5 ;;Quote: Peano's axioms for number concern 0, unique successors, and induction

7 ;;Quote: any progression matches Peano's five axioms for numbers; so the axioms do not define '0', 'number' and 'successor'

7+;;Quote: the concept of number should apply in the right way to common objects

12 ;;Quote: can define a class by extension (enumerate its members) or intension (a defining property); can reduce extension to intension

13 ;;Quote: the difference between a class and its definition is that a class has many defining characteristics but only one set of members

15 ;;Quote: a number is the number of some class, i.e., the class of those classes that are similar to it; discovered by Frege

15+;;Quote: two classes are similar when there is a oneone relation between them

21 ;;Quote: the natural numbers are the posterity of 0 w.r.t. immediate predecessor; i.e., all larger numbers, by hereditary; includes Peano's 0 and induction

27 ;;Quote: for Russell, mathematical induction is a definition, not a principle as for Poincare

27+;;Quote: we define the natural numbers as those to which proofs by mathematical induction can be applied

27 ;;Quote: mathematical induction is the essential characteristic that distinguishes the finite from the infinite

31 ;;Quote: an ordering relation has three properties: x not y<x; x<y and y x<z; and, either x<y or y<x

35 ;;Quote: order the inductive numbers by: m<n if n possesses every hereditary property possessed by the successor of m

63 ;;Quote: different types of numbers do not include simpler types; e.g., the positive integers are not the same as the natural numbers

64 ;;Quote: the positive and negative integers are relations instead of a class of classes; e.g., n is the relation of n to n+m

64 ;;Quote: the fraction m/n is the relation that holds of inductive numbers x,y when xn=ym; leads to ordering relation on fractions

72 ;;Quote: construct the real numbers as segments of series of rational ratios in order of magnitude; an irrational number is a segment without boundary

169 ;;Quote: while Shakespeare's imagination and thoughts are real, there is not an objective Hamlet

169+;;Quote: reality is vital to logic; should not allow Hamlet as another kind of reality

172;;Quote: 'Socrates is a man' identifies a named object with an ambiguously described object; i.e., the ambiguously described object actually exists

172 ;;Quote: definite descriptionsits component words determine its meaning; compare with names, e.g., 'Scott is the author of Waverly'

172+;;Quote: a name is a simple symbol that directly designates an individual

176 ;;Quote: although x=x is always true for names, it is not always true for definite descriptions; e.g., 'the present King of France'

178 ;;Quote: a name must name something, otherwise it is not a name; in 'Did Homer exist?', Homer is an abbreviation for a definite description

178+;;Quote: a description has significance even if it describes nothing; its meaning is derived from its constituent symbols

197 ;;Quote: formal reasoning concerns statements that do not mention actual things; e.g., if all alphas are betas and x is an alpha than x is a beta

199 ;;Quote: logic and mathematics are only concerned with the truth value of forms; i.e., that which remains unchanged when every constituent is changed
