## AbstractContext: 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 ## CitationGwiazda J. (2012) On Infinite Number and Distance. Constructivist Foundations 7(2): 126–130. Available at http://www.univie.ac.at/constructivism/journal/7/2/126.gwiazda Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS) ## ReferencesConsciousness, philosophy, and mathematics. In: Benacerraf B. & Putnam H. (eds.) Philosophy of mathematics: Selected readings. Second edition. Cambridge University Press, Cambridge: 90–96. Contributions to the founding of the theory of transfinite numbers. Dover, New York. 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. The foundations of mathematics in the theory of sets. Cambridge University Press: Cambridge. Predicative arithmetic. Princeton University Press. Princeton NJ. Existence and feasibility in arithmetic. The Journal of Symbolic Logic 36: 494–508. 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/ Non-standard analysis. Princeton University Press: Princeton NJ. The collected papers of Bertrand Russell. Volume 3. George Allen & Unwin: London. | ||