this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of consistency, borrowed primarily from the rigors of modern developments in logic, has prevented latter day twentieth century philosophers from producing philosophical systems of the type produced in earlier times.

We use the sequence version of the completeness axiom and the definition of real number as stated in that paper, and we also invoke propositions proved early on in that paper. The proofs are presented here with an eye toward their use in a Calculus One classroom. We aim to introduce students to a mathematically correct idea of real number and also get them to practice thinking mathematically rather than simply adopting propositions because they “feel right. ” On the other hand, while we do want to expose students to a mild level of rigor, we don’t want to overwhelm them with it. We allow as given all of the usual facts about the natural numbers (including at least an intuitive understanding of the wellordering principle) and the rationals, saving the rigor for the truly new idea, that of real number. And to further avoid belaboring the point, we won’t worry about details such as just when two rational sequences yield the same real number, or bother to verify basic properties of reals such as commutativity, associativity, and trichotomy. Those issues can still be examined outside of class