Some Remarks on Indiscernibility

If α and α’ are distinct variables and ϕ and ϕ’ are open sentences of some language, where ϕ’ is the result of replacing one or more free occurrences of a in α with free occurrences of α’ in ϕ’, then a universal closure of ⌜(α=α’ → (ϕ → ϕ’))⌝, is an indiscernibility principle of that language. For instance, (1) is an indiscernibility principle.The existence of opaque constructions falsifies the familiar unrestricted principle of substitution which affirms that co-referential expressions are intersubstitutable in all contexts without change of truth-value. But indiscernibility principles are another matter. Not every counter-example to the unrestricted principle of substitution is a counter-example to some indiscernibility principle. Indeed, it is likely to be thought that there is no counter-example to any indiscernibility principle, and that the semantics of variables and objectual quantification ensures that all indiscernibility principles are true.

The optimality of induction as an axiomatization of arithmetic

By induction for a formula φ we mean the schema(where the terms in brackets are implicitly substituted for some fixed variable, with the usual restrictions). Let be the schema IAφ for φ in Πn (i.e. ); similarly for . Each instance of is Δn+2, and each instance of is Σn+1 Thus the universal closure of an instance α is Πn+2 in either case. Charles Parsons [72] proved that and are equivalent over Z0, where Z0 is essentially Primitive Recursive Arithmetic augmented by classical First Order Logic [Parsons 70].Theorem. For each n > 0 there is a Πn formula π for whichis not derivable in Z0from(i) true Πn+1sentences; nor even(ii) Πn+1sentences consistent withZ0.

On an Ackermann-type set theory

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.