The Ordinal Interpretation of the Integers and Its Use in Number Theory

Abstract

The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography; and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.