[FOM] Extending the Language of Set Theory - addendum

[Aatu Koskensilta]

On Mar 1, 2005, at 9:34 AM, I wrote:

> Again, this is expressed as a rule of inference: if we have established
> that B is a class ordinal, then we can infer the above. In addition, we
> add the following rule of inference:
>
>       Q_1A_1...Q_nA_n( <V,L[A]_B,A,A_1,...,A_n>|=phi)
>   --------------------------------------------------------------
>   Q_1a_1...Q_na_n( <V_kappa, L[a]_alpha, a, a_1,...,a_n> |= phi)
>
> Where Q_i is a sequence of alternating quantifiers and A_i are class
> variables and a_i are set variables.

I forgot to say anything about why one should accept this rule of  
inference! The motivation is this: <V_kappa, L[a]_alha,a> should "look  
like" <V,L[A]_B,A>. In particular, since proper classes are collections  
which are not sets and from the "point of view" of V_kappa the  
collections that are not in V_kappa are "proper classes" whatever we  
can show to hold about proper classes in relation to V should hold  
about sets not in V_kappa in relation to V_kappa. The reason we can't  
formulate this as an axiom rather than as a rule of inference is that  
it's not obvious that the totality of non-set collections is in any  
sense determinate - in fact, the opposite is to be expected! Thus we  
can't just assume that e.g. "for all classes A, Phi(A)" makes sense at  
all. However, we might be able to show that no matter what proper  
classes there happen to be - or to put it in another way, which  
properties of sets and collections make sense ("are determinate") -  
it's impossible that there should be a proper class A, s.t. ~Phi(A),  
and thus establish that for all A Phi(A). For example, whatever proper  
classes there turns out to be, the intersection of a class and a set is  
a set. As an another example whatever proper classes there are, none is  
a non-empty subcollection of On u {On} with no epsilon-minimal element  
since any such class would show that On is not well-ordered.

It's also noteworthy that the quantifiers Q_1a_1 ... Q_na_n are  
unrestricted, i.e. they are not restricted to e.g. the powerset of  
V_kappa. This must be so because we have not assumed that non-set  
collections are collections of sets. Rather we have assumed that  
collections of collections of collections ... are legitimate - it  
certainly makes sense to speak of them, and such talk is reducible to  
iterated truth predicates. But this means that there is no natural  
choice for the restricting set for the quantifiers Q_1 ... Q_n, and  
thus we are lead to choose no restriction at all. For example, if  
V_kappa is to look like V and for all collections A, Phi(V,A), then  
certainly for any *set* a we should have Phi(V_kappa,a).

A possible strenghtening of the rule of inference is as follows:

                 Ord(C) & Q_1A_1...Q_nA_n(<V,L[A]_B,A,A_1,...,A_n>|=phi)
        
------------------------------------------------------------------------ 
-------
         Q_1A_1...Q_nA_n( A_i in L[V]_C -->  
<V_kappa,L[a]_alpha,a,A_1,...,A_n>|=phi)

Stretching the similarity of the structure a to the structure A up to  
L[V]_C whenever we have proved that C is a (class) ordinal.

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus