We construct a countable set \( S \) that possesses the essential properties of the real numbers:... more We construct a countable set \( S \) that possesses the essential properties of the real numbers: it is everywhere dense in \( \mathbb{R} \) and Dedekind complete, with no gaps. Each element of \( S \) is part of a unique triplet \( (a_\alpha, x_\alpha, b_\alpha) \), where \( a_\alpha \) and \( b_\alpha \) are rational endpoints, and \( x_\alpha \) is a real number confined between them. By construction, every rational pair \( (a_\alpha, b_\alpha) \) is used, but each is used exactly once, ensuring the countability of \( S \) by prohibiting multiple \( x_\alpha \) values within the same interval. This restriction does not alter \( S \)'s density (it remains everywhere dense in \( \mathbb{R} \)) or its Dedekind completeness (no gaps are introduced, as no numbers are excluded), thereby preserving properties that make \( S \) equivalent to the set of all real numbers.
Through rigorous proofs, we demonstrate that \( S \) encompasses all real numbers by recognizing each within a nested sequence of intervals, corresponding to an element \( x_\alpha \) in \( S \). Furthermore, we establish that any Dedekind cut in \( S \) does not produce gaps, thereby confirming \( S \) as a complete and countable representation of the real line.
The cardinality of real numbers is contingent upon the chosen axiomatic system and the specific m... more The cardinality of real numbers is contingent upon the chosen axiomatic system and the specific models of set theory. According to the diagonal argument within the Zermelo-Fraenkel (ZF) framework, real numbers are non-denumerable. However, their precise cardinality remains a topic of debate, varying with different extensions of the ZF axioms. For example, under the inner-model axiom 'V-ultimate L', the Continuum Hypothesis (CH) holds true. In contrast, it is negated if Martin's Axiom is adopted. Additionally, there is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. This model-dependent viewpoint is underlined by the Löwenheim-Skolem theorem, which asserts that no theory can have only non-denumerable models. Accepting any form of axiomatization implies that terms like 'denumerable' and 'nondenumerable' are relative. This necessitates the acceptance of the relativization of cardinal numbers, as initially suggested by Skolem and Carnap.
Constructive Mathematics: Foundations and Practice, 2023
It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradi... more It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradicts the diagonal argument or non-denumerability of real numbers. To show that, we start with a set of all algebraic numbers S and define a function f , which inserts a transcendental number between every pair of algebraic numbers within S. To show that such an expanded set denoted S , which in addition to algebraic numbers includes transcendental numbers generated by the function f , contains all real numbers used is a counterexample. Defined is an arbitrarily chosen missing number T x , not included in set S , with a sequence of algebraic pairs (intervals with algebraic endpoints) converging to T x. Then, AC is used to show that exactly the same sequence of closed intervals that defines the number T x exists in the set S. Subsequently, the Nested Intervals Property is applied to show that the recognized sequence of closed intervals within S has a nonempty intersection and contains the number T x. Given that number T x is chosen arbitrarily and is in set S , it implies that all numbers are in the set S , which is also obviously denumerable.
Constructive Mathematics: Foundations and Practice: , 2023
It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradi... more It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradicts the diagonal argument or non-denumerability of real numbers. To show that, we start with a set of all algebraic numbers S and define a function f, which inserts a transcendental number between every pair of algebraic numbers within S. To show that such an expanded set denoted S, which in addition to algebraic numbers includes transcendental numbers generated by the function f, contains all real numbers used is a counterexample. Defined is an arbitrarily chosen missing number Tx, not included in set S, with a sequence of algebraic pairs (intervals with algebraic endpoints) converging to Tx. Then, AC is used to show that exactly the same sequence of closed intervals that defines the number Tx exists in the set S. Subsequently, the Nested Intervals Property is applied to show that the recognized sequence of closed intervals within S has a nonempty intersection and contains the number Tx. Given that number Tx is chosen arbitrarily and is in set S, it implies that all numbers are in the set S , which is also obviously denumerable.
The cardinality of real numbers depends on the axioms that are implemented. For instance, the con... more The cardinality of real numbers depends on the axioms that are implemented. For instance, the continuum hypothesis in the inner-model axiom {\textquoteleft V-ultimate L\textquoteright} is true while if Martin's axiom is correct, then the continuum hypothesis is false. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the L\"{o}wenheim-Skolem theorem, no theory can have only nondenumerable models. Accepting axiomatization in whatever shape means the notions {\textquoteleft denumerable\textquoteright} and {\textquoteleft nondenumerable\textquoteright} turn out to be relative, and it requires to accept the relativization of cardinals as Skolem and Carnap first point it out.
In this article, we provide a surprising result that the denumerability of real numbers is possible using the Axiom of Choice. The proof is sketched here and elaborated further in the text. First, we consider an interval $[a,b]$, where $a<b$, that contains only algebraic numbers and define a set $S$, which by definition contains all possible algebraic intervals $I_n=[a_n, b_n]$, and $a_n \leqslant b_n$, where $a_n$ and $b_n$, are algebraic numbers from $[a,b]$. Then defined is a function $f$ that for each interval $I_n$ assigns a transcendental number $t_n$, so that $t_n$ is within the interval on which function $f$ is acting, i.e. $f(I_n)=t_n$, and $a_n \leqslant t_n \leqslant b_n$. It is also required that $t _n$ is in all intervals containing that interval, i.e. in all superintervals. The numbers $t_n$ are then added to the set $S$, and to all algebraic intervals $I_n$ of the set defining the set $S’$, which is countable and complete. The proof is simple and based on contra example. Assume that there is a missing transcendental number $T_x$ that is different from all $t_n$ numbers generated by the function $f$ and therefore from all numbers in the set $S’$, and define that missing number $T_x$ with a sequence of nested closed intervals with algebraic endpoints $\alpha_n$ and $\beta_n$, so $T_x=[\alpha_1, \beta_1]\supset [\alpha_2, \beta_2] \supset [\alpha_3, \beta_3]… \supset [\alpha_n, \beta_n]…$. Using the Axiom of Choice, one can recognize in the set $S’$ closed intervals $I_n=[a_n,b_n]$ that have the same endpoints as the closed intervals $[\alpha_n, \beta_n]$ in the sequence that defines the missing number $T_x$. So, one can find in the set $S’$ all closed intervals $I_n$ with the same endpoints $a_n=\alpha_n$ and $b_n=\beta_n$, which can be placed together to create a sequence $T_x=[a_1,b_1]\supset [a_2,b_2] \supset [a_3,b_3]… \supset [a_n,b_n]…$ So, using the AC, one can recognize in the set $S’$ sequence of intervals $[a_n,b_n]$ that is exactly the same as the sequence of intervals $[\alpha_n, \beta_n]$ that represents the missing number, i.e. it has closed intervals with the same endpoints. To show that set $S'$ is complete, that there is no missing number $T_x$, that $T_x$ is in the set $S’$, it is enough to show that the intersection of the sequence of closed intervals $[a_n,b_n]$ is not empty, that it contains the number $T_x$. It is shown using the Intersection Property Theorem and applying it to the sequence of intervals $[\alpha_n, \beta_n]$ that defines the $T_x$ number. To satisfy the requirement that the function $f$ acted on all $I_n$ intervals in the set $S$, the same $T_x$ number must be in the sequence with the $[a_n,b_n]$ intervals that are from the set $S'$. The power of AC is that it does not require to define sequence of any real number. Instead, it allows to recognize in the set $S'$ any sequence representing any real number. The sequence of any real number is implicitly in the set $S'$ because all possible intervals with algebraic endpoints are in the set $S'$. Since by AC in the set $S'$ can be recognized any arbitrary sequence representing any number, and since these sequences have nonempty intersections, the set $S'$ is complete.
Assuming the validity of AC and its consistency with ZF axioms, the discrepancy with the diagonal argument and other proofs of nondenumerability must originate in logical or technical problems common for these proofs, which we discuss in the second part of the article. In each of them, there is either a logical problem that the inability to assign in a sequence a specific n-th place for a number implies that that number is not in the sequence or a technical problem related to ignoring the Intersection Property Theorem in some elements of the proofs, or both of these problems.
The proof of denumerability of real numbers provided in the first part of the article does not specify the sequence of real numbers since numbers $t_n$ are chosen arbitrarily. In the third part of the article, we provide an explicit sequence of all real numbers. It is done by modifying the G\"{o}del constructible model universe and generating a set of real numbers, starting with a set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's $\Sigma$ constructive model, which is different from ZF, result in a set of all real numbers. For the first time, provided is an explicit example that satisfies Wang's model.
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel’s and Cohen’s proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms there are efforts of adding at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom ‘V-ultimate L’ that builds a universe of sets from the ground up and Martin’s axiom that allows to expand it outward in all directions. If Martin’s axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löw...
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel’s and Cohen’s proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms there are efforts of adding at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom ‘V-ultimate L’ that builds a universe of sets from the ground up and Martin’s axiom that allows to expand it outward in all directions. If Martin’s axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löw...
A new computational method for solving the configuration-space Faddeev equations for a three-nucl... more A new computational method for solving the configuration-space Faddeev equations for a three-nucleon system has been developed. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The s-wave calculations of the inelasticity and phase-shift, as well as of the breakup amplitudes for nd and pd breakup scattering for lab energies 14.1 and 42.0 MeV have been performed using the Malfliet-Tjon MT I-III potential. In the case of nd breakup scattering, the results are in good agreement with those of the benchmark solution (J. L. Friar et al., Phys. Rev. C 42 (1990) 1838 and J. L. Friar et al., Phys. Rev. C 51 (1995) 2356). In the case of pd quartet breakup scattering, disagreement for the inelasticities reaches up to 6% when compared with the results of the Pisa group (A. Kievsky et al., Phys. Rev. C 64 (2001) 024002). The calculated pd amplitudes fulfill the optical theorem with a good precision.
A new computational method for solving the nucleon-deuteron breakup scattering problem has been a... more A new computational method for solving the nucleon-deuteron breakup scattering problem has been applied to study the elastic neutronand proton-deuteron scattering on the basis of the configuration-space Faddeev-Noyes-Noble-Merkuriev equations. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The Merkuriev-Gignoux-Laverne approach has been generalized for arbitrary nucleonnucleon potentials and with an arbitrary number of partial waves. The nucleondeuteron observables at the incident nucleon energy 3 MeV have been calculated using the charge-independent AV14 nucleon-nucleon potential including the Coulomb force for the proton-deuteron scattering. Results have been compared with those of other authors and with experimental proton-deuteron scattering data.
2021 IEEE 32nd International Conference on Microelectronics (MIEL)
It is of great interest to determine the relation between biophysical and condensed matter system... more It is of great interest to determine the relation between biophysical and condensed matter systems, because this biomimetical approach provides new frontiers for further microelectronic circuits integrations. Molecular and submolecular particles are identical in living and nonliving organisms, which implies the possibility of considering these particles motion as the biunivocal phenomenon. Brownian motion and its fractal nature, as a general phenomenon existing within both systems, is an integrative property based on which we can establish the relation between these two systems. We used some experimental data, based on the bacterial motion experiments, regarding influence of different energy impulses on bacterial trajectories. The goal of this research is to introduce the asymptotic approaching of biophysical and condensed matter systems, based on Brownian motion fractal nature similarities, presented in mathematical analytical forms.
