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**Real Numbers Definition( Real Analysis )**

**.99.. vs 1 a Real Problem**

e**Real Number Definition (Real Analysis):** In this context, mathematical generalists claim that the completeness(*) of the real number set is a proven fact. They demonstrate, through several steps, that there are no "gaps" between real numbers because of other defined properties , thus establishing their equality. It is important not to accept unproven definitions as facts and to understand that proving the completeness of the real numbers depends on the assumption that all infinities are equal.

* Real Number Definition : In mathematical analysis, the definition of real numbers often involves the concept of completeness. Completeness means there are no "gaps" or "missing points" on the real number line, unlike the rational numbers, which have gaps at irrational values. In decimal notation, completeness implies that any infinite decimal string represents a real number. Completeness can be formulated as an axiom (the completeness axiom) . Two notable forms of completeness are Dedekind completeness and Cauchy completeness.

Dedekind Completeness: This means that every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) within the real numbers. This property is not shared by the rational numbers.

*Ironically, Dedekind using math created a "math-free zone" between the set of rational numbers and hyperreal numbers. This is used to explain the claim that there must be a set where we can include π, because it is not in the set of rational numbers and thus cannot be complete without real numbers. However, the same argument should not be applied to the existence hyperreal set. This double standard is uncanny and a definition of contradiction( flat argument) . If 0.99… is considered a real number, one could argue that π should be considered a rational number simply because one can imagine that a deviation of two arbitrarily long infinite(...) digits could yield π. *It is ironic that no one asserts that 1 and 2 are equal, as there are no natural numbers between them when discussing natural numbers. However, we must acknowledge that 0.99... is indeed equal to 1, since there are no real numbers between them.

The idea that 0.99... equals 1 in the real number system is well-established in today's mathematics. However, arguments against this equality are compelling these days. One such argument is that while 1 x 1 x ... = 1, 0.99... x 0.99... doesn't converge to 1, therefore 0.99... cannot be equal to 1.

This is essentially the final step before concluding that we cannot consistently define the real value of .99... At this stage, it becomes clear that defining .99... as a real number is problematic. We need to set aside the Archimedean property, which applies to the smallest infinity, and understand that the reciprocals of all real numbers are smaller than this smallest infinity and thus smaller than infinitely many larger infinities (e.g., aleph-one, aleph-two, etc.).

In simpler terms, a real number like .99... could theoretically have an aleph-null (the smallest infinity) number of nines, but it never actually reaches one because there are infinitely many hyperreal numbers in between. Therefore, we must accept that .99... is a real number and that the gap between .99... and 1 is filled with hyperreal numbers. Otherwise, claiming that .99... is not a real number would be incorrect. Thus, asserting that the real number .99... is not equal to 1 is false.

**At the time of publishing ****The Naked Emperor**** playlist, we can see the misconception start with simply claim 1/3 = 0.33..., and as we go step by step forward, we discover the error and find a better and more apparat logically rigorous proof. However, we end up leaving a trail of misunderstanding for others. This means that for superficial observers who do not delve deeply, the erroneous repetitive false proofs are enough to confuse and accept, triggering a false positive feedback loop similar to the story of 'The Emperor's New Clothes,' hence the name 'The Naked Emperor.'**

Pretty much after this level, most mathematicians accept the idea of hyperreal numbers and the existence of infinitesimals, which provide ample proof that .99... is NOT equal to 1. However, as someone who has read this, you might understand that it is quite rare for people to reach this level of mathematical understanding. Some may not progress to this advanced level of math. You might also understand why many people resist this idea and cling to outdated notions, similar to flat Earth theories, which seem more intuitive and simpler.

Perhaps you can also see why mathematicians at this level often don’t bother correcting others on such issues. It may seem unnecessary to argue about something that could take an unreasonable amount of time to explain, especially if the person is unlikely to change their mind. Instead, mathematicians might prefer to focus on their own work and avoid frustration over disputes that may not be worth the effort. However, we believe this is a real problem that holds back mathematical progress. Addressing misconceptions like these can free up time and resources that could be better spent on advancing mathematical development. Correcting misunderstandings helps to clear away false or misleading ideas and paves the way for genuine advancements. By resolving these issues, we not only open new doors to innovation but also prevent the perpetuation of outdated or incorrect concepts. This process is crucial for ensuring that mathematical inquiry and research are based on sound principles, ultimately leading to more accurate and impactful discoveries.

