Arithmetical Sentences Research Articles

The de Jongh property for Basic Arithmetic

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p 1,..., p n ