Construction of Ordinal Numbers and Arithmetic of Ordinal Numbers

Denik Agustito; Krida Singgih Kuncoro; Istiqomah Istiqomah; Agus Hendriyanto

doi:10.31764/jtam.v7i3.15039

Construction of Ordinal Numbers and Arithmetic of Ordinal Numbers

Denik Agustito, Krida Singgih Kuncoro, Istiqomah Istiqomah, Agus Hendriyanto

Abstract

The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include the methods of constructing both finite and transfinite ordinal numbers using two generation principles. Both finite and transfinite ordinal numbers are defined as well-orderings that are also transitive sets. Arithmetic of finite ordinal numbers is well-known, but the arithmetic of transfinite ordinal numbers will be introduced in this paper, including addition, multiplication, and exponentiation.

Keywords

Absolut; Aleph; Arithmetic; Cardinality; Ordinal; Transfinite.

Full Text:

DOWNLOAD [PDF]

References

Augustine, A. (2015). The City of God. Rosenberg: Xist Publishing.

Clark, J. R. (2017). Transfinite Ordinal Arithmetic. University Park.

Dasgupta, A. (2014). Set theory: with an introduction to real point sets. New York, NY: Birkhäuser. https://doi.org/10.1007/978-1-4614-8854-5

Dauben, J. W. (1979). Cantor’s Philosophy of the Infinite. In His Mathematics and Philosophy of the Infinite. Georg Cantor (pp. 120–148). New Jersey: Princeton University Press. https://doi.org/10.2307/j.ctv10crfh1.9

Der Veen, J., & Horsten, L. (2013). Cantorian infinity and philosophical concepts of God. European Journal for Philosophy of Religion, 5(3), 117–138. https://doi.org/10.24204/ejpr.v5i3.222

Drozdek, A. (1995). Beyond Infinity: Augustine and Cantor. Laval Theologique Et Philosophique, 51, 127–140. https://doi.org/10.7202/400897ar

Engler, J. P. (2023). Transcendental and mathematical infinity in Kant’s first antinomy. Inquiry, 1–25. https://doi.org/10.1080/0020174X.2023.2170912

Goldrei, D. C. (1996). Classic Set Theory: For Guided Independent Study. New York: Chapman & Hall/CRC Press Company.

Grim, P. (1991). The incomplete universe: totality, knowledge, and truth. Mit Press.

Jech, T. J. (2006). Lectures in set theory: with particular emphasis on the method of forcing (Vol. 217; H. A. Dold & Z. B. Eckmann, Eds.). New York: Springer.

Jourdain, P. E. B. (1910). Transfinite Numbers and the Principles of Mathematics. The Monist, 20(1), 93–118. https://doi.org/10.5840/monist191020131

Lobo, M. P., & Cardoso, M. P. (2020). Subtraction of Transfinite Ordinals. Open Journal of Mathematics and Physics, 2(91), 1–6. https://doi.org/10.31219/osf.io/yvrf3

Oppy, G. (2014). Describing gods: an investigation of divine attributes. Cambridge: Cambridge University Press.

Rauff, J. V. (2002). Transfinite numbers and native American cosmology: A course about infinity. PRIMUS, 12(1), 1–10. https://doi.org/10.1080/10511970208984013

Read, R., & Greiffenhagen, C. (2023). Finite and Infinite: On Not Making ‘Them’ Different Enough. In C. Coseru (Ed.), Reasons and Empty Persons: Mind, Metaphysics, and Morality: Essays in Honor of Mark Siderits (pp. 307–324). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-031-13995-6_16

Roszak, P., & Huzarek, T. (2019). Seeing God: Thomas aquinas on divine presence in the world. Bogoslovni Vestnik/Theological Quarterly, 79(3), 739–749. https://doi.org/10.34291/BV2019/03/Roszak

Russell, R. J. (2011). God and infinity: Theological insights from Cantor’s mathe-matics. In M. Heller & W. H. Woodin (Eds.), Infinity. New research frontiers(pp. 275–289). Cambridge: Cambridge University Press.

Sambin, G. (2019). Dynamics in Foundations: What Does It Mean in the Practice of Mathematics? In S. Centrone, D. Kant, & D. Sarikaya (Eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts (pp. 455–494). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-15655-8_21

Sheppard, B. (2014). The logic of infinity. Cambridge: Cambridge University Press.

Steinhart, E. C. (2009). A Mathematical Model of Divine Infinity. Theology and Science, 7(3), 261–274. https://doi.org/10.1080/14746700903036528

Thomas-Bolduc, A. R. (2016). Cantor, god, and inconsistent multiplicities. Studies in Logic, Grammar and Rhetoric, 44(57), 133–146. https://doi.org/10.1515/slgr-2016-0008

Williams, J. K. (2018). A Comprehensive Review of Seven Steps to a Comprehensive Literature Review. The Qualitative Report, 23(2), 345+. https://link.gale.com/apps/doc/A533000662/AONE?u=anon~f5492d27&sid=googleScholar&xid=815f2772

Zermelo, E. (2012). Ernst Zermelo - Collected Works/Gesammelte Werke (1st ed.; H.-D. Ebbinghaus, C. G. Fraser, & A. Kanamori, Eds.). Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-79384-7

DOI: https://doi.org/10.31764/jtam.v7i3.15039