Forging Companion Sets of Individuals

When undertaking the construction of a standard universe, in practice one starts with a pre-defined, infinite basis—say the set R of all real numbers—whose elements may have an inner structure that prevents their direct use as individuals. If so, how can we conceal their structure? We need a technique for converting a set s' whatsoever into a set s" so that Ur (s") holds and there is a one-one correspondence between s' and s".

One plainly sees that Ur (s") cannot hold if any set of finite rank belongs to s"; on the other hand, imposing that 0 / s" and that all elements of elements of s" share the same infinite rank r suffices to ensure that Ur (s") holds—one shows inductively, in fact, that each stage originating from s" is the union of a set of finite rank with a set whose elements have ranks exceeding r. This observation makes it rather easy to conceive an injection ur whose domain is the given s' and whose set of values, s" = {urx : x e s'}, can serve as basis in place of s' in the construction of the standard superstructure. Should any rationale arise for doing so, we can even tune the range of s" by means of an auxiliary ‘gauge’ set c', as suggested by the interface of the Theory urification in Fig. 10.6.

This Theory receives sets s', c' such that s' U c'—and hence rk (s' U c') —is infinite; it manufactures and produces in output a function ur0 sending injectively each x e s' to a set ur0 (x) all of whose elements have rank rk (s' U c')+ = rk (s' U c7) U

Fig. 10.6 Gauged transformation of a set s'

whatsoever into a set of individuals

{rk (s' U c')}, where R+ =Def R U {R}. The definition of ur0 —internally hidden, insofar as immaterial outside the Theory urification—could well be

What really counts to us is that Ur ({ur0 (x) : x e s'}) holds, as we aimed at.

To see a more sophisticated exploitation of the Theory at hand, suppose next that we are given a set s along with an infinite set i' that we want to use as index set for enlarging s, seen as a standard set of individuals, into a set w of nonstandard individuals. To ease the discussion, we momentarily dismiss the concurrence issue debated in Sect. 10.4; we will content ourselves with an ultrafilter none of whose elements is a finite set, over (a counterpart i" of) i'.

First move. Convert i' into a set i" so that all indices j in i" have the same infinite rank r, exceeding the rank of s, and there is a one-one correspondence u(X) between i' and i ":

Second move. Observe that when W is a set of functions from i" to s then each element of U W is an ordered pair (j, x) = [1], whose rank is infinite. Trivially 0 e W and hence Ur (W) holds.

Third move. Introduce an ultrafilter ж such that

and at this point specify W as follows:

Now regard this W and its subset

respectively, as the ‘wide’ and the ‘small’ set of all nonstandard individuals and of the standard ones: it should be clear that S can act as a counterpart of the original s, in view of the natural correspondence between the two.

What precedes has offered clues about how to implement the Theory whose interface is shown in the lower part of Fig. 10.7.

Transformation of a set s into a set w

Fig. 10.7 Transformation of a set s into a set w

 
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