Editing Principia Mathematica (section)

==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 (or more precisely a version of it where the axiom of separation has all quantifiers bounded).
* The system of propositional logic and predicate calculus in PM is essentially the same as that used now, except that the notation and terminology has changed.
* The most obvious difference between PM and set theory is that in PM all objects belong to one of a number of disjoint types. This means that everything gets duplicated for each (infinite) type: for example, each type has its own ordinals, cardinals, real numbers, and so on. This results in a lot of bookkeeping to relate the various types with each other.
* In ZFC functions are normally coded as sets of ordered pairs. In PM functions are treated rather differently. First of all, "function" means "propositional function", something taking values true or false. Second, functions are not determined by their values: it is possible to have several different functions all taking the same values (for example, one might regard 2''x''+2 and 2(''x''+1) as different functions on grounds that the computer programs for evaluating them are different). The functions in ZFC given by sets of ordered pairs correspond to what PM call "matrices", and the more general functions in PM are coded by quantifying over some variables. In particular PM distinguishes between functions defined using quantification and functions not defined using quantification, whereas ZFC does not make this distinction.
* PM has no analogue of the [[axiom of replacement]], though this is of little practical importance as this axiom is used very little in mathematics outside set theory.
* PM emphasizes relations as a fundamental concept, whereas in modern mathematical practice it is functions rather than relations that are treated as more fundamental; for example, category theory emphasizes morphisms or functions rather than relations. (However, there is an analogue of categories called [[Allegory (category theory)|allegories]] that models relations rather than functions, and is quite similar to the type system of PM.)
* In PM, cardinals are defined as classes of similar classes, whereas in ZFC cardinals are special ordinals. In PM there is a different collection of cardinals for each type with some complicated machinery for moving cardinals between types, whereas in ZFC there is only 1 sort of cardinal. Since PM does not have any equivalent of the axiom of replacement, it is unable to prove the existence of cardinals greater than ℵ<sub>ω</sub>.
* In PM ordinals are treated as equivalence classes of well-ordered sets, and as with cardinals there is a different collection of ordinals for each type. In ZFC there is only one collection of ordinals, usually defined as [[von Neumann ordinal]]s. One strange quirk of PM is that they do not have an ordinal corresponding to 1, which causes numerous unnecessary complications in their theorems. The definition of ordinal exponentiation α<sup>β</sup> in PM is not equivalent to the usual definition in ZFC and has some rather undesirable properties: for example, it is not continuous in β and is not well ordered (so is not even an ordinal).
* The constructions of the integers, rationals and real numbers in ZFC have been streamlined considerably over time since the constructions in PM.