Now that we have developed some familiarity with structures, we can turn our attention to implication. In the propositional logic setting, we defined two ways to say that a set $\Gamma \subseteq {\text{Form}}_P$ implies a formula $\phi \in {\text{Form}}_P$. In that case, the semantic approach involved examining truth assignments, and the syntactic approach relied on the development of a formal proof system. We then established, in the Soundness and Completeness Theorems, that these two concepts coincided. By jumping between the two approaches, we found the Compactness Theorem as a nice corollary.

Chapter 8 focused on describing first-order axiomatic set theory, and then showing how to embed mathematics within that theory. But is there anything more that the set-theoretic perspective provides to the mathematical tool kit beyond a unifying foundation and cute diagonal arguments?

One of the successful results of such a program is the ability to study mathematical language and reasoning using mathematics itself. For example, we will eventually give a precise mathematical definition of a formal proof, and to avoid confusion with our current intuitive understanding of what a proof is, we will call these objects deductions. One can think of our eventual definition of a deduction as analogous to the precise mathematical definition of continuity, which replaces the fuzzy “a graph that can be drawn without lifting your pencil.” Once we have codified the notion in this way, we will have turned deductions into precise mathematical objects, allowing us to prove mathematical theorems about deductions using normal mathematical reasoning. For example, we will open up the possibility of proving that there is no deduction of certain mathematical statements.

One of the major triumphs of early twentieth-century logic was the formulation of several (equivalent) precise definitions for what it means to say that a function $f:\mathbb{N}\to \mathbb{N}$ is computable. In our current age, many people have an intuitive sense of this concept through experience with computer programming. However, it is challenging to turn such intuition into a concise formal mathematical treatment that is susceptible to a rigorous mathematical analysis.

Set theory originated in an attempt to understand and somehow classify small, or negligible, sets of real numbers. Cantor’s early explorations into the realm of the transfinite were motivated by a desire to understand the points of convergence of trigonometric series. The basic ideas quickly became a fundamental part of analysis, in addition to permeating many other areas of mathematics. Since then, set theory has become a way to unify mathematical practice, and the way in which mathematicians grapple with the infinite in all areas of mathematics.

In many areas of mathematics (like partial orderings, groups, or graphs), we write down some axioms and immediately have several different models of these axioms in mind. In the setting of first-order logic, this corresponds to writing down a set Σ of sentences in a language and looking at the elementary class $\text{Mod}(\sum )$. Since $\text{Mod(Σ)}=\text{Mod(Cn(Σ))}$ by Proposition 6.5.3, and Cn(Σ) is a theory by Proposition 6.5.4, we can view this situation as looking at the (elementary) class of models of a theory.

Our development of a formal definition of computability in the previous chapter might have seemed out of place. We used our generation template and some simple references to propositional connectives and (bounded) quantifiers, but otherwise there was seemingly little connection to logic. In this chapter, we establish that computability and logic are fundamentally intertwined.

We now embark on a careful study of propositional logic. As described in Chapter 1, in this setting, we start with an arbitrary set P, which we think of as our collection of primitive statements. From here, we build up more complicated statements by repeatedly applying connectives. The corresponding process generates a set of syntactic objects that we call formulas. In order to assign meaning to these formulas, we introduce truth assignments, which are functions on P that propagate upward through formulas of higher complexity.

Many of our powerful results about first-order logic, such as the Löwenheim–Skolem Theorem and the Łoś-Vaught Test, focused on countable structures in countable languages. Now that we have a well-developed theory of infinite cardinalities, we can extend these results into the uncountable realm. In addition to the satisfaction we obtain through such generalizations, we will be able to argue that some other important theories are complete, and further refine our intuition about the inability of first-order logic to delineate between infinite cardinalities.

Suppose that we have a (first-order) language $ℒ$. As emphasized in , the elements of ${\text{Form}}_{ℒ}$ are just syntactic sequences of symbols, and we only attach meaning to these formulas once we provide an $ℒ$-structure together with a variable assignment. The fundamental separation between syntactic formulas and semantic structures is incredibly important, because it opens up an interesting way to find both commonalities and differences across structures. That is, given two structures with variable assignments $(ℳ{}_1,s_1)$ and $(ℳ{}_2,s_2)$, we can compare the two sets $\{\phi \in {\text{Form}}_{ℒ}:(ℳ{}_1,s_1)\models \phi \}$ and $\{\phi \in {\text{Form}}_{\mathcal{L}}:(\mathcal{M}{}_2,s_2)\models \phi \}$. Although the two structures and variable assignments likely live in different worlds, these two sets both live inside the same set ${\text{Form}}_{ℒ}$. In other words, the syntactic nature of the formulas provides a shared substrate where we can perform comparisons.

Proofs by induction and definitions by recursion are fundamental tools when working with the natural numbers. However, there are many other places where variants of these ideas apply. In fact, more delicate and exotic proofs by induction and definitions by recursion are two central tools in mathematical logic. We will eventually develop transfinite versions of these ideas in Chapter 9 to give us ways to continue into exotic, infinite realms, and these techniques are essential in both set theory and model theory. In this chapter, we develop the more modest tools of induction and recursion along structures that are generated by one-step processes, like the natural numbers. Occasionally, these types of induction and recursion are called structural.