author={Volker Halbach and Albert Visser},
  journal={The Review of Symbolic Logic},
  pages={671 - 691}
Abstract A Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin’s problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing… 


Abstract The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to


  • L. Picollo
  • Philosophy
    The Review of Symbolic Logic
  • 2018
This paper introduces adequate notions of reference for the language of first-order arithmetic, which are shown to be fruitful for addressing the aforementioned issues in metamathematics.

A NewModel for the Liar

A newmodel forL t pa (the language of arithmetic enhanced by the unary truth predicate T ) is presented, which extends Kripke’s minimal fixed point. The latter, it will be argued, does not adequately

Some Remarks on True Undecidable Sentences

In this paper I try to discuss the question of the truth-value of Godel-type undecidable sentences in a framework which keeps into due account the idea that mathematical inquiry develops in a

Axiomatic Theories of Partial Ground II

This theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground and is shown to be a proof-theoretically conservative extension of the ramified theory of positive truth up to 𝜖0 and thus is consistent.


In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings,

Yablo Without Gödel

We prove Yablo’s paradox without the diagonal lemma or the recursion theorem. Only a disquotation schema and axioms for a serial and transitive ordering are used in the proof. e consequences for the

HYPER-REF: A General Model of Reference for First-Order Logic and First-Order Arithmetic

Abstract In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic (FOL). I also explain how this model can be used to determine the referent

On the Arithmetical Truth of Self-Referential Sentences

We take an argument of Godel's from his ground-breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: the sentence $G$ says about itself that it is not



Self-reference and the languages of arithmetic

I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of

Languages in which self reference is possible

This paper treats of semantical systems S of sufficient strength so that for any set W definable in S, there must exist a sentence X which is true in S if and only if it is an element of W .

On Godel Sentences and What They Say

Proofs of Godel's First Incompleteness Theorem are often accompanied by claims such as that the godel sentence constructed in the course of the proof says of itself that it is unprovable and that it

A System of Complete and Consistent Truth

PA is the theory containing all defining equations of the primitive recursive functions and all the induction axioms in the full language LT and it is shown that for every formula φ ∈ LT that: PA $ ¬.

Intensionality and the Gödel Theorems

Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to

Redundancies in the Hilbert-Bernays derivability conditions for Gödel's second incompleteness theorem

Three generalizations of the Second Incompleteness Theorem of Godel are presented which apply to a broader class of formal systems than previous generalizations and show that the provability of the consistency statement implies ⊢¬ φ, and hence that consistency is unprovable.

Yablo's paradox

Stephen Yablo has given an ingenious liar-style paradox that, he claims, avoids self-reference, even of an indirect kind, one that is, in fact, 'not in any way circular' (Yablo 1993, his italics). He

Solution of a Problem of Leon Henkin

  • M. H. Lob
  • Mathematics, Philosophy
    J. Symb. Log.
  • 1955
This note presents a solution of the previous problem with respect to the system Z μ, and, more generally, to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable.

Jumping in Arithmetic

In this paper we study a new relation between sentences: the jump relation. The idea of the jump relation is based on an analysis of Feferman's Theorem that the inconsistency of a theory U is