The consistency of system Q

Extract

It will be shown that system Q of [1], with minor corrections, is free from contradiction and can be strengthened in various ways without destroying its consistency.

By the system C*A will be meant the system that results in place of CA if rule 3.4.10 is omitted in [2] from the rules 3.4 defining PA and RA for an arbitrary class A of equations. We let P*A be the theorems of C*A and R*A be the antitheorems of C*A, just as PA and RA are respectively the theorems and antitheorems of CA in [2]. Similarly Γ* is the class that is defined exactly like the class Γ in 4.18 of [2] except that systems of the form C*A replace those of the form CA. The terminology and conventions of [2] will be assumed in what follows.

Using the methods that were used to show in 5.9 of [2] that CΓ is free from contradiction, it is easy to show that C*A is free from contradiction for every class A of equations. (The fact that rule 3.4.10 is omitted causes the proof of consistency to work even for cases where A is not systematically closed in the sense of 4.11 of [2].)

Just as it was shown in 4.19 of [2] that Γ is systematically closed (in the sense of 4.11 of [2]), so also it clearly can be shown that Γ* is systematically closed. This means, in effect, that the rule of substitution of equals for equals holds in the system C*Γ*.