What is the Size of the Hilbert Hotel's Computer?

. The Hilbert's Hotel is a hotel with countably infinitely many rooms. The size of its hypothetical computer was the pretext in order to think about whether it makes sense and what would be log 2 (  0 ). Thus, at the road of this journey, this little paper demonstrates – surprisingly – that there exist countably infinite sets strictly smaller than ℕ (the natural numbers), with very elementary mathematics, so shockingly stating the inconsistency of the Zermelo-Fraenkel Set Theory with the Axiom of Choice


Introduction
This paper proves, utilizing the suitable axioms and rules strictly within ZFC, the inconsistency of the proper ZFC. [7,16] The proof relies on the construction of a countably infinite set strictly smaller than ℕ, which would be impossible, by the Axiom of Countable Choice or Axiom of Denumerable Choice (ACω) [8] , hence this axiom is unfortunately contradictory with ZFC, which implies that the ZFC is inconsistent, regrettably.
Proof. There exists an injective function f : Ƥ(X → Y. We can see it defining the sets Xr and Yr below and demonstrating constructively that always |Ƥ(Xr| ≤ |Yr|, for every r, and then as r approaches 0, this shall necessarily lead to |Ƥ(X| ≤ |Y|. So, there exists an injective function fr : Ƥ(Xr → Yr for every r. We can demonstrate it defining that function as f r(Ø) = 0, and for each nonempty subset s = {k1, k2, …, km, …} of Ƥ(Xr (the power set of Xr) [1] , fr (s) = 1/(log22 k 1 + 2 k 2 + … + 2 k m + …) + 1). Note that that s can be an either finite or infinite subset of Ƥ(Xr.

Schema of Definition 2.2: Restricted Sets
Then, we can prove that that fr is really injective by construction, where for every member p of Ƥ(Xrthere exists one single member y of Yr, that is if fr(p) = y, and fr(q) = y, then p = q. This happens because we need double r in order to generate only one new value to log2r, which in its turn will double the sizes of Yr and of Ƥ(Xr, nearly equalizing their sizes (|Yr| and |Ƥ(Xr|), since that if km ∈ Xr, then Yr contains necessarily at least 2 k m elements (or members), which implies, as Xr ⸦ ℕ + , that |Yr| ≥ |Ƥ(Xr| for all the r varying from 1 up to 0, as shown in the symbolical constructive completed infinite table below:  Hence, for every finite or infinite subset {k1, k2, …, km, …} of Ƥ(X, there exists a definite and distinct value 1/(log22 k 1 + 2 k 2 + … + 2 k m + …) + 1) of Y: So, there is an injective function f : Ƥ(X → Y, and then |Y| = |ℕ| ≥ |Ƥ(X)|, thus we can define log2(0) = |X|  |ℕ| = 0, because |X| is strictly less than |Ƥ(X)|, since always |w| < |Ƥ(w)| (every set is strictly smaller than its power set) for every [finite or infinite] set w, by the Cantor's Theorem [13] .
Verify that the Cantor's diagonal argument [13] is not valid here in order to attempt to prove that |Ƥ(X)| > |ℕ|, since log2(0)  0, so a supposed anti-diagonal sequence from a countably infinite (supposed exhaustive) 0-enumeration cannot generate another indicator function (or characteristic function) different from all the other ones of this 0-enumeration, since the enumeration is 0-length, but that supposed anti-diagonal is only log2(0)-length, as shown constructively in the symbolical table below, where all the supposed anti-diagonal sequences can be in that 0-enumeration without being different from any position of their diagonal sequences (otherwise, then it would lead to a contradiction to the exhaustiveness assumption, and then it would prove that |Ƥ(X)| > |ℕ|, after all, as in that Cantor's argument): In order to better understanding of the infinite construction above, let W be a set very similar to X, but a finite set instead of an infinite one, for instance, W = {log2n : n ∈ {2, 3, 4, 5, 6, 7, 8}}. What would be |W| here? |W| = |{1, 2, 3}| = 3.
Therefore, four questions loom about that set X, which are readily answered here: 1. "-Is X really a well-defined set within ZFC?" -Yes, X is very well-defined, since its definition results from ZFC, plainly.
2. "-Aprioristically, X could even be a finite set; so, is X actually infinite?" -Yes, it is infinite, since for every number log2r, there is another one greater than it log2r+r = log2r + 1 (see by the way that we "need" r more in the "input" in order to get only 1 more in the "output", which even assist to explain why that set X "raises" so sluggishly).
3. "-Then, isn't X in fact a traditional countably infinite set, as ℕ, with cardinality equal to 0 (that is isn't simply |X| = 0)?" -No, X cannot be 0-sized, since its cardinality, log2(0), must be strictly less than 0, as proven within the completed infinite construction shown in the Tab. 2.1 above, unless we conclude otherwise that |Ƥ(X)| = 2 0 = 0, which would be even very very worse to ZFC. (See within that construction above that r more steps (numbers) are necessary in order to insert only 1 more member to X, which even helps to clarify why X "grows" so slowly (logarithmically) on the number r of "steps" or table rows in that construction, and hence it cannot "reach" 0.) 4. "-In truth, isn't 2 n : ℕ → ℕ an injective [total] function?" -No, 2 n , neither every increasing exponential function in n, cannot even be just a [total] function from ℕ to ℕ, since 2 0 > 0. On the other hand, every polynomial in n is so, because k.0 i ≤ 0 [18] , for every positive finite numbers i, k. (But 2 n : log -1 -ω → ℕ is an injective [total] function, as log -1 -ω is defined herein.) Barbosa panfinite number). -1 is simply another symbol (or name, or label) to represent log2(0), which leads to -1 = log2(0), and 2 -1 = 0.
Note that that concept of Barbosa panfinite numbers encompasses the Beth numbers (infinite cardinal numbers represented by the symbol ℶj, where ℶj+1 = 2 ℶj , for all j  ℕ) [2] , since i can be non-positive in the Def. 2.5 above, where ℶj = j, for all the integers j ≥ 0, entailing that the Beth numbers are just a proper subset of the Barbosa panfinite numbers.
Notice also that the countably infinite recursive process above generates countably infinite cardinalities -i, where all ones are strictly greater than every positive finite number n. See yet that 0 = 0 = ℶ0, hence there is herein a kind of positive-negative natural symmetry generalizing from Beth numbers to Barbosa panfinite numbers.

What is the size of the Hilbert Hotel's Computer?
With the definitions above, we can already easily answer that question "-What is the size of the Hilbert Hotel's computer?" -It is equal to -1. See the construction of this answer in the symbolical infinite table below (note that an -sized binary register or RAM cell can store one and only one number from exactly 2  distinct ones: 0…2  -1) [14] :  In fact, we now can answer innumerable theoretical questions of same kind, such as: 1. "-What is the [theoretical] length of a sequence of symbols that represents the cardinality of ℕ (0) in a base b [into a numeral system] [3] strictly greater than 1?" -It is equal to logb(0) = -1. (Notice that if that base b was equal to 1 (unary base), then that length would be equal to 0, instead of -1.) 2. "-How many months should we [theoretically] invest our savings at [positive] fixed rate of interest, [4] in order to get an 0-moneyed account?" 6 -We should do it by -1 months.
3. "-How many times should you [theoretically] bend an infinitely malleable paper sheet, in order to get an 0-lengthy thread?" -You should do it -1 times.

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-What is the [theoretical] maximal size of an NFA that can be converted into an exponentially larger DFA?" [6] -It is equal to -1.
7. "-How many terms are there in the infinite sum that is used as a representation of some Zeno's Paradoxes: 1/2 + 1/4 + … + 1/2 n + … = 1?" [17] -If we consider sensibly that all those terms are rational numbers, then 2 n is upper bounded by 0, hence there are -1 terms in that sum.
8. "-Can the Mathematical Induction be used in order to establish a given statement for all 0 natural numbers?" [18] -No, in general, it cannot; it can do it only for the first -j natural numbers, where that statement is proven for all ones only when j = 0, since only that 0 = 0. The maximum increasing rate (polynomial, exponential, etc.) of the integer formulas that occur within each particular induction shall determine that particular j. For instance, the inductive proof that 2 n > n 3 (for n ≥ 10) is valid only for the first -1 natural numbers, not for all 0 ones, as 2 n is not integer for n beyond -1, because naturally 2 0 > 0, and 2 -1 = 0.
Thus, as a preliminary result, the cardinalities in this paper can be strictly ordered by magnitude, as simply outlined below:

Related Work
The main result of this paper unfortunately asserts that the Axiom of Countable Choice or Axiom of Denumerable Choice (ACω) [8] (that states that 0 is smaller than every other transfinite cardinal number) is inconsistent with ZFC (so, the axiom of choice, a stronger version of that one) [9] , which implies that the ZFC is inconsistent, lamentably.
Therefore, I think we need build a new foundational frame to support and unify the Axiomatic Mathematics, either fixing or replacing the ZFC.