On Infinite Number and Distance.

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 + (&ohgr; * + &ohgr; ) &agr; + &ohgr; *. 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 &ohgr; + (&ohgr; * + &ohgr; ) &agr;+ &ohgr; *; in contrast, there are "indefinitely many" natural numbers. Constructivist content · The constructivist perspective of the paper is a form of strict finitism. [ABSTRACT FROM AUTHOR]