The Problem of Countability of Highest Ordinals

Alexey Pavlovich Kulaichev

doi:doi:10.11648/j.pamj.20160504.14

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The Problem of Countability of Highest Ordinals

Alexey Pavlovich Kulaichev

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4)

Received: 16 March 2016 Accepted: 25 March 2016 Published: 28 June 2016

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Abstract

In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal is seen as the union for any j=1, n-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation . The first form corresponds to the geometric representation of ordinal as an infinite n-dimensional matrix. According traditional formulation , thus is -countable union of countable ordinals so is countable. According to alternate formulation for any n, thus is -union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit -union; 2) seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of -exponentiation of , i.e. , ,..., correspond to consecutive , , , ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 4)
DOI	10.11648/j.pamj.20160504.14
Page(s)	108-112
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Set Theory, Ordinals, Mathematical Induction, Absolute Truth

References

[1]	Georg Cantor, 1878, Ein beitrag zur mannigfaltigkeitslehre. Journal fr die reine und angewandte Mathematik, 84, 242-258.
[2]	Georg Cantor, 1883, Grundlagen einer allge. meinen Mannigfaltigkeitslehre: Ein mathematisch-philo-sophischer Versuch in der Lehre des Unendlichen. Leipzig: Teubner.
[3]	Deza, E., Deza, M., 2011, Figurate Numbers. World Scientific.
[4]	Felix Hausdorff, 1914, Grundzuge der Mengenlehre. Veit and Company, Leipzig.
[5]	Stephen Cole Kleene, 1967, Mathematical logic. N. Y.
[6]	Morris Kline, 1980, Mathematics. The loss of certainty. N. Y.
[7]	Kuratowski K., Mostowski A., 1967, Set theory. Amsterdam.
[8]	Lisi M., 2007, Some remarks on the Cantor pairing function. Le Matematiche, 62(1), 55-65.
[9]	Maydim A. Malkov, 2014, Stairs of Natural Set Theories, Pure and Applied Mathematics Journal. 3(3), 49-65.
[10]	Vereshchagin N. K., Shen A., 2002, Lectures on mathematical logic and theory of algorithms. Moscow, MCNME.

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Alexey Pavlovich Kulaichev. (2016). The Problem of Countability of Highest Ordinals. Pure and Applied Mathematics Journal, 5(4), 108-112. https://doi.org/10.11648/j.pamj.20160504.14

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Alexey Pavlovich Kulaichev. The Problem of Countability of Highest Ordinals. Pure Appl. Math. J. 2016, 5(4), 108-112. doi: 10.11648/j.pamj.20160504.14

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Alexey Pavlovich Kulaichev. The Problem of Countability of Highest Ordinals. Pure Appl Math J. 2016;5(4):108-112. doi: 10.11648/j.pamj.20160504.14

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@article{10.11648/j.pamj.20160504.14,
  author = {Alexey Pavlovich Kulaichev},
  title = {The Problem of Countability of Highest Ordinals},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {108-112},
  doi = {10.11648/j.pamj.20160504.14},
  url = {https://doi.org/10.11648/j.pamj.20160504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.14},
  abstract = {In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal  is seen as the union  for any j=1, n-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation . The first form corresponds to the geometric representation of ordinal  as an infinite n-dimensional matrix. According traditional formulation , thus  is -countable union of countable ordinals so  is countable. According to alternate formulation  for any n, thus  is -union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit -union; 2) seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of -exponentiation of , i.e. , ,..., correspond to consecutive , , , ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.},
 year = {2016}
}

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TY  - JOUR
T1  - The Problem of Countability of Highest Ordinals
AU  - Alexey Pavlovich Kulaichev
Y1  - 2016/06/28
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.14
DO  - 10.11648/j.pamj.20160504.14
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JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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EP  - 112
PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.pamj.20160504.14
AB  - In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal  is seen as the union  for any j=1, n-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation . The first form corresponds to the geometric representation of ordinal  as an infinite n-dimensional matrix. According traditional formulation , thus  is -countable union of countable ordinals so  is countable. According to alternate formulation  for any n, thus  is -union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit -union; 2) seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of -exponentiation of , i.e. , ,..., correspond to consecutive , , , ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.
VL  - 5
IS  - 4
ER  -

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Author Information

Alexey Pavlovich Kulaichev

Department of System Analysis, Moscow State University, Moscow, Russia

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APA Style

Alexey Pavlovich Kulaichev. (2016). The Problem of Countability of Highest Ordinals. Pure and Applied Mathematics Journal, 5(4), 108-112. https://doi.org/10.11648/j.pamj.20160504.14

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ACS Style

Alexey Pavlovich Kulaichev. The Problem of Countability of Highest Ordinals. Pure Appl. Math. J. 2016, 5(4), 108-112. doi: 10.11648/j.pamj.20160504.14

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AMA Style

Alexey Pavlovich Kulaichev. The Problem of Countability of Highest Ordinals. Pure Appl Math J. 2016;5(4):108-112. doi: 10.11648/j.pamj.20160504.14

Copy | Download

@article{10.11648/j.pamj.20160504.14,
  author = {Alexey Pavlovich Kulaichev},
  title = {The Problem of Countability of Highest Ordinals},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {108-112},
  doi = {10.11648/j.pamj.20160504.14},
  url = {https://doi.org/10.11648/j.pamj.20160504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.14},
  abstract = {In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal  is seen as the union  for any j=1, n-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation . The first form corresponds to the geometric representation of ordinal  as an infinite n-dimensional matrix. According traditional formulation , thus  is -countable union of countable ordinals so  is countable. According to alternate formulation  for any n, thus  is -union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit -union; 2) seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of -exponentiation of , i.e. , ,..., correspond to consecutive , , , ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Problem of Countability of Highest Ordinals
AU  - Alexey Pavlovich Kulaichev
Y1  - 2016/06/28
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.14
DO  - 10.11648/j.pamj.20160504.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 108
EP  - 112
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.14
AB  - In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal  is seen as the union  for any j=1, n-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation . The first form corresponds to the geometric representation of ordinal  as an infinite n-dimensional matrix. According traditional formulation , thus  is -countable union of countable ordinals so  is countable. According to alternate formulation  for any n, thus  is -union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit -union; 2) seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of -exponentiation of , i.e. , ,..., correspond to consecutive , , , ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.
VL  - 5
IS  - 4
ER  -

Copy | Download