in PM that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems. For all that, PM notations are not widely used today: probably the foremost reason for this is that practicing mathematicians tend to assume that the background Foundation is a form of the system of Zermelo–Fraenkel set theory. Nonetheless, the scholarly, historical, and philosophical interest in PM is great and ongoing: for example, the Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century. There are also multiple articles on the work in the peer-reviewed Stanford Encyclopedia of Philosophy and academic researchers continue working with Principia, whether for the historical reason of understanding the text or its authors, or for mathematical reasons of understanding or developing Principias logical system. Scope of foundations laid The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third. Theoretical basis As noted in the criticism of the theory by Kurt Gödel (below), unlike a formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Furthermore in the theory, it is almost immediately observable that interpretations (in the sense of model theory) are presented in terms of truth-values for the behaviour of the symbols "⊢" (assertion of truth), "~" (logical not), and "V" (logical inclusive OR).Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw (pure) formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of "values", a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the "interpretation" of what the symbols commonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)".