Classes and truths in set theory

Deflationism beyond arithmetic

“…Most, if not all, of the axiomatic theories of truth over arithmetic presented so far become conservative when we restrict arithmetical induction to L N . The same applies to those over set theory or many other subject matters; for instance, if we restrict the axiom schemata of set theory to L ∈ , the language of first-order set theory, then the resulting theories of truth are also conservative in many cases (see Fujimoto 2012).…”

“…Hence, even if B is an L N -theory, CT B + L + N -Ind is sometimes conservative over B, when B contains axioms of 'higher-order' and non-purely arithmetical nature in the sense of Isaacson (1987). 32 For instance, the Kripke-Feferman theories over ZF and Z 2 are conservative over their bases, when only syntactic induction is extended to the whole language and all the other axiom schemata are restricted to the language of the base theory (see Fujimoto 2012). 33 The crucial point of the proofs of Theorems 1 and 2 is that the constructed transitive models and ω-models keep the (first-order) arithmetical part unchanged.…”

“…Now, some base theories, such as PA, contain axiom schemata, such as the schema of arithmetical induction, 7 If we formulate a theory of truth in terms of the truth predicate T over a base theory B whose language L B does not contain enough names for the objects of the domain of discourse of B, then we sometimes need a stronger theory of syntax than the ordinary finitary one, which needs to encode an expanded language L ∞ B with constant symbols for all objects of the domain of discourse; a typical example of such a case is found in the definition of CT ZF in Fujimoto (2012). In contrast, if we use the satisfaction predicate Sat, we only need to assume that base theories interpret some fixed weak fragment of arithmetic such as I 1 , independently of our choice of subject matter, and thereby we can give a uniform treatment to theories of truth across different subject matters.…”

“…However, I doubt the validity of this extrapolation and suspect that theories of truth over arithmetic do not constitute such a generic case. Recent research in formal logic, such as (Fujimoto 2012(Fujimoto , 2017, reveals certain significant dissimilarities between axiomatic theories of truth over arithmetic and those over set theory, and we will see a further example of such dissimilarity in this article. These formal results indicate that theories of truth over arithmetic do not constitute such a generic case.…”

Full and hat inductive definitions are equivalent in NBG

“…or from second order number theory have played central roles, those into or from second order set theory play the same roles in the new theory, as in [5]. The notions of truth over the theories of reals and of functions also seem worthy to investigate.…”