step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs . . . The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens . . . it is chiefly the rule of substitution which would have to be proved" (Gödel 1944:124) Contents Part I Mathematical logic. Volume I ✸1 to ✸43 This section describes the propositional and predicate calculus, and gives the basic properties of classes, relations, and types. Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97 This part covers various properties of relations, especially those needed for cardinal arithmetic. Part III Cardinal arithmetic. Volume II ✸100 to ✸126 This covers the definition and basic properties of cardinals. A cardinal is defined to be an equivalence class of similar classes (as opposed to ZFC, where a cardinal is a special sort of von Neumann ordinal). Each type has its own collection of cardinals associated with it, and there is a considerable amount of bookkeeping necessary for comparing cardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite cardinals. ✸120.03 is the Axiom of infinity. Part IV Relation-arithmetic. Volume II ✸150 to ✸186 A "relation-number" is an equivalence class of isomorphic relations. PM defines analogues of addition, multiplication, and exponentiation for arbitrary relations. The addition and multiplication is similar to the usual definition of addition and multiplication of ordinals in ZFC, though the definition of exponentiation of relations in PM is not equivalent to the usual one used in ZFC. Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276 This covers series, which is PM's term for what is now called a totally ordered set. In particular it covers complete series, continuous functions between series with the order topology (though of course they do not use this terminology), well-ordered series, and series without "gaps" (those with a member strictly between any two given members). Part VI Quantity. Volume III ✸300 to ✸375 This section constructs the ring of integers, the fields of rational and real numbers, and "vector-families", which are related to what are now called torsors over abelian groups. Comparison with set theory This section compares