Volume 7 · Number 2 · Pages 126–130 < Previous Paper · Next Paper > On Infinite Number and Distance Jeremy Gwiazda Download the full text in PDF (305 kB) Share on

Abstract

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism.

Key words: Cantor, infinite number, infinity, ordinals, infinite distance

Citation

Gwiazda J. (2012) On Infinite Number and Distance. Constructivist Foundations 7(2): 126–130. Available at http://constructivist.info/7/2/126

Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS)

Similar articles

References

Brouwer L. E. J. (1983) Consciousness, philosophy, and mathematics. In: Benacerraf B. & Putnam H. (eds.) Philosophy of mathematics: Selected readings. Second edition. Cambridge University Press, Cambridge: 90–96. << Google Scholar

Cantor G. (1955) Contributions to the founding of the theory of transfinite numbers. Dover, New York. << Google Scholar

Mancosu P. (2009) Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? The Review of Symbolic Logic 2: 612–646. << Google Scholar

Mayberry J. P. (2000) The foundations of mathematics in the theory of sets. Cambridge University Press: Cambridge. << Google Scholar

Nelson E. (1986) Predicative arithmetic. Princeton University Press. Princeton NJ. << Google Scholar

Parikh R. (1971) Existence and feasibility in arithmetic. The Journal of Symbolic Logic 36: 494–508. << Google Scholar

Parker M. W. (2009) Philosophical method and Galileo’s paradox of infinity. In: Van Kerkhove B. (ed.) New Perspectives on mathematical practices. World Scientific, New Jersey: 76–113. Preprint available at http://philsci-archive.pitt.edu/4276/ << Google Scholar

Robinson A. (1996) Non-standard analysis. Princeton University Press: Princeton NJ. << Google Scholar

Russell B. (1983) The collected papers of Bertrand Russell. Volume 3. George Allen & Unwin: London. << Google Scholar