"God, grant me the serenity to accept the things I cannot change, courage to change the things I can, and wisdom to know the difference." The Serenity Prayer, Reinhold Niebuhr
Level B1 is where the language stops being descriptive and becomes a tool for competent thought: propositional and predicate logic get built rigorously, symbol by symbol, and models turn syntax into something that can be checked against a semantics — culminating in Gödel's two most famous theorems.
The formalization of logic itself — turning it into an object of mathematical study — was a defining project at the turn of the 20th century: Frege, Russell and Whitehead's Principia Mathematica, and Hilbert's program, all aimed at rigorous axiomatic systems for logic strong enough to rebuild the whole of mathematics. At Level B1, the book follows the same path: propositional formulas are defined by exactly three rules (every variable is a formula; combinations under $\neg,\land,\lor,\to$ are formulas; nothing else is), and everything from there — derivability, soundness, completeness — is built strictly on top of that syntax.
The payoff is a genuine shift in what the language can do. Up through Level A2, $\Math$ described things. Now it can certify that a description follows from other descriptions — the difference between being able to speak a language and being able to reason competently in it.
A theory is consistent if no formula and its negation are both derivable from it; it is satisfiable if it has a model — some structure in which all of its theorems come out true. These sound like different properties: one is about proofs, the other about semantics. Gödel's Completeness Theorem for predicate logic says they're the same property, stated as a single criterion.
The two directions of this criterion have very different weights. Gödel proved the substantial one, left-to-right — consistent $\Rightarrow$ satisfiable — by building a model out of nothing but the language's own terms: every constant symbol goes into the universe, and every true existential formula contributes a "witness" term expressible in the language itself. The converse, right-to-left, is comparatively trivial: it follows directly from the soundness of predicate calculus with respect to models — if every axiom of a theory is true in some structure, then by induction on the length of a derivation, every formula derivable from those axioms is true there too, so a theory with a model can never derive both $\varphi$ and $\neg\varphi$.
An equivalent formulation of the same criterion: $\Gamma\Vdash\varphi$ is equivalent to $\Gamma\vdash\varphi$ — semantic consequence and syntactic derivability turn out to be exactly the same relation in classical predicate logic.
Incompleteness then draws the boundary: pure predicate logic, the theory of orders, of equality, of groups — all consistent and complete relative to the metatheory. The moment a theory can talk about its own provability (as Peano Arithmetic can, via Gödel numbering), completeness is lost — a fact independent formulas like the Axiom of Choice relative to $\mathsf{ZF}$, or Euclid's fifth postulate relative to absolute geometry, turn out to instantiate again and again.
The first five exercises from the chapter's problem set, from the English edition.
Level B1 has 20 exercises in total, each with a full worked solution in the book's appendix.