Hello everyone,

I am currently studying set theory on my own on the book *Set Theory, an Introduction to Large Cardinals* by Frank R. Drake and I have a couple of serious doubts.

Drake first introduces the usual (not formal) definition (due to Tarski) of *satisfaction* on a given collection A and then the usual definition of a *model* of some theory $S$ ($M \models S$). Then he gives (more or less) the the following definition:

Given a theory $S$ and a formula $\phi$ we write $S \vdash \phi$ iff for all structures $M$ we have $$M \models S \text{ iff } M \models \phi.$$

Of course this is not the usual definition of $S \vdash \phi$ (there exists a finite sequence of formulas, etc..) but (I think) the two are equivalent because of the Completeness Theorem (even for classes in the sense below). This is a minor point but I feel it could be somewhat related to the major problems below.

After the presentation of some of the axiomatic development of ZF, Drake starts to discuss (now informally) some model theory and the notion of absoluteness between structures (again he uses the word "collection" referring to the domain of the structures). Then he returns to formality and explains how we can define the notion of satisfaction within ZF, i.e. how we can formally write the statement

$ZF \vdash x \models \phi$, for some $\phi$,

where of course $x$ is a set.

Then he says that the whole properties about absoluteness could be easily formalized within ZF so that we should have, e.g.

$ZF \vdash x \models \phi \leftrightarrow y \models \phi$, if $x$ and $y$ are both transitive models of ZF and $\phi$ is a $\Delta_1^{ZF}$ formula.

I am quite convinced about this. Then problems arise:

Drake defines the universe of the constructible sets, $L$, and, in order to prove consistency results, uses strongly (for example) the following fact:

$V \models \phi $ iff $ L \models \phi$, if $\phi$ is a $\Delta_1^{ZF}$ formula.

I've been thinking about this for days and I arrived to the conclusion that this is only a short form for

$ZF \vdash \phi$ iff $ZF \vdash \phi^L$, if $\phi$ is a $\Delta_1^{ZF}$ formula.

i.e., from the Completeness Theorem and the definition of satisfaction,

$ZF \vdash \phi \leftrightarrow \phi^L$, if $\phi$ is a $\Delta_1^{ZF}$ formula.

So, if you have had the kindness to read up to this point, my questions are the following:

a) Is my reasoning correct?

b) If my reasoning is correct, is $ZF \vdash \phi^M$ a "good" definition of $M \models \phi$ (M proper class)? "Good" meaning "consistent with Tarski's definition of satisfaction.

c) If my reasoning is correct, is there a easy way to show that the absoluteness properties remain the same even for a model which is a proper class?

d) How can we speak about such a thing as $V$? I am really uncomfortable about it.