Principia Mathematica


The Principia Mathematica is a three-volume work on the foundations of mathematics written by the philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✸9 and all-new Appendix B and Appendix C. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM was originally conceived as a sequel volume to Russell's 1903 Principles, but as PM states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."
PM, according to its introduction, had three aims: to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions and axioms, and inference rules; to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; to solve the paradoxes that plagued logic and set theory at the turn of the 20th-century, like Russell's paradox.
This third aim motivated the adoption of the theory of types in PM. The theory of types adopts grammatical restrictions on formulas that rules out the unrestricted comprehension of classes, properties, and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of PM.
There is no doubt that PM is of great importance in the history of mathematics and philosophy: as Irvine has noted, it sparked interest in symbolic logic and advanced the subject by popularizing it; it showcased the powers and capacities of symbolic logic; and it showed how advances in philosophy of mathematics and symbolic logic could go hand-in-hand with tremendous fruitfulness. Indeed, PM was in part brought about by an interest in logicism, the view on which all mathematical truths are logical truths. It was in part thanks to the advances made in PM that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems.
For all that, PM is not widely used today: probably the foremost reason for this is its reputation for typographical complexity. Somewhat infamously, several hundred pages of
PM precede the proof of the validity of the proposition 1+1=2. Contemporary mathematicians tend to use a modernized 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.

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, unlike a formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that almost immediately in the theory, interpretations are presented in terms of truth-values for the behaviour of the symbols "⊢", "~", and "V".
Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw 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., "¬ ". But this is not a pure Formalist theory.

Contemporary construction of a formal theory

The following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows:
  1. Symbols used: This set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols: "→", "&", "V", "¬", "∀", "∃" ; predicate symbol "=" ; function symbols "+", "∙", "'" ; individual symbol "0" ; variables "a", "b", "c", etc.; and parentheses "".
  2. Symbol strings: The theory will build "strings" of these symbols by concatenation.
  3. Formation rules: The theory specifies the rules of syntax usually as a recursive definition that starts with "0" and specifies how to build acceptable strings or "well-formed formulas". This includes a rule for "substitution" of strings for the symbols called "variables".
  4. Transformation rule: The axioms that specify the behaviours of the symbols and symbol sequences.
  5. Rule of inference, detachment, modus ponens : The rule that allows the theory to "detach" a "conclusion" from the "premises" that led up to it, and thereafter to discard the "premises". If this were not the case, then substitution would result in longer and longer strings that have to be carried forward. Indeed, after the application of modus ponens, nothing is left but the conclusion, the rest disappears forever.

    Construction

The theory of PM has both significant similarities, and similar differences, to a contemporary formal theory. Kleene states that "this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intended to be believed, or at least to be accepted as plausible hypotheses concerning the world". Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, PM introduces the notion of "truth-values", i.e., truth and falsity in the real-world sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory :
  1. Variables
  2. Uses of various letters
  3. The fundamental functions of propositions: "the Contradictory Function" symbolised by "~" and the "Logical Sum or Disjunctive Function" symbolised by "∨" being taken as primitive and logical implication defined as
p ⊃ q .=. ~ p ∨ q Df.
and logical product defined as
p . q .=. ~ Df.
  1. Equivalence: Logical equivalence, not arithmetic equivalence: "≡" given as a demonstration of how the symbols are used, i.e., "Thus ' p ≡ q ' stands for ' . '.". Notice that to discuss a notation PM identifies a "meta"-notation with "... ":
Logical equivalence appears again as a definition:
p ≡ q .=. . ,
Notice the appearance of parentheses. This grammatical usage is not specified and appears sporadically; parentheses do play an important role in symbol strings, however, e.g., the notation "" for the contemporary "∀x".
  1. Truth-values: "The 'Truth-value' of a proposition is truth if it is true, and falsehood if it is false" .
  2. Assertion-sign: "'⊦. p may be read 'it is true that'... thus '⊦: p .⊃. q ' means 'it is true that p implies q ', whereas '⊦. p .⊃⊦. q ' means ' p is true; therefore q is true'. The first of these does not necessarily involve the truth either of p or of q, while the second involves the truth of both".
  3. Inference: PM 's version of modus ponens. " '⊦. p ' and '⊦ ' have occurred, then '⊦ . q ' will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of '⊦.' q ' ".
  4. The use of dots
  5. Definitions: These use the "=" sign with "Df" at the right end.
  6. Summary of preceding statements: brief discussion of the primitive ideas "~ p" and "p ∨ q" and "⊦" prefixed to a proposition.
  7. Primitive propositions: the axioms or postulates. This was significantly modified in the second edition.
  8. Propositional functions: The notion of "proposition" was significantly modified in the second edition, including the introduction of "atomic" propositions linked by logical signs to form "molecular" propositions, and the use of substitution of molecular propositions into atomic or molecular propositions to create new expressions.
  9. The range of values and total variation
  10. Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned in the second edition. In particular, the distinction between the concepts defined in sections 15. Definition and the real variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.
  11. Formal implication and formal equivalence
  12. Identity
  13. Classes and relations
  14. Various descriptive functions of relations
  15. Plural descriptive functions
  16. Unit classes

