# SELF-REFERENCE IN ARITHMETIC I

@article{Halbach2014SELFREFERENCEIA, title={SELF-REFERENCE IN ARITHMETIC I}, author={Volker Halbach and Albert Visser}, journal={The Review of Symbolic Logic}, year={2014}, volume={7}, 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…

## 35 Citations

### ON THE INVARIANCE OF GÖDEL’S SECOND THEOREM WITH REGARD TO NUMBERINGS

- MathematicsThe Review of Symbolic Logic
- 2020

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…

### REFERENCE IN ARITHMETIC

- PhilosophyThe 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.

### Henkin sentences and local reflection principles for Rosser provability

- Philosophy, MathematicsAnn. Pure Appl. Log.
- 2016

### A NewModel for the Liar

- Philosophy
- 2017

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

- Philosophy
- 2018

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

- PhilosophyJ. Philos. Log.
- 2018

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.

### SELF-REFERENCE UPFRONT: A STUDY OF SELF-REFERENTIAL GÖDEL NUMBERINGS

- PhilosophyThe Review of Symbolic Logic
- 2021

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

- Philosophy
- 2016

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

- PhilosophyKRITERION – Journal of Philosophy
- 2022

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

- PhilosophyTheoria
- 2018

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…

## References

SHOWING 1-10 OF 48 REFERENCES

### Self-reference and the languages of arithmetic

- Philosophy
- 2006

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

- PhilosophyJournal of Symbolic Logic
- 1957

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

- Philosophy
- 2007

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

- PhilosophyNotre Dame J. Formal Log.
- 1994

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

- Philosophy
- 1985

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

- MathematicsJournal of Symbolic Logic
- 1973

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

- Philosophy
- 1997

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

- Mathematics, PhilosophyJ. 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

- Philosophy
- 2014

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…