Jeremy Gwiazda received his Ph.D. in philosophy from the City University of New York Graduate Center. One area of his current research is the infinite, where he is interested in such questions as: What is the difference between the potential and the actual infinite? Is it logically possible to complete infinitely many tasks in a finite time? And, which objects are the infinite numbers?
Gwiazda J. (2012) On Infinite Number and Distance. Constructivist Foundations 7(2): 126–130. Fulltext at https://cepa.info/255
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.