    Primitive ideas

Cf. PM 1962:90–94, for the first edition:
The first edition begins with a definition of the sign "⊃"
✸1.01. p ⊃ q .=. ~ p ∨ q. Df.
✸1.1. Anything implied by a true elementary proposition is true. Pp modus ponens
✸1.2. ⊦: p ∨ p .⊃. p. Pp principle of tautology
✸1.3. ⊦: q .⊃. p ∨ q. Pp principle of addition
✸1.4. ⊦: p ∨ q .⊃. q ∨ p. Pp principle of permutation
✸1.5. ⊦: p ∨ .⊃. q ∨. Pp associative principle
✸1.6. ⊦:. q ⊃ r .⊃: p ∨ q .⊃. p ∨ r. Pp principle of summation
✸1.7. If p is an elementary proposition, ~p is an elementary proposition. Pp
✸1.71. If p and q are elementary propositions, p ∨ q is an elementary proposition. Pp
✸1.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨ ψp is an elementary proposition. Pp
Together with the "Introduction to the Second Edition", the second edition's Appendix A abandons the entire section ✸9. This includes six primitive propositions ✸9 through ✸9.15 together with the Axioms of reducibility.
The revised theory is made difficult by the introduction of the Sheffer stroke to symbolise "incompatibility", the contemporary logical NAND. In the revised theory, the Introduction presents the notion of "atomic proposition", a "datum" that "belongs to the philosophical part of logic". These have no parts that are propositions and do not contain the notions "all" or "some". For example: "this is red", or "this is earlier than that". Such things can exist ad finitum, i.e., even an "infinite enumeration" of them to replace "generality". PM then "advance to molecular propositions" that are all linked by "the stroke". Definitions give equivalences for "~", "∨", "⊃", and ".".
The new introduction defines "elementary propositions" as atomic and molecular positions together. It then replaces all the primitive propositions ✸1.2 to ✸1.72 with a single primitive proposition framed in terms of the stroke:
The new introduction keeps the notation for "there exists" and "for all". Appendix A strengthens the notion of "matrix" or "predicative function" and presents four new Primitive propositions as ✸8.1–✸8.13.
✸88. Multiplicative axiom
✸120. Axiom of infinity

