 To clarify this last point, let’s look at a non-mathematical example, one which neither Hilbert nor Frege explicitly dealt with. Consider the set of sentences {Jones had a nightmare, Jones didn’t have a dream}, or equivalently its first-order rendition, \(\{Nj, {{\sim}Dj}\}\). The set is clearly consistent in the sense used by Hilbert in FG; it is a straightforward matter to provide interpretations of “Jones”, “x had a nightmare” and “x had a dream” (or of “j”, “N”, and “D”) such that the sentences, so interpreted, express truths. (Consider, for example, an interpretation on which “j” is assigned the number 7, “N” the set of prime numbers, and “D” the set of numbers greater than 12.) But from the Fregean point of view, the thoughts expressed are arguably inconsistent, since part of what it is to have a nightmare is to have a dream. The inconsistency from Frege’s point of view can be demonstrated by providing an analysis of the thoughts expressed, and noting that the results of this analysis yield the set {Jones had a disturbing dream, Jones didn’t have a dream}.