This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.

Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

This is the introductory half of a lecture given at the Association for Symbolic Logic Annual Meeting in Philadelphia in March, 2001. Our goal is to give logicians and advanced students who are unfamiliar with inner model theory a taste of what the subject is about at a level where it is possible to say something meaningful about the intuitions and proofs. At the same time, we wish to avoid overwhelming the reader, so we will leave out many well known closely related theorems and give only short proof sketches. Neither the results nor the proofs found here are due to the author. We will begin near the beginning but still touch on some modern aspects of the theory. The proofs and historical notes that we leave out can be found in the books and articles listed in the references. Our approach to the theory of 0# is a combination of that taken by Dodd in his book and by Steel in his lectures to students. Dodd's introduction is one of the author's favorite essays on inner model theory; another is the introduction to Martin-Steel [6]. In Section 2, we review the theory of 0# in a form that touches on some of the techniques that have been and continue to be generalized. These combine what is commonly called iteration and fine structure. In Section 3, we give a superficial description of what comes after 0#. Those interested in learning about more recent developments in inner model theory are encouraged to see the articles Lowe-Steel [5] and Steel [8], both of which are currently available online at http://www.math.berkeley.edu/~ steel.

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.