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Axiomatizing Kripke’s theory of truth. (English) Zbl 1101.03005
Kripke’s fixed point models were formalized in classical logic by S. Feferman in a quite natural way [see “Reflecting on incompleteness”, J. Symb. Log. 56, 1–49 (1991; Zbl 0746.03046)]. That axiomatization is commonly known as $$\mathbf{KF}$$ system and is equivalent to the system $$\mathbf{RA}_{<\varepsilon_0}$$ of ramified analysis up to any ordinal level smaller than $$\varepsilon_0$$. Reinhardt has posed the problem whether $$\mathbf{KF}$$ can be viewed as a tool for producing theorems that would also be derivable in a direct formalization of Kripke’s original theory in partial logic, if one focuses on the sentences that are provably true in $$\mathbf{KF}$$ [see W. Reinhardt, “Remarks on significance and meaningful applicability”, in: L. P. de Alcantara (ed.), Mathematical logic and formal systems, Coll. Pap. Hon. N. C. A. da Costa, Lect. Notes Pure Appl. Math. 94, 227–242 (1985; Zbl 0611.03004), and “Some remarks on extending and interpreting theories with a partial predicate for truth”, J. Philos. Logic 15, 219–251 (1986; Zbl 0629.03002)].
The authors solve Reinhardt’s problem negatively and present an axiomatization $$\mathbf{PKF}$$ of Kripke’s theory of truth in partial logic. They claim that any natural axiomatization of Kripke’s theory in Strong Kleene logic will be equivalent to their system $$\mathbf{PKF}$$. The proof-theoretic strength of $$\mathbf{PKF}$$ is determined as that of $$\mathbf{RA}_{<\omega^\omega}$$ of ramified analysis up to $$\omega^\omega$$ in contrast to the much stronger $$\mathbf{KF}$$. This result shows that axiomatizing Kripke’s theory in the most natural way leads to a system that is much weaker than the classical system $$\mathbf{KF}$$. In particular, the arithmetical content of both theories is far from identical. This proof-theoretic analysis sheds some light on the classification of axiomatizations of Kripke’s theory with respect to other theories of truth as well.

MSC:
 03A05 Philosophical and critical aspects of logic and foundations 03F03 Proof theory in general (including proof-theoretic semantics)
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