Two points need to be addressed before defining the intended models for the first order language Li. The first point concerns the composition of the class H of homogeneous i-collections. In the last section of Chapter 2, we have defined the notion of ‘homogeneity’ as follows: a given i-collection x is homogeneous if and only if it only contains indiscernible objects, and all the objects it contains are of the same type. This informal definition can now be stated formally:
H :={x:∀y(y /∈x)∧ ∀i((τi ∈x)→ ∀j(τj ∈x↔j =i))},
where the variablesxandyrange over the entire universeU, and the formula ‘∀i((τi ∈ x) → ∀j(τj ∈ x ↔ j = i))’ is an abbreviation of the well-formed
formula:
((τ1 ∈x)→(τ2, ..., τn ∈/ x))∧ ((τ2 ∈x)→(τ1, τ3, ..., τn ∈/ x))∧ ...
... ∧((τn∈x)→(τ1, ..., τn−1 ∈/ x)).
Clearly, the empty set is an element of H. From the formal definition of the classHof homogeneous i-collections, and from the assumption that there are only finitely many types of i-objects, it follows that it is possible to define a sequence of collections: T1,T2, ...,Tn as follows:
for all i≤n ∈N: Ti :={x∈ H :τi ∈x}.29
Clearly,T1,T2, ...,Tn are subcollections ofH, and they are mutually disjoint.
Furthermore, together with the empty set, they exhaust H. In fact, by definition of H, given an homogeneous i-collection x, eitherx=∅ orτ1 ∈x
or τ2 ∈x or ... or τn ∈x: that is, either x =∅ orx∈ T1 orx ∈ T2 or ... or
x∈ Tn. It follows that the class of homogeneous i-collection can be thought
of as the union of the classes :
H={∅} ∪ T1∪ T2∪...∪ Tn,
under the assumption that n is the number of types of i-objects allowed in our domain of quantification. In constructing a model for Li, we will make use of the following fact: that for any i ≤ n, m ∈ N there exists only one homogeneous i-collection xsuch that x∈ Ti andκ(x) = m. This fact, which
is of the outmost importance to understand the structure of the class H of homogeneous collections, will be shown as a theorem at the end of the chap- ter.
The second point concerns the nature of i-objects, and the nature of any formal language that can be consistently used to talk about them. In Chapter 2 we have shown that, under the assumption that the definition of individu- ality provided by Lowe (2016) is considered correct, i-objects arenonindivid- uals. This conclusion undermines the possibility to individuate such objects: it is not possible to single out any i-object, and as a consequence it is not possible to say anything meaningful concerning one i-object, that has not already been said of any other of them. As a consequence, no i-object can be consistently defined as to be the output of any zfc-function, for it is mean- ingless to ask of two i-objects of the same type if they are the same object. 29The assumption of the finiteness of the number of types of i-objects is equivalent to the assumption that, given a model = for the language Li there exists a finite natural number n ∈Nsuch that, for any m > n there is no collectionx∈ U such that τm ∈x. We remind the reader that the notion of ‘type’ is taken as primitive.
It follows that no i-object can be assigned to an individual constant, and that no individual variable can be thought of as ranging over nonindividual objects. The problem that we need to solve to be able to talk about nonin- dividuals is thence twofold: on one side, we need to provide symbols which are neither individual constants nor variables, and on the other side we need to find a way to interpret them. In an attempt to solve the syntactic part of the problem, we have introduced in the alphabet of the language Li the symbolsτ1, τ2, ..., τnas nonindividual constants, and we have informally stip-
ulated the meaning of these symbols to be, respectively ‘something of type 1’, ‘something of type 2’, ..., ‘something of typen’, where the use of the word ‘something’ is such that it does not admit as meaningful any question of the form: “which one —?”. These symbols are taken to be indefinite descriptions, and still no arbitrary reference can serve as their interpretation, since no i- object can be in principle referred to by any imaginable function, as defined withinzfc. The problem of how to assign objects to these symbols will then be solved by referring to them indirectly. We will construct a model with two different domains of quantification. The first domain will be the class U+, containing all i-objects, all the i-collections, all the
zfc-collections, all
the collections of i-collections, all the collections of collections of i-collections, and so on.30 Both individual variables and individual constants, as well as
all the predicates of any degrees having collections as their argument will be assigned objects in U ⊆ U+. The second domain of quantification, V=, will be used as a figurehead domain, and the nonindividual constantsτ1, τ2, ..., τn
will be assigned individual objects in V=. The truth conditions for state- ments of the form ‘τm ∈ x’ (for some number m ≤ n ∈ N and some x ∈ U)
will be defined with respect to collections in H, since it will be shown that any i-collection inU can be defined as the union of homogeneous i-collections and zfu-collections.
30The difference between the classesU andU+ is thatU only contains collections, while
U+ contains all the collections in U plus the i-objects, according to the structure of the Model.