Ramified types and the axiom of reducibility

In simple type theory objects are elements of various disjoint "types". Types are implicitly built up as follows. If τ1,...,τm are types then there is a type that can be thought of as the class of propositional functions of τ1,...,τm. In particular there is a type of propositions, and there may be a type ι of "individuals" from which other types are built. Russell and Whitehead's notation for building up types from other types is rather cumbersome, and the notation here is due to Church.
In the ramified type theory of PM all objects are elements of various disjoint ramified types. Ramified types are implicitly built up as follows. If τ1,...,τm,σ1,...,σn are ramified types then as in simple type theory there is a type of "predicative" propositional functions of τ1,...,τm,σ1,...,σn. However, there are also ramified types that can be thought of as the classes of propositional functions of τ1,...τm obtained from propositional functions of type by quantifying over σ1,...,σn. When n=0 these propositional functions are called predicative functions or matrices. This can be confusing because current mathematical practice does not distinguish between predicative and non-predicative functions, and in any case PM never defines exactly what a "predicative function" actually is: this is taken as a primitive notion.
Russell and Whitehead found it impossible to develop mathematics while maintaining the difference between predicative and non-predicative functions, so they introduced the axiom of reducibility, saying that for every non-predicative function there is a predicative function taking the same values. In practice this axiom essentially means that the elements of type can be identified with the elements of type, which causes the hierarchy of ramified types to collapse down to simple type theory.
In Zermelo set theory one can model the ramified type theory of PM as follows. One picks a set ι to be the type of individuals. For example, ι might be the set of natural numbers, or the set of atoms or any other set one is interested in. Then if τ1,...,τm are types, the type is the power set of the product τ1×...×τm, which can also be thought of informally as the set of functions from this product to a 2-element set. The ramified type can be modeled
as the product of the type with the set of sequences of n quantifiers indicating which quantifier should be applied to each variable σi.

Notation

One author observes that "The notation in that work has been superseded by the subsequent development of logic during the 20th century, to the extent that the beginner has trouble reading PM at all"; while much of the symbolic content can be converted to modern notation, the original notation itself is "a subject of scholarly dispute", and some notation "embodies substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism".
Kurt Gödel was harshly critical of the notation:
This is reflected in the example below of the symbols "p", "q", "r" and "⊃" that can be formed into the string "p ⊃ q ⊃ r". PM requires a definition of what this symbol-string means in terms of other symbols; in contemporary treatments the "formation rules" would have prevented the formation of this string.
Source of the notation: Chapter I "Preliminary Explanations of Ideas and Notations" begins with the source of the elementary parts of the notation :
PM changed Peano's Ɔ to ⊃, and also adopted a few of Peano's later symbols, such as ℩ and ι, and Peano's practice of turning letters upside down.
PM adopts the assertion sign "⊦" from Frege's 1879 Begriffsschrift:
Thus to assert a proposition p PM writes:
Most of the rest of the notation in PM was invented by Whitehead.

An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)

PM 's dots are used in a manner similar to parentheses. Each dot represents either a left or right parenthesis or the logical symbol ∧. More than one dot indicates the "depth" of the parentheses, for example, ".", ":" or ":.", "::". However the position of the matching right or left parenthesis is not indicated explicitly in the notation but has to be deduced from some rules that are complex and at times ambiguous. Moreover, when the dots stand for a logical symbol ∧ its left and right operands have to be deduced using similar rules. First one has to decide based on context whether the dots stand for a left or right parenthesis or a logical symbol. Then one has to decide how far the other corresponding parenthesis is: here one carries on until one meets either a larger number of dots, or the same number of dots next that have equal or greater "force", or the end of the line. Dots next to the signs ⊃, ≡,∨, =Df have greater force than dots next to, and so on, which have greater force than dots indicating a logical product ∧.
Example 1. The line
corresponds to
The two dots standing together immediately following the assertion-sign indicate that what is asserted is the entire line: since there are two of them, their scope is greater than that of any of the single dots to their right. They are replaced by a left parenthesis standing where the dots are and a right parenthesis at the end of the formula, thus:
The first of the single dots, standing between two propositional variables, represents conjunction. It belongs to the third group and has the narrowest scope. Here it is replaced by the modern symbol for conjunction "∧", thus
The two remaining single dots pick out the main connective of the whole formula. They illustrate the utility of the dot notation in picking out those connectives which are relatively more important than the ones which surround them. The one to the left of the "⊃" is replaced by a pair of parentheses, the right one goes where the dot is and the left one goes as far to the left as it can without crossing a group of dots of greater force, in this case the two dots which follow the assertion-sign, thus
The dot to the right of the "⊃" is replaced by a left parenthesis which goes where the dot is and a right parenthesis which goes as far to the right as it can without going beyond the scope already established by a group of dots of greater force. So the right parenthesis which replaces the dot to the right of the "⊃" is placed in front of the right parenthesis which replaced the two dots following the assertion-sign, thus
Example 2, with double, triple, and quadruple dots:
stands for
Example 3, with a double dot indicating a logical symbol :
stands for
where the double dot represents the logical symbol ∧ and can be viewed as having the higher priority as a non-logical single dot.
Later in section ✸14, brackets "" appear, and in sections ✸20 and following, braces "" appear. Whether these symbols have specific meanings or are just for visual clarification is unclear. Unfortunately the single dot is also used to symbolise "logical product".
Logical implication is represented by Peano's "Ɔ" simplified to "⊃", logical negation is symbolised by an elongated tilde, i.e., "~", the logical OR by "v". The symbol "=" together with "Df" is used to indicate "is defined as", whereas in sections ✸13 and following, "=" is defined as "identical with", i.e., contemporary mathematical "equality". Logical equivalence is represented by "≡" ; "elementary" propositional functions are written in the customary way, e.g., "f", but later the function sign appears directly before the variable without parenthesis e.g., "φx", "χx", etc.
Example, PM introduces the definition of "logical product" as follows:
Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitive translation can be given. However, because of criticisms such as that of Kurt Gödel below, the best contemporary treatments will be very precise with respect to the "formation rules" of the formulas.
The first formula might be converted into modern symbolism as follows:
alternately
alternately
etc.
The second formula might be converted as follows:
But note that this is not equivalent to nor to, and these two are not logically equivalent either.

An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)

These sections concern what is now known as predicate logic, and predicate logic with identity.
Section ✸10: The existential and universal "operators": PM adds "" to represent the contemporary symbolism "for all x " i.e., " ∀x", and it uses a backwards serifed E to represent "there exists an x", i.e., "", i.e., the contemporary "∃x". The typical notation would be similar to the following:
Sections ✸10, ✸11, ✸12: Properties of a variable extended to all individuals: section ✸10 introduces the notion of "a property" of a "variable". PM gives the example: φ is a function that indicates "is a Greek", and ψ indicates "is a man", and χ indicates "is a mortal" these functions then apply to a variable x. PM can now write, and evaluate:
The notation above means "for all x, x is a man". Given a collection of individuals, one can evaluate the above formula for truth or falsity. For example, given the restricted collection of individuals the above evaluates to "true" if we allow for Zeus to be a man. But it fails for:
because Russell is not Greek. And it fails for
because Zeus is not a mortal.
Equipped with this notation PM can create formulas to express the following: "If all Greeks are men and if all men are mortals then all Greeks are mortals".
Another example: the formula:
means "The symbols representing the assertion 'There exists at least one x that satisfies function φ' is defined by the symbols representing the assertion 'It's not true that, given all values of x, there are no values of x satisfying φ'".
The symbolisms ⊃x and "≡x" appear at ✸10.02 and ✸10.03. Both are abbreviations for universality that bind the variable x to the logical operator. Contemporary notation would have simply used parentheses outside of the equality sign:
PM attributes the first symbolism to Peano.
Section ✸11 applies this symbolism to two variables. Thus the following notations: ⊃x, ⊃y, ⊃x, y could all appear in a single formula.
Section ✸12 reintroduces the notion of "matrix", the notion of logical types, and in particular the notions of first-order and second-order functions and propositions.
New symbolism "φ ! x" represents any value of a first-order function. If a circumflex "＾" is placed over a variable, then this is an "individual" value of y, meaning that "ŷ" indicates "individuals" ; this distinction is necessary because of the matrix/extensional nature of propositional functions.
Now equipped with the matrix notion, PM can assert its controversial axiom of reducibility: a function of one or two variables where all its values are given is equivalent to some "predicative" function of the same variables. The one-variable definition is given below as an illustration of the notation :
✸12.1 ⊢: : φx .≡x. f ! x Pp;
This means: "We assert the truth of the following: There exists a function f with the property that: given all values of x, their evaluations in function φ is logically equivalent to some f evaluated at those same values of x. ". In other words: given a matrix determined by property φ applied to variable x, there exists a function f that, when applied to the x is logically equivalent to the matrix. Or: every matrix φx can be represented by a function f applied to x, and vice versa.
✸13: The identity operator "=" : This is a definition that uses the sign in two different ways, as noted by the quote from PM:
means:
The not-equals sign "≠" makes its appearance as a definition at ✸13.02.
✸14: Descriptions:
From this PM employs two new symbols, a forward "E" and an inverted iota "℩". Here is an example:
This has the meaning:

