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Introduction

Mathematics strives for the most powerful and most general system possible. In the past this has led to inconsistent systems. Constructivists maintain that the only way to insure the correctness of mathematical reasoning about infinite sets is through constructive proofs. Proof by contradiction or the excluded middle is not allowed. This means that one cannot assume that an ideal computer program must either halt or not halt. Each step of an ideal computer program is logically determined. Assuming that the program will either halt in a finite time or run forever seems unproblematic. In general if a property of a computer program is determined by events enumerated in the execution of that program than the property would seem to be meaningful and objectively either true or false.

Central to this issue is the ontological status of mathematical abstractions. Finite sets can at least in theory exist as physical structures. In what sense are infinite sets meaningful? Properties of recursive processes or computer programs are objectively meaningful if they are determined by recursively enumerable finite events. All of the events could be physically realized in a potentially infinite universe.

This philosophical point of view does not see these properties as existing in some ideal state. They are created by human endeavor yet they are objectively determined. The recognition that these properties are objectively meaningful is a creative human act.


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