ℵ₀ (aleph-null) is the cardinality of the set of natural numbers , Also it is the smallest infinite cardinal number and greatest element of set of natural number ( at the same time) . Since we can always find natural numbers larger than any real number (when rounding up), ℵ₀ is the largest element in the set of real numbers as well. The cardinality of the real numbers is 2^ℵ₀ or ℵ₁ according to Cantor’s Set theory. The largest value we can find in a set of real numbers is still smaller than ℵ₀, and we know the cardinality of the set of real numbers is greater than ℵ₀ but that is not a problem. For example, the cardinality of the set {1/2, 1/3, 1/4} is 3 it is ordinal 8, but the largest value is 1/2( and they don't have to be the same). **Since we know the ℵ family of 'numbers' is well-ordered, we use it to support our argument against .999... = 1 by considering the well-ordered reciprocals of these numbers.**** **We don't need to use ℵ₀ and rely on Cantor here as Carol Wood said assume it as K. https://www.youtube.com/watch?v=BBp0bEczCNg

Let us consider the following proposition: Definitions do not serve as proof. To illustrate this, let Represent the set of individuals who accept definitions as proof. We can construct a scenario where we define P such that P={x ∣ x accepts definitions as proof}

Now, let us introduce the proposition Q: Individuals in P are mistaken. By defining P in this manner, we can assert that if an individual x ∈ P, then according to our definition, x holds a belief that is, in fact, incorrect. Thus, we can conclude that:

∀ x ∈ P, x is mistaken

This illustrates that the act of accepting definitions as proof is fundamentally flawed. Therefore, by our constructed definitions, we can assert that such individuals are wrong in their belief.

Theorem 1 ℵ₀ ∈ N ∈ R . ( ℵ₀ is a real number)

Proof :The cardinality of the set {1} is 1, the cardinality of the set {1, 2} is 2, and the cardinality of the set {1, 2, 3, ..., N} is N. Let N* = N→ ∞ {1, 2, 3, ..., N} where it includes an exhaustive list of all finite elements of set natural number therefore N*-N is equal to cardinality of the set of natural numbers or ℵ₀ (aleph-null). Thus, ℵ₀ is a natural number because N* is the set of all finite numbers; however, we need at least one element, the supremum, to be infinite in order to have a complete set of natural numbers. Otherwise, the cardinality of the set of natural numbers would be a finite number N we know that all natural number are real number . Alternatively if you say N* has an infinite element, then we reach the conclusion that ℵ₀ ∈ N*= N hence a natural number . Thus in either case ℵ₀ ∈ N ∈ R . This completes the proof.

Because ℵ₀ is a number in the real number system, 1/ ℵ₀ is a real number as well ( not zero) . Since ℵ₀ < ℵ₁, it follows that 1/ ℵ₀ > 1/ℵ₁, which implies that the claim of no gap property in the real numbers is false. Therefore, there must be a gap between 0.99... and 1; thus, they cannot be equal in the real number system.

Note that, by the strict definition of real numbers, there is no number between 0.99... and 1. However, we have proven that there is a gap, and in reality, this gap is filled with infinitely many numbers. This emphasizes that the absence of a real number between 0.99... and 1 does not imply that they are equal, as there is an obvious gap between them.

In other words, the completeness of the real number set might be viewed as a mathematical delusion. Just as some might not recognize the existence of other numbers and claim that 1 equals 2 because there is no natural number between them, a similar misunderstanding can occur regarding the completeness of the real numbers. The set of real numbers is complete in itself, and this completeness is limited to its definition.

Below are our personal notes ( share with public) outlining a framework to define a number system that combines all sets of numbers. The analogy we used is that we previously understood numbers like different parts of an elephant; now it is time to combine these parts and show what a truly global number system should be. This is not a complete work; it is a starting point. It is meant to be used in the context of ambiguous number theory, chromo number theory, and supersymmetry. Understanding this may not require knowledge of 0.999... being NOT equal to one, but it will be necessary to grasp why modern science produces contradictory results, such as in quantum mechanics.

