Incompleteness and Mathematical Truth

Curtis Franks

In this essay I defend a brand of mathematical realism against several problems that a mathematical realist must face when dealing with incompleteness in formal systems. To emphasize the scope of my argument, I discuss reasons why mathematical realism corresponds with the most attractive general theories of truth, and I make precise the notion of formal incompleteness by comparing it to other types of undecidability. Throughout the paper, I use the continuum hypothesis as the model incomplete statement. From more general arguments I suggest that the continuum hypothesis has a precise truth value, and within the specific context of set theory I suggest both that it is false and how it might be shown to be.