Introduction to the notation of the theory of classes and relations

The text leaps from section ✸14 directly to the foundational sections ✸20 GENERAL THEORY OF CLASSES and ✸21 GENERAL THEORY OF RELATIONS. "Relations" are what is known in contemporary set theory as sets of ordered pairs. Sections ✸20 and ✸22 introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" ; "⊂" signifies "is contained in", "is a subset of"; "∩" signifies the intersection of classes ; "∪" signifies the union of classes ; "–" signifies negation of a class ; "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse.
Small Greek letters represent classes :
When applied to relations in section ✸23 CALCULUS OF RELATIONS, the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example: "⊍", "∸".
The notion, and notation, of "a class" : In the first edition PM asserts that no new primitive ideas are necessary to define what is meant by "a class", and only two new "primitive propositions" called the axioms of reducibility for classes and relations respectively. But before this notion can be defined, PM feels it necessary to create a peculiar notation "ẑ" that it calls a "fictitious object".
At least PM can tell the reader how these fictitious objects behave, because "A class is wholly determinate when its membership is known, that is, there cannot be two different classes having the same membership". This is symbolised by the following equality, i.e.,
This has the reasonable meaning that "IF for all values of x the truth-values of the functions φ and ψ of x are equivalent, THEN the function ƒ of a given φẑ and ƒ of ψẑ are equivalent." PM asserts this is "obvious":
Observe the change to the equality "=" sign on the right. PM goes on to state that will continue to hang onto the notation "ẑ", but this is merely equivalent to φẑ, and this is a class..

Consistency and criticisms

According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.
Beyond the status of the axioms as logical truths, one can ask the following questions about any system such as PM:
Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms of set theory. Russell and Whitehead suspected that the system in PM is incomplete: for example, they pointed out that it does not seem powerful enough to show that the cardinal ℵω exists. However, one can ask if some recursively axiomatizable extension of it is complete and consistent.

Gödel 1930, 1931

In 1930, Gödel's completeness theorem showed that first-order predicate logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.
Gödel's incompleteness theorems cast unexpected light on these two related questions.
Gödel's first incompleteness theorem showed that no recursive extension of Principia could be both consistent and complete for arithmetic statements. According to the theorem, within every sufficiently powerful recursive logical system, there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.
Gödel's second incompleteness theorem shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system.

Wittgenstein 1919, 1939

By the second edition of PM, Russell had removed his axiom of reducibility to a new axiom. Gödel 1944:126 describes it this way:
This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as "All 'x' are blue" now has to list all of the 'x' that satisfy the proposition, listing them in a possibly infinite conjunction: e.g. x1 ∧ x2 ∧... ∧ xn ∧.... Ironically, this change came about as the result of criticism from Wittgenstein in his 1919 Tractatus Logico-Philosophicus. As described by Russell in the Introduction to the Second Edition of PM:
In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense cannot be described by the new theory proposed in PM Second Edition.
Wittgenstein in his Lectures on the Foundations of Mathematics, Cambridge 1939 criticised Principia on various grounds, such as:
Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.

Gödel 1944

In his 1944 Russell's mathematical logic, Gödel offers a "critical but sympathetic discussion of the logicistic order of ideas":

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. 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, well-ordered series, and series without "gaps".

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 the system in PM with the usual mathematical foundations of ZFC. The system of PM is roughly comparable in strength with Zermelo set theory.
Apart from corrections of misprints, the main text of PM is unchanged between the first and second editions. The main text in Volumes 1 and 2 was reset, so that it occupies fewer pages in each. In the second edition, Volume 3 was not reset, being photographically reprinted with the same page numbering; corrections were still made. The total number of pages in the first edition is 1,996; in the second, 2,000. Volume 1 has five new additions:
In 1962, Cambridge University Press published a shortened paperback edition containing parts of the second edition of Volume 1: the new introduction, the main text up to *56, and Appendices A and C.

Footnotes