Theorem 2 ℵ₀ =ω ∈ N ∈ R . ( **ω** is a real number)

Proof : Cardinality of the set {1, 2, 3, ..., N} is N and it is equal to it ordinal value . Since N = limit N→ ∞ {1, 2, 3, ..., N} thus ℵ₀ =ω

Theorem 1 (/ 2) proof # 2

Let P(S) denote a property of a set S, where P(S) is true if the cardinality/ordinality of S is a natural number, and P(S) is false otherwise. we have P({1}) is true. The cardinality/ordinality of the set containing only the number one is precisely 1, which is a natural number. Hence the base case holds. 1 For the successor case, we assume that P({1,…,n}) is true and wish to prove that P({1,…n,n+1}) is true. Notice that we can express the set {1,…,n,n+1} as the union of the two disjoint sets {1,…,n} and {n+1}. By the inductive hypothesis, P({1,…,n}) is true — thus the cardinality of {1,…,n} is a natural number. The cardinality/ordinality of {n+1} is 1. Adding 1 to a natural number always yields another natural number, so the cardinality/ordinality of {1,…,n,n+1} is a natural number as well, and P({1,…n,n+1}) is true. Thus P({1,… n,n+1}) = P({1,…,n})+ P{n+1}= Ture &(+) True= True completes the proof.

Convention (ω₀ =ω, ℵ₁ =ω₁,... ) ,

Infinity: In mathematics, particularly in set theory, infinity represents a quantity that is larger than any finite number. It is often used to describe limits that exceed all finite bounds.

Infinitesimals: An infinitesimal is a quantity that is greater than 0 but smaller than any positive real number. This concept allows for rigorous treatment of values that are "infinitely close" to standard real numbers.

Expression 1 - 1/ω: When considering ω as an infinite quantity, we can analyze this expression. As ω approaches infinity, 1/ω approaches 0. Therefore, 1 - 1/ω approaches 1 - 0, which equals 1. However, if we treat 1/ω as an infinitesimal, then 1 - 1/ω represents a number that is infinitesimally less than 1.

Summary: Smallest Infinity represents a quantity larger than any finite number. 1/ω= ε is the largest infinitesimal that gets closer to zero as ω increases. Thus, 1 - 1/ω is a value that is infinitesimally less than 1, supporting the idea that 0.999... (which can be viewed as 1 - 1/ω) is not equal to 1 in this mathematical contexts.

So, in short, ANT is about numbers in their original framework of infinities. Number behavior is ambiguous simply because the infinities that define them are ambiguous.

- The number 0.999... with ω digits is a real number, while the number 0.999...;9 with ω + 1 digits is not a real number; it is a hyperreal number.
- Understanding 0.999...: The infinite decimal 0.999... is a real number. It can be shown that 1 - 0.999... equals 1/ω , which can be considered as 1/100... (where the infinite zeros follow the decimal). Both 0.999... and 1000... have an infinite number of digits.
- Representation of Infinite Digits: Some numbers cannot be exactly represented with an infinite decimal expansion due to the nature of their digits. The smallest infinite sequence is denoted as having ω digits. Numbers with ω + 1 digits are not typically represented as real numbers.
- Hyperreal Numbers: In the context of hyperreal numbers, 0.999...;9 can be understood as having ω + 1 digits, allowing it to represent a rank-one hyperreal number.
- Limitations of Real Numbers: The set of real numbers is not sufficiently large to encompass all numbers with infinite digits. At best, it includes numbers where all digits after ω are zero. This is analogous to the set of natural numbers, which can contain some, but not all, real numbers.
- Decimal Expansions: Properly defining real numbers reveals that some can be represented as a decimal expansion , while others, such as 1/3, cannot. This is due to the perpetual remainder in their decimal representation. (1/3=0.333…;333…;333…;… and 0.333…;0 is just round number rank ω )
- The Dedekind cut of 0.99... = 1 - 1/ω shows that on the left we have all real numbers smaller than 1( with expectation of .99...) , and on the right we have 1 and all hyperreal numbers smaller than 1 (for example, 0.999...;9 with ω+1 9s) that are greater than 1 - 1/ω. This collectively shows that the no-gap property of real numbers is a huge misunderstanding simplify .

Ultimately, ANT is a framework that predicts number behavior based on their infinity. Numbers will have groups property of ℵ and ranks of ω, with numbers within the same group behaving consistently. The rank serves as a classifier to help us disambiguate them and perform non-contradictory calculations. We believe this will be the cornerstone of technology for infinite machines. For example, it could aid in the calculation of infinities in quantum computing, particularly regarding singularities like the Big Bang or black holes. To put it simply, ANT, along with the other pillars of CNT and SS, will enhance our ability to transcend singularities.

**The last Words: The set of real numbers, in no way or form, includes all possible sets of numbers, and there is no set that can encompass all possible sets of numbers and, It is predicable that no single set can include all possible sets. Although a proof that such a set does not exist has yet to be provided, However, considering the incompleteness theorem, such a proof may well be on the horizon. The proof of such a theorem would falsify ( there must be infinity many number between 1 and 2, 1 and .99... , and , **ℵ₀ and