# [FOM] Extending the Language of Set Theory - addendum

Aatu Koskensilta aatu.koskensilta at xortec.fi
Wed Mar 2 05:44:20 EST 2005

```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

```