Editing Logicism (section)

===The definition of the natural numbers===
In the Prinicipia, the natural numbers derive from ''all'' propositions that can be asserted about ''any'' collection of entities. Russell makes this clear in the second (italicized) sentence below.

:"In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. ''In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collection of trios in the world'', for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the third place, we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)

To illustrate, consider the following finite example: Suppose there are 12 families on a street. Some have children, some do not. To discuss the names of the children in these households requires 12 propositions asserting "''childname'' is the name of a child in family F''n''" applied to this collection of households on the particular street of families with names F1, F2, . . . F12. Each of the 12 propositions regards whether or not the "argument" ''childname'' applies to a child in a particular household. The children's names (''childname'') can be thought of as the ''x'' in a propositional function ''f''(''x''), where the function is "name of a child in the family with name F''n''".<ref>To be precise both ''childname'' = variable ''x'' and family name ''Fn'' are variables. ''Childname''{{'}}s domain is "all childnames", and family name ''Fn'' has a domain consisting of the 12 families on the street.</ref>{{or|date=August 2018}}

Whereas the preceding example is finite over the finite propositional function "''childnames'' of the children in family F''n'''" on the finite street of a finite number of families, Russell apparently intended the following to extend to all propositional functions extending over an infinite domain so as to allow the creation of all the numbers. 

Kleene considers that Russell has set out an [[impredicativity|impredicative]] definition that he will have to resolve, or risk deriving something like the [[Russell paradox]]. "Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence" (Kleene 1952:44). The problem will appear, even in the finite example presented here, when Russell deals with the unit class (cf. Russell 1903:517).

The question arises what precisely a "class" ''is'' or should be. For Dedekind and Frege, a class is a distinct entity in its own right, a 'unity' that can be identified with all those entities ''x'' that satisfy some propositional function ''F''. (This symbolism appears in Russell, attributed there to Frege: "The essence of a function is what is left when the ''x'' is taken away, i.e in the above instance, 2( )<sup>3</sup> + ( ). The argument ''x'' does not belong to the function, but the two together make a whole (ib. p. 6 [i.e. Frege's 1891 ''Function und Begriff'']" (Russell 1903:505).) For example, a particular "unity" could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:
:{ a, b, c }<sub>Fα</sub>

This notion of collection or class as object, when used without restriction, results in [[Russell's paradox]]; see more below about [[impredicativity|impredicative definitions]]. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments ''x'' do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic" (First edition of ''Principia Mathematica'' 1927:24).

Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions": {{or|date=August 2018}}
:"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than ''symbolic fictions''. And if we can find any way of dealing with them as ''symbolic fictions'', we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions.  . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)

And in the second edition of ''PM'' (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.

'''Step 2: Collect "similar" classes into 'bundles' ''': These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by '''≈''', i.e. one-one correspondence of the elements,<ref>"If the predicates are partitioned into classes with respect to equinumerosity in such a way that all predicates of a class are equinumerous to one another and predicates of different classes are not equinumerous, then each such class represents the ''Number'', which applies to the predicates that belong to it" (Bernays 1930-1 in Mancosu 1998:240.</ref> and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and  so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes" (Russell 1919:14).

'''Step 3: Define the null class''': Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection. 

The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? In ''PM'' Russell says that "A class is said to ''exist'' when it has at least one member . . . the class which has no members is called the "null class"  . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in Russell's 1903 work.<ref name=:0>Cf. sections 487ff (pages 513ff in the Appendix A).</ref> After he discovered the paradox in Frege's ''Grundgesetze'' he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types"; see more about the unit class, the problem of [[impredicativity|impredicative definitions]] and Russell's "vicious circle principle" below.<ref name=:0/>

'''Step 4: Assign a "numeral" to each bundle''': For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary. 

'''Step 5: Define "0"''' Following Frege, Russell picked the empty or ''null'' class of classes as the appropriate class to fill this role, this being the class of classes having no members.  This null class of classes may be labelled "0"

'''Step 6: Define the notion of "successor"''': Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number ''n'', it also belongs to ''n''+1, the successor of ''n''". (1903:21). He asserts that "the natural numbers are the ''posterity'' – the "children", the inheritors of the "successor" – of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).

Note Russell has used a few words here without definition, in particular "number series", "number ''n''", and "successor". He will define these in due course. ''Observe in particular that Russell does not use the unit class of classes "1" to construct the successor''. The reason is that, in Russell's detailed analysis,<ref>1909 Appendix A</ref> if a unit class becomes an entity in its own right, then it too can be an element in its own proposition; this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, ''unit classes'' must be defined so as not to assume that we know what is meant by ''one'' (1919:181).

For his definition of successor,  Russell will use for his "unit" a single entity or "term" as follows:
: "It remains to define "successor". Given any number ''n'' let ''α'' be a class which has ''n'' members, and let ''x'' be a term which is not a member of ''α''. Then the class consisting of ''α'' with ''x'' added on will have ''+1'' members. Thus we have the following definition:
:''the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class''." (1919:23)

Russell's definition requires a new "term" which is "added into" the collections inside the bundles.

'''Step 7: Construct the successor of the null class'''.

'''Step 8: For every class of equinumerous classes, create its successor'''.

'''Step 9: Order the numbers''': The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "''S''", between the various "numerals". "We must now consider the ''serial'' character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the ''class'' of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)

Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of [[asymmetric relation|asymmetry]] i.e. given the relation such as ''S'' (" . . . is the successor of  . . . ") between two terms ''x'' and ''y'': ''x S y'' ≠ ''y S x''. Second, he defines the notion of [[transitive relation|transitivity]] for three numerals ''x'', ''y'' and ''z'': if ''x S y'' and ''y S z'' then ''x S z''. Third, he defines the notion of [[connected relation|connected]]: "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)

He concludes: ". . . [natural] number ''m'' is said to be less than another number ''n'' when ''n'' possesses every hereditary property possessed by the successor of ''m''. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers]." (1919:35)