above number is  finite (has a numerical limit), but digit-count is infinite?

An analogy will be: The head is in the Finite, but the tail ventures / goes into an Infinite... (And the sise /cross section of the tail gets smaller and smaller, the father it gets from the head...

Giving a new meaning to: Hold Infinity in the palm of your hand (Auguries of Innocense...). Aka hold a ℝ number like π on the palm of your hand...

[Side note :Yet another format to denote Real numbers could / would be.

 ℝ : (±)? (ℕ)* ([∞↻ℕ])? (ℕ)+ (. (ℕ)* ([∞↻ℕ])? (ℕ)+)? 

Where (ℕ)*, denotes zero or more Natural number digits.]

And to conclude this article here is a small food for thought...

Take the following inequality: 

n * n > n  (for all n > 1)

Via mathematical induction, the above can be proven without much difficulty.

Now what will happen when n→∞

Does this prove that:

∞ * ∞ > ∞

Is it possible to justify or negate this answer via the 2D Hilbert Hotel analogy / construct?

Ly DeSandaru.

Note: Following Articles have further discussions on this topic...

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

https://projectsofacademicinterests.blogspot.com/2025/07/lets-take-another-look-at-infinity.html

P.S.

The following is a review given by ChatGPT on the above article...

https://lydesandarureviews.blogspot.com/2025/03/review-challenging-cantor-diagonal-rule.html