I'm reading Kunen's "Set Theory" (Revised edition 2013).

On page 108 he defines for axiomatic set theories $\Lambda, \Gamma$ which are strong enough to formalize finitistic arguments (e.g. ZFC, Z, BST...)

$$\Gamma \triangleleft \Lambda \quad\text{ :iff }\quad \Lambda \vdash Con(\Gamma) $$

and

$$\Gamma \leq \Lambda \quad\text{ :iff }\quad \text{we have a finitistic proof that } Con(\Lambda) \rightarrow Con(\Gamma)$$

Then he states

Lemma II.1.3

- $\Gamma \triangleleft \Lambda \rightarrow \Gamma \leq \Lambda$
- $\Gamma \leq \Lambda \ \wedge\ \Lambda \triangleleft \Theta \rightarrow \Gamma \triangleleft \Theta$
- /4. ...

Proof.For (1): Assuming $\Lambda \vdash Con(\Gamma)$, we may argue finitistically that if we had a formal proof of contradiction from $\Gamma$, then $\Lambda \vdash \neg Con(\Gamma)$, which would give us a contradiction in $\Lambda$. So, $Con(\Lambda) \rightarrow Con(\Gamma)$.

For (2): Assuming $\Gamma \leq \Lambda \ \wedge\ \Lambda \triangleleft \Theta$, then working in $\Theta$, we can prove $Con(\Lambda)$ and $Con(\Lambda) \rightarrow Con(\Gamma)$, and hence $Con(\Gamma)$.

At the beginning of this chapter Kunen says:

The definitions and lemmas in this section are

informal, and are intended to give an overview of the types of consistency results you will find in this book and in other works on set theory. The reader who is familiar with proof theory will see how to make them precise and formal. These remarks take place in the metatheory, using just finitistic reasoning.

**My question:** What is a precise formulation of 1. and 2.?

I see two possibilities.

Let $\vdash_{fin}$ denote the syntactic consequence relation of a finitistic system like PRA (Primitive Recursive Arithmetic). We write $\ulcorner \varphi \urcorner$ for the statement $\varphi$ encoded into PRA and let $\square \varphi$ be an abbreviation for $\ulcorner \vdash_{fin} \varphi \urcorner$.

Option a):

- $\vdash_{fin} \ulcorner\Lambda \vdash \neg Incon(\Gamma) \urcorner \rightarrow \square( Incon(\Gamma) \rightarrow Incon(\Lambda) )$
- $\vdash_{fin} \square( Incon(\Gamma) \rightarrow Incon(\Lambda) ) \ \wedge\ \ulcorner\Theta \vdash \neg Incon(\Lambda) \urcorner \rightarrow \ulcorner\Theta \vdash \neg Incon(\Gamma) \urcorner$

Option b):

- $\vdash_{fin} \ulcorner\Lambda \vdash \neg Incon(\Gamma) \urcorner \rightarrow ( Incon(\Gamma) \rightarrow Incon(\Lambda) )$
- $\vdash_{fin} ( Incon(\Gamma) \rightarrow Incon(\Lambda) ) \ \wedge\ \ulcorner\Theta \vdash \neg Incon(\Lambda) \urcorner \rightarrow \ulcorner\Theta \vdash \neg Incon(\Gamma) \urcorner$

In case of option a) I have a problem in proving 1, in b) in proving 2.

I use: Since we can assume that $Incon(\Gamma)$ is $\Sigma_1$, we have $\vdash_{fin} Incon(\Gamma) \rightarrow \square Incon(\Gamma)$. But for $Incon(\Gamma) \rightarrow Incon(\Lambda)$ this doesn't work.

Can anyone prove option a) or b)? Or am I completely missing the point here?