Valid arithmetical principles

Some principles are easily seen to hold in a general arithmetical setting. From now on we will only study arithmetical theories that are reasonable. As mentioned before, reasonable can be read as ``containing $I\Delta_0 + \mathrm{SUPEXP}$''. We will treat some principles which hold in every reasonable arithmetical theory. Precisely these principles are later collected to be studied in a modal setting.

• We call J1 the principle of $\Box (\alpha \rightarrow \beta )\rightarrow \alpha \rhd \beta$. This reflects the fact that the identity function is a special case of an interpretation. If one takes J in (+) to be the identity, a tautology arises. (The relativizing formula can in this example just be taken x=x.)

• The principle J2 reads $\alpha \rhd \beta \wedge \beta \rhd \gamma \rightarrow \alpha \rhd \gamma$. We already encountered a plea for this principle when we viewed it as a statement of models. By more elementary means one can also see this principle to be true. The observation that the composition of two interpretations is again an interpretation, more or less explains the principle. Finally one should convince oneself that this reasoning can be done within the theory. (Here one has to use technical facts like $\vdash \Box (\beta \rightarrow \gamma^I) \rightarrow \Box(\beta^J\rightarrow \gamma^{J\circ I})$.)

• The principle J3 is $\alpha \rhd \gamma \wedge \beta \rhd \gamma \rightarrow \alpha \vee \beta \rhd \gamma$. Reading this as a statement of models legitimizes it immediately. If in any model of A, and in any model of B, one can define a submodel for C, well then in any model of $A\vee B$ one can define a submodel for C. For if $A\vee B$ holds, either one of them holds, but in both cases a model of C can be defined. In terms of interpretations J3 reflects that one can choose which interpretation to use, depending on $\alpha$ or $\neg \alpha$ to hold.

• Another principle, J4: $\alpha \rhd \beta \rightarrow (\Diamond \alpha \rightarrow \Diamond \beta)$, reflects that interpretations yield relative consistency results. One can also see J4 to be a direct consequence of J1 and J2. For this we write the consequent as $\Box \neg \beta \rightarrow \Box \neg \alpha$. Now suppose $\alpha \rhd \beta$ and $\Box \neg \beta$. J1 gives that $\Box \neg \beta = \Box (\beta \rightarrow \bot )$ implies $\beta \rhd \bot$. By transitivity , (J2), one gets $\alpha \rhd \bot$. But as the translation of $\bot$under any interpretation is always $\bot$, we automatically get $\Box \neg \alpha$.

• The principle J5 reads $\Diamond \alpha \rhd \alpha$. It can be seen as an arithmetized version of the completeness theorem. The consistency of a statement implies the existence of a model on which this statement holds. As the theories we consider are strong enough to encode the whole Henkin construction, one can prove within that very theory $T+{\mathit Con}_T(\alpha )\rhd T+\alpha$. This coding can be done in a really weak theory, $T_0\subseteq T$, so actually something stronger can be proven as well: $T_0+{\mathit Con}_T(\alpha )\rhd T+\alpha$. This strengthening of J5 is essentially used in establishing the arithmetical validity of the new principle P₀ in paragraph 6.5.

All these principles are collected together in a modal logic. This modal logic is properly defined in chapter 3. It will be referred to as $\mathit{IL}$.