... 316319 INE Beam Interactions with materials fl Atoms Probing the threenucleon force using nuc... more ... 316319 INE Beam Interactions with materials fl Atoms Probing the threenucleon force using nucleondeuteron breakup reactions CR Howell a, , HR Setze a, RT Braun a, DE Gonzalez Trotter a, AH Hussein a,l, CD Roper a, F. Salinas a, 1 ... Cornelius and W. Gl6ckle, FewBody Sys. ...
Crystal structure and morphology of mechanically activated nanocrystalline Fe/BaTiO 3 was investi... more Crystal structure and morphology of mechanically activated nanocrystalline Fe/BaTiO 3 was investigated using a combination of spectroscopic and microscopic methods. These show that mechanical activation led to the creation of new surfaces and the comminution of the initial powder particles. Prolonged milling resulted in formation of larger agglomerates of BaTiO 3 and bimodal particle size distribution, where BaTiO 3
We point out a weak side of the commonly used determination of scalar cosmological perturbations ... more We point out a weak side of the commonly used determination of scalar cosmological perturbations lying in the fact that their average values can be nonzero for some matter distributions. It is shown that introduction of the finite-range gravitational potential instead of the infinite-range one resolves this problem. The concrete illustrative density profile is investigated in detail in this connection.
We construct a countable set \( S \) that possesses the essential properties of the real numbers:... more We construct a countable set \( S \) that possesses the essential properties of the real numbers: it is everywhere dense in \( \mathbb{R} \) and Dedekind complete, with no gaps. Each element of \( S \) is part of a unique triplet \( (a_\alpha, x_\alpha, b_\alpha) \), where \( a_\alpha \) and \( b_\alpha \) are rational endpoints, and \( x_\alpha \) is a real number confined between them. By construction, every rational pair \( (a_\alpha, b_\alpha) \) is used, but each is used exactly once, ensuring the countability of \( S \) by prohibiting multiple \( x_\alpha \) values within the same interval. This restriction does not alter \( S \)'s density (it remains everywhere dense in \( \mathbb{R} \)) or its Dedekind completeness (no gaps are introduced, as no numbers are excluded), thereby preserving properties that make \( S \) equivalent to the set of all real numbers.
Through rigorous proofs, we demonstrate that \( S \) encompasses all real numbers by recognizing each within a nested sequence of intervals, corresponding to an element \( x_\alpha \) in \( S \). Furthermore, we establish that any Dedekind cut in \( S \) does not produce gaps, thereby confirming \( S \) as a complete and countable representation of the real line.
The cardinality of real numbers is contingent upon the chosen axiomatic system and the specific m... more The cardinality of real numbers is contingent upon the chosen axiomatic system and the specific models of set theory. According to the diagonal argument within the Zermelo-Fraenkel (ZF) framework, real numbers are non-denumerable. However, their precise cardinality remains a topic of debate, varying with different extensions of the ZF axioms. For example, under the inner-model axiom 'V-ultimate L', the Continuum Hypothesis (CH) holds true. In contrast, it is negated if Martin's Axiom is adopted. Additionally, there is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. This model-dependent viewpoint is underlined by the Löwenheim-Skolem theorem, which asserts that no theory can have only non-denumerable models. Accepting any form of axiomatization implies that terms like 'denumerable' and 'nondenumerable' are relative. This necessitates the acceptance of the relativization of cardinal numbers, as initially suggested by Skolem and Carnap.
Constructive Mathematics: Foundations and Practice, 2023
It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradi... more It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradicts the diagonal argument or non-denumerability of real numbers. To show that, we start with a set of all algebraic numbers S and define a function f , which inserts a transcendental number between every pair of algebraic numbers within S. To show that such an expanded set denoted S , which in addition to algebraic numbers includes transcendental numbers generated by the function f , contains all real numbers used is a counterexample. Defined is an arbitrarily chosen missing number T x , not included in set S , with a sequence of algebraic pairs (intervals with algebraic endpoints) converging to T x. Then, AC is used to show that exactly the same sequence of closed intervals that defines the number T x exists in the set S. Subsequently, the Nested Intervals Property is applied to show that the recognized sequence of closed intervals within S has a nonempty intersection and contains the number T x. Given that number T x is chosen arbitrarily and is in set S , it implies that all numbers are in the set S , which is also obviously denumerable.
Constructive Mathematics: Foundations and Practice: , 2023
It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradi... more It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradicts the diagonal argument or non-denumerability of real numbers. To show that, we start with a set of all algebraic numbers S and define a function f, which inserts a transcendental number between every pair of algebraic numbers within S. To show that such an expanded set denoted S, which in addition to algebraic numbers includes transcendental numbers generated by the function f, contains all real numbers used is a counterexample. Defined is an arbitrarily chosen missing number Tx, not included in set S, with a sequence of algebraic pairs (intervals with algebraic endpoints) converging to Tx. Then, AC is used to show that exactly the same sequence of closed intervals that defines the number Tx exists in the set S. Subsequently, the Nested Intervals Property is applied to show that the recognized sequence of closed intervals within S has a nonempty intersection and contains the number Tx. Given that number Tx is chosen arbitrarily and is in set S, it implies that all numbers are in the set S , which is also obviously denumerable.
The cardinality of real numbers depends on the axioms that are implemented. For instance, the con... more The cardinality of real numbers depends on the axioms that are implemented. For instance, the continuum hypothesis in the inner-model axiom {\textquoteleft V-ultimate L\textquoteright} is true while if Martin's axiom is correct, then the continuum hypothesis is false. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the L\"{o}wenheim-Skolem theorem, no theory can have only nondenumerable models. Accepting axiomatization in whatever shape means the notions {\textquoteleft denumerable\textquoteright} and {\textquoteleft nondenumerable\textquoteright} turn out to be relative, and it requires to accept the relativization of cardinals as Skolem and Carnap first point it out.
In this article, we provide a surprising result that the denumerability of real numbers is possible using the Axiom of Choice. The proof is sketched here and elaborated further in the text. First, we consider an interval $[a,b]$, where $a<b$, that contains only algebraic numbers and define a set $S$, which by definition contains all possible algebraic intervals $I_n=[a_n, b_n]$, and $a_n \leqslant b_n$, where $a_n$ and $b_n$, are algebraic numbers from $[a,b]$. Then defined is a function $f$ that for each interval $I_n$ assigns a transcendental number $t_n$, so that $t_n$ is within the interval on which function $f$ is acting, i.e. $f(I_n)=t_n$, and $a_n \leqslant t_n \leqslant b_n$. It is also required that $t _n$ is in all intervals containing that interval, i.e. in all superintervals. The numbers $t_n$ are then added to the set $S$, and to all algebraic intervals $I_n$ of the set defining the set $S’$, which is countable and complete. The proof is simple and based on contra example. Assume that there is a missing transcendental number $T_x$ that is different from all $t_n$ numbers generated by the function $f$ and therefore from all numbers in the set $S’$, and define that missing number $T_x$ with a sequence of nested closed intervals with algebraic endpoints $\alpha_n$ and $\beta_n$, so $T_x=[\alpha_1, \beta_1]\supset [\alpha_2, \beta_2] \supset [\alpha_3, \beta_3]… \supset [\alpha_n, \beta_n]…$. Using the Axiom of Choice, one can recognize in the set $S’$ closed intervals $I_n=[a_n,b_n]$ that have the same endpoints as the closed intervals $[\alpha_n, \beta_n]$ in the sequence that defines the missing number $T_x$. So, one can find in the set $S’$ all closed intervals $I_n$ with the same endpoints $a_n=\alpha_n$ and $b_n=\beta_n$, which can be placed together to create a sequence $T_x=[a_1,b_1]\supset [a_2,b_2] \supset [a_3,b_3]… \supset [a_n,b_n]…$ So, using the AC, one can recognize in the set $S’$ sequence of intervals $[a_n,b_n]$ that is exactly the same as the sequence of intervals $[\alpha_n, \beta_n]$ that represents the missing number, i.e. it has closed intervals with the same endpoints. To show that set $S'$ is complete, that there is no missing number $T_x$, that $T_x$ is in the set $S’$, it is enough to show that the intersection of the sequence of closed intervals $[a_n,b_n]$ is not empty, that it contains the number $T_x$. It is shown using the Intersection Property Theorem and applying it to the sequence of intervals $[\alpha_n, \beta_n]$ that defines the $T_x$ number. To satisfy the requirement that the function $f$ acted on all $I_n$ intervals in the set $S$, the same $T_x$ number must be in the sequence with the $[a_n,b_n]$ intervals that are from the set $S'$. The power of AC is that it does not require to define sequence of any real number. Instead, it allows to recognize in the set $S'$ any sequence representing any real number. The sequence of any real number is implicitly in the set $S'$ because all possible intervals with algebraic endpoints are in the set $S'$. Since by AC in the set $S'$ can be recognized any arbitrary sequence representing any number, and since these sequences have nonempty intersections, the set $S'$ is complete.
Assuming the validity of AC and its consistency with ZF axioms, the discrepancy with the diagonal argument and other proofs of nondenumerability must originate in logical or technical problems common for these proofs, which we discuss in the second part of the article. In each of them, there is either a logical problem that the inability to assign in a sequence a specific n-th place for a number implies that that number is not in the sequence or a technical problem related to ignoring the Intersection Property Theorem in some elements of the proofs, or both of these problems.
The proof of denumerability of real numbers provided in the first part of the article does not specify the sequence of real numbers since numbers $t_n$ are chosen arbitrarily. In the third part of the article, we provide an explicit sequence of all real numbers. It is done by modifying the G\"{o}del constructible model universe and generating a set of real numbers, starting with a set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's $\Sigma$ constructive model, which is different from ZF, result in a set of all real numbers. For the first time, provided is an explicit example that satisfies Wang's model.
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel’s and Cohen’s proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms there are efforts of adding at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom ‘V-ultimate L’ that builds a universe of sets from the ground up and Martin’s axiom that allows to expand it outward in all directions. If Martin’s axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löw...
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel’s and Cohen’s proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms there are efforts of adding at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom ‘V-ultimate L’ that builds a universe of sets from the ground up and Martin’s axiom that allows to expand it outward in all directions. If Martin’s axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löw...
A new computational method for solving the configuration-space Faddeev equations for a three-nucl... more A new computational method for solving the configuration-space Faddeev equations for a three-nucleon system has been developed. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The s-wave calculations of the inelasticity and phase-shift, as well as of the breakup amplitudes for nd and pd breakup scattering for lab energies 14.1 and 42.0 MeV have been performed using the Malfliet-Tjon MT I-III potential. In the case of nd breakup scattering, the results are in good agreement with those of the benchmark solution (J. L. Friar et al., Phys. Rev. C 42 (1990) 1838 and J. L. Friar et al., Phys. Rev. C 51 (1995) 2356). In the case of pd quartet breakup scattering, disagreement for the inelasticities reaches up to 6% when compared with the results of the Pisa group (A. Kievsky et al., Phys. Rev. C 64 (2001) 024002). The calculated pd amplitudes fulfill the optical theorem with a good precision.
A new computational method for solving the nucleon-deuteron breakup scattering problem has been a... more A new computational method for solving the nucleon-deuteron breakup scattering problem has been applied to study the elastic neutronand proton-deuteron scattering on the basis of the configuration-space Faddeev-Noyes-Noble-Merkuriev equations. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The Merkuriev-Gignoux-Laverne approach has been generalized for arbitrary nucleonnucleon potentials and with an arbitrary number of partial waves. The nucleondeuteron observables at the incident nucleon energy 3 MeV have been calculated using the charge-independent AV14 nucleon-nucleon potential including the Coulomb force for the proton-deuteron scattering. Results have been compared with those of other authors and with experimental proton-deuteron scattering data.
2021 IEEE 32nd International Conference on Microelectronics (MIEL)
It is of great interest to determine the relation between biophysical and condensed matter system... more It is of great interest to determine the relation between biophysical and condensed matter systems, because this biomimetical approach provides new frontiers for further microelectronic circuits integrations. Molecular and submolecular particles are identical in living and nonliving organisms, which implies the possibility of considering these particles motion as the biunivocal phenomenon. Brownian motion and its fractal nature, as a general phenomenon existing within both systems, is an integrative property based on which we can establish the relation between these two systems. We used some experimental data, based on the bacterial motion experiments, regarding influence of different energy impulses on bacterial trajectories. The goal of this research is to introduce the asymptotic approaching of biophysical and condensed matter systems, based on Brownian motion fractal nature similarities, presented in mathematical analytical forms.
... 316319 INE Beam Interactions with materials fl Atoms Probing the threenucleon force using nuc... more ... 316319 INE Beam Interactions with materials fl Atoms Probing the threenucleon force using nucleondeuteron breakup reactions CR Howell a, , HR Setze a, RT Braun a, DE Gonzalez Trotter a, AH Hussein a,l, CD Roper a, F. Salinas a, 1 ... Cornelius and W. Gl6ckle, FewBody Sys. ...
Crystal structure and morphology of mechanically activated nanocrystalline Fe/BaTiO 3 was investi... more Crystal structure and morphology of mechanically activated nanocrystalline Fe/BaTiO 3 was investigated using a combination of spectroscopic and microscopic methods. These show that mechanical activation led to the creation of new surfaces and the comminution of the initial powder particles. Prolonged milling resulted in formation of larger agglomerates of BaTiO 3 and bimodal particle size distribution, where BaTiO 3
We point out a weak side of the commonly used determination of scalar cosmological perturbations ... more We point out a weak side of the commonly used determination of scalar cosmological perturbations lying in the fact that their average values can be nonzero for some matter distributions. It is shown that introduction of the finite-range gravitational potential instead of the infinite-range one resolves this problem. The concrete illustrative density profile is investigated in detail in this connection.
The cardinality of real numbers and validity of continuum hypothesis depend on the axioms that ar... more The cardinality of real numbers and validity of continuum hypothesis depend on the axioms that are implemented. In the inner-model axiom 'V-ultimate L' it is true while if Martin's axiom is correct, then the continuum hypothesis is false. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals as Skolem and Carnap first point it out. To show the impact of axiomatization, and validity of Löwenheim-Skolem theorem, that every nondenumerable set becomes denumerable in a higher system, considered is a nondenumerable set in ZF model, showing that it is denumerable in ZF model expanded with AC axiom, or in an absolute sense. It is done by considering the set containing all algebraic pairs and transcendental numbers obtained by a function f that between each of the paris places one transcendental number and showing that in ZFC model that set is equivalent to set of real numbers. That supports the Löwenheim-Skolem theorem because such a set is denumerable. To underline obtained result we will modify the Gödel constructible model universe and generate set of real numbers, starting with set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's Σ constructive model, which is different from ZF, result in set of all real numbers. For the first time, provided will be an explicit example that satisfies Wang's model. Discussed will be appeared disagreement with the diagonal argument that is explained by the intersection theorem and infinite induction method.
The cardinality of real numbers and validity of continuum hypothesis depend on the axioms that ar... more The cardinality of real numbers and validity of continuum hypothesis depend on the axioms that are implemented. In the inner-model axiom 'V-ultimate L' it is true while if Martin's axiom is correct, then the continuum hypothesis is false. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals as Skolem and Carnap first point it out. To show the impact of axiomatization, and validity of Löwenheim-Skolem theorem, that every nondenumerable set becomes denumerable in a higher system, considered is a nondenumerable set in ZF model, showing that it is denumerable in ZF model expanded with AC axiom, or in an absolute sense. It is done by considering the set containing all algebraic pairs and transcendental numbers obtained by a function f that between each of the paris places one transcendental number and showing that in ZFC model that set is equivalent to set of real numbers. That supports the Löwenheim-Skolem theorem because such a set is denumerable. To underline obtained result we will modify the Gödel constructible model universe and generate set of real numbers, starting with set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's Σ constructive model, which is different from ZF, result in set of all real numbers. For the first time, provided will be an explicit example that satisfies Wang's model. Discussed will be appeared disagreement with the diagonal argument that is explained by the intersection theorem and infinite induction method.
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel's and Cohen's proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms, there have been efforts to add at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom 'V-ultimate L' that builds a universe of sets from the ground up and Martin's axiom that allows to expand it outward in all directions. If Martin's axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models; if a theory has a nondenumerable model, it must have denumerably infinite ones as well. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals Skolem and Carnap first point it out. To show the impact of axiomatization, consider will be as an example a set created by a successive generation of algebraic pairs, each filled with a single transcendental number, and examined will be the impact of the axiom of choice on the determining the property of such set. It will be shown that the given example supports the Löwenheim-Skolem theorem. To underline the uncertainty implied by that theorem considered will be Wang's Σ model that generates a set in a constructive way, different from ZF. For the first time, provided will be an explicit example that satisfies Wang's model. In our example, a set of algebraic numbers will correspond to the 0-th Wang's layer, the 1-st layer will be a set of transcendental numbers generated by diagonal cuts applied to the 0 − th layers, and successive layers will be produced by diagonal cuts applied on the previously generated union of layers. It will be shown that this model also supports the Löwenheim-Skolem theorem.
This work is partially motivated by the recent articles related to the continuum hypothesis and t... more This work is partially motivated by the recent articles related to the continuum hypothesis and the forcing axioms. Ever since Gödel's and Cohen's proofs that the ZFC cannot be used to disprove or approve continuum hypothesis, i.e., that it is independent of the ZFC axioms, there have been efforts to add at least one new axiom to ZFC that should illuminate the structure of infinite sets. The two main contenders for addition to ZFC are the inner-model axiom 'V-ultimate L' that builds a universe of sets from the ground up and Martin's axiom that allows to expand it outward in all directions. If Martin's axiom is correct, then the continuum hypothesis is false, and if the inner-model axiom is right, then the continuum hypothesis is true. There is also an orthogonal view assuming that there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false, but all equally worth of exploring. That model dependent view is underlined with the Löwenheim-Skolem theorem, no theory can have only nondenumerable models; if a theory has a nondenumerable model, it must have denumerably infinite ones as well. Accepting axiomatization in whatever shape means the notions 'denumerable' and 'nondenumerable' turn out to be relative, and it requires to accept the relativization of cardinals Skolem and Carnap first point it out. To show the impact of axiomatization, consider will be as an example a set created by a successive generation of algebraic pairs, each filled with a single transcendental number, and examined will be the impact of the axiom of choice on the determining the property of such set. It will be shown that the given example supports the Löwenheim-Skolem theorem. To underline the uncertainty implied by that theorem considered will be Wang's Σ model that generates a set in a constructive way, different from ZF. For the first time, provided will be an explicit example that satisfies Wang's model. In our example, a set of algebraic numbers will correspond to the 0-th Wang's layer, the 1-st layer will be a set of transcendental numbers generated by diagonal cuts applied to the 0 − th layers, and successive layers will be produced by diagonal cuts applied on the previously generated union of layers. It will be shown that this model also supports the Löwenheim-Skolem theorem.
The cardinality of real numbers depends on the chosen axiomatic system and specific models of set... more The cardinality of real numbers depends on the chosen axiomatic system and specific models of set theory. Under the ZF framework, real numbers are nondenumerable, but their exact cardinality is debated, depending on extensions of the ZF axioms. We explored this by analyzing sets in the ZFC model and Wang's $\Sigma$ model. Set $S'$, within the ZFC framework, includes all pairs of algebraic numbers with a transcendental number inserted between each pair by a function $f$, which inserts an arbitrarily chosen transcendental number into every interval with algebraic endpoints. This method allows for the representation of any number using these intervals. Employing the AC, one can identify intervals from any Cauchy sequence within $S'$. Since each interval is non-empty due to the action of the function $f$, the intersections of these intervals contain numbers defined by these sequences, making $S'$ both complete and countable.
The second set, derived by inserting specifically defined transcendental numbers between algebraic numbers, begins with the sequence of algebraic numbers as the 0-th layer in Wang's $\Sigma$ model. A diagonal procedure applied to that layer creates a set of transcendental numbers, which combined with the 0-th layer form the $\Sigma_1$ layer. Continuing this recursive procedure results in a set equivalent to set $S'$. This suggests that all constructive real numbers are countable, challenging the traditional diagonal argument. The application of the AC is crucial for selecting transcendental numbers and identifying intervals and sequences, making the set $S'$ complete and countable. This might resonate with constructivists who oppose the AC. However, further exploration revealed conflicts in non-denumerability proofs with the Nested Interval Property, indicating a need for reevaluation of the countability of real numbers.
The cardinality of real numbers depends on the chosen axiomatic system and specific models of set... more The cardinality of real numbers depends on the chosen axiomatic system and specific models of set theory. Under the ZF framework, real numbers are nondenumerable, but their exact cardinality is debated, depending on extensions of the ZF axioms. We explored this by analyzing sets in the ZFC model and Wang's $\Sigma$ model. Set $S'$, within the ZFC framework, includes all pairs of algebraic numbers with a transcendental number inserted between each pair by a function $f$, which inserts an arbitrarily chosen transcendental number into every interval with algebraic endpoints. This method allows for the representation of any number using these intervals. Employing the AC, one can identify intervals from any Cauchy sequence within $S'$. Since each interval is non-empty due to the action of the function $f$, the intersections of these intervals contain numbers defined by these sequences, making $S'$ both complete and countable.
The second set, derived by inserting specifically defined transcendental numbers between algebraic numbers, begins with the sequence of algebraic numbers as the 0-th layer in Wang's $\Sigma$ model. A diagonal procedure applied to that layer creates a set of transcendental numbers, which combined with the 0-th layer form the $\Sigma_1$ layer. Continuing this recursive procedure results in a set equivalent to set $S'$. This suggests that all constructive real numbers are countable, challenging the traditional diagonal argument. The application of the AC is crucial for selecting transcendental numbers and identifying intervals and sequences, making the set $S'$ complete and countable. This might resonate with constructivists who oppose the AC. However, further exploration revealed conflicts in non-denumerability proofs with the Nested Interval Property, indicating a need for reevaluation of the countability of real numbers.
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Papers by B. Vlahović
Through rigorous proofs, we demonstrate that \( S \) encompasses all real numbers by recognizing each within a nested sequence of intervals, corresponding to an element \( x_\alpha \) in \( S \). Furthermore, we establish that any Dedekind cut in \( S \) does not produce gaps, thereby confirming \( S \) as a complete and countable representation of the real line.
In this article, we provide a surprising result that the denumerability of real numbers is possible using the Axiom of Choice. The proof is sketched here and elaborated further in the text. First, we consider an interval $[a,b]$, where $a<b$, that contains only algebraic numbers and define a set $S$, which by definition contains all possible algebraic intervals $I_n=[a_n, b_n]$, and $a_n \leqslant b_n$, where $a_n$ and $b_n$, are algebraic numbers from $[a,b]$. Then defined is a function $f$ that for each interval $I_n$ assigns a transcendental number $t_n$, so that $t_n$ is within the interval on which function $f$ is acting, i.e.
$f(I_n)=t_n$, and $a_n \leqslant t_n \leqslant b_n$. It is also required that $t _n$ is in all intervals containing that interval, i.e. in all superintervals. The numbers $t_n$ are then added to the set $S$, and to all algebraic intervals $I_n$ of the set defining the set $S’$, which is countable and complete. The proof is simple and based on contra example. Assume that there is a missing transcendental number $T_x$ that is different from all $t_n$ numbers generated by the function $f$ and therefore from all numbers in the set $S’$, and define that missing number $T_x$ with a sequence of nested closed intervals with algebraic endpoints $\alpha_n$ and $\beta_n$, so
$T_x=[\alpha_1, \beta_1]\supset [\alpha_2, \beta_2] \supset [\alpha_3, \beta_3]… \supset [\alpha_n, \beta_n]…$.
Using the Axiom of Choice, one can recognize in the set $S’$ closed intervals $I_n=[a_n,b_n]$ that have the same endpoints as the closed intervals $[\alpha_n, \beta_n]$ in the sequence that defines the missing number $T_x$. So, one can find in the set $S’$ all closed intervals $I_n$ with the same endpoints $a_n=\alpha_n$ and $b_n=\beta_n$,
which can be placed together to create a sequence
$T_x=[a_1,b_1]\supset [a_2,b_2] \supset [a_3,b_3]… \supset [a_n,b_n]…$
So, using the AC, one can recognize in the set $S’$ sequence of intervals $[a_n,b_n]$ that is exactly the same as the sequence of intervals $[\alpha_n, \beta_n]$ that represents the missing number, i.e. it has closed intervals with the same endpoints. To show that set $S'$ is complete, that there is no missing number $T_x$, that $T_x$ is in the set $S’$, it is enough to show that the intersection of the sequence of closed intervals $[a_n,b_n]$ is not empty, that it contains the number $T_x$. It is shown using the Intersection Property Theorem and applying it to the sequence of intervals $[\alpha_n, \beta_n]$ that defines the $T_x$ number. To satisfy the requirement that the function $f$ acted on all $I_n$ intervals in the set $S$, the same $T_x$ number must be in the sequence with the $[a_n,b_n]$ intervals that are from the set $S'$.
The power of AC is that it does not require to define sequence of any real number. Instead, it allows to recognize in the set $S'$ any sequence representing any real number. The sequence of any real number is implicitly in the set $S'$ because all possible intervals with algebraic endpoints are in the set $S'$. Since by AC in the set $S'$ can be recognized any arbitrary sequence representing any number, and since these sequences have nonempty intersections, the set $S'$ is complete.
Assuming the validity of AC and its consistency with ZF axioms, the discrepancy with the diagonal argument and other proofs of nondenumerability must originate in logical or technical problems common for these proofs, which we discuss in the second part of the article. In each of them, there is either a logical problem that the inability to assign in a sequence a specific n-th place for a number implies that that number is not in the sequence or a technical problem related to ignoring the Intersection Property Theorem in some elements of the proofs, or both of these problems.
The proof of denumerability of real numbers provided in the first part of the article does not specify the sequence of real numbers since numbers $t_n$ are chosen arbitrarily. In the third part of the article, we provide an explicit sequence of all real numbers. It is done by modifying the G\"{o}del constructible model universe and generating a set of real numbers, starting with a set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's $\Sigma$ constructive model, which is different from ZF, result in a set of all real numbers. For the first time, provided is an explicit example that satisfies Wang's model.
Through rigorous proofs, we demonstrate that \( S \) encompasses all real numbers by recognizing each within a nested sequence of intervals, corresponding to an element \( x_\alpha \) in \( S \). Furthermore, we establish that any Dedekind cut in \( S \) does not produce gaps, thereby confirming \( S \) as a complete and countable representation of the real line.
In this article, we provide a surprising result that the denumerability of real numbers is possible using the Axiom of Choice. The proof is sketched here and elaborated further in the text. First, we consider an interval $[a,b]$, where $a<b$, that contains only algebraic numbers and define a set $S$, which by definition contains all possible algebraic intervals $I_n=[a_n, b_n]$, and $a_n \leqslant b_n$, where $a_n$ and $b_n$, are algebraic numbers from $[a,b]$. Then defined is a function $f$ that for each interval $I_n$ assigns a transcendental number $t_n$, so that $t_n$ is within the interval on which function $f$ is acting, i.e.
$f(I_n)=t_n$, and $a_n \leqslant t_n \leqslant b_n$. It is also required that $t _n$ is in all intervals containing that interval, i.e. in all superintervals. The numbers $t_n$ are then added to the set $S$, and to all algebraic intervals $I_n$ of the set defining the set $S’$, which is countable and complete. The proof is simple and based on contra example. Assume that there is a missing transcendental number $T_x$ that is different from all $t_n$ numbers generated by the function $f$ and therefore from all numbers in the set $S’$, and define that missing number $T_x$ with a sequence of nested closed intervals with algebraic endpoints $\alpha_n$ and $\beta_n$, so
$T_x=[\alpha_1, \beta_1]\supset [\alpha_2, \beta_2] \supset [\alpha_3, \beta_3]… \supset [\alpha_n, \beta_n]…$.
Using the Axiom of Choice, one can recognize in the set $S’$ closed intervals $I_n=[a_n,b_n]$ that have the same endpoints as the closed intervals $[\alpha_n, \beta_n]$ in the sequence that defines the missing number $T_x$. So, one can find in the set $S’$ all closed intervals $I_n$ with the same endpoints $a_n=\alpha_n$ and $b_n=\beta_n$,
which can be placed together to create a sequence
$T_x=[a_1,b_1]\supset [a_2,b_2] \supset [a_3,b_3]… \supset [a_n,b_n]…$
So, using the AC, one can recognize in the set $S’$ sequence of intervals $[a_n,b_n]$ that is exactly the same as the sequence of intervals $[\alpha_n, \beta_n]$ that represents the missing number, i.e. it has closed intervals with the same endpoints. To show that set $S'$ is complete, that there is no missing number $T_x$, that $T_x$ is in the set $S’$, it is enough to show that the intersection of the sequence of closed intervals $[a_n,b_n]$ is not empty, that it contains the number $T_x$. It is shown using the Intersection Property Theorem and applying it to the sequence of intervals $[\alpha_n, \beta_n]$ that defines the $T_x$ number. To satisfy the requirement that the function $f$ acted on all $I_n$ intervals in the set $S$, the same $T_x$ number must be in the sequence with the $[a_n,b_n]$ intervals that are from the set $S'$.
The power of AC is that it does not require to define sequence of any real number. Instead, it allows to recognize in the set $S'$ any sequence representing any real number. The sequence of any real number is implicitly in the set $S'$ because all possible intervals with algebraic endpoints are in the set $S'$. Since by AC in the set $S'$ can be recognized any arbitrary sequence representing any number, and since these sequences have nonempty intersections, the set $S'$ is complete.
Assuming the validity of AC and its consistency with ZF axioms, the discrepancy with the diagonal argument and other proofs of nondenumerability must originate in logical or technical problems common for these proofs, which we discuss in the second part of the article. In each of them, there is either a logical problem that the inability to assign in a sequence a specific n-th place for a number implies that that number is not in the sequence or a technical problem related to ignoring the Intersection Property Theorem in some elements of the proofs, or both of these problems.
The proof of denumerability of real numbers provided in the first part of the article does not specify the sequence of real numbers since numbers $t_n$ are chosen arbitrarily. In the third part of the article, we provide an explicit sequence of all real numbers. It is done by modifying the G\"{o}del constructible model universe and generating a set of real numbers, starting with a set of algebraic numbers and expanding it iteratively by adding in each step transcendental numbers and creating bigger and bigger sets of real numbers. That iterative process that by diagonal procedure creates in each step a new layer of transcendental numbers will cumulatively, similarly to Wang's $\Sigma$ constructive model, which is different from ZF, result in a set of all real numbers. For the first time, provided is an explicit example that satisfies Wang's model.
The second set, derived by inserting specifically defined transcendental numbers between algebraic numbers, begins with the sequence of algebraic numbers as the 0-th layer in Wang's $\Sigma$ model. A diagonal procedure applied to that layer creates a set of transcendental numbers, which combined with the 0-th layer form the $\Sigma_1$ layer. Continuing this recursive procedure results in a set equivalent to set $S'$. This suggests that all constructive real numbers are countable, challenging the traditional diagonal argument. The application of the AC is crucial for selecting transcendental numbers and identifying intervals and sequences, making the set $S'$ complete and countable. This might resonate with constructivists who oppose the AC. However, further exploration revealed conflicts in non-denumerability proofs with the Nested Interval Property, indicating a need for reevaluation of the countability of real numbers.
The second set, derived by inserting specifically defined transcendental numbers between algebraic numbers, begins with the sequence of algebraic numbers as the 0-th layer in Wang's $\Sigma$ model. A diagonal procedure applied to that layer creates a set of transcendental numbers, which combined with the 0-th layer form the $\Sigma_1$ layer. Continuing this recursive procedure results in a set equivalent to set $S'$. This suggests that all constructive real numbers are countable, challenging the traditional diagonal argument. The application of the AC is crucial for selecting transcendental numbers and identifying intervals and sequences, making the set $S'$ complete and countable. This might resonate with constructivists who oppose the AC. However, further exploration revealed conflicts in non-denumerability proofs with the Nested Interval Property, indicating a need for reevaluation of the countability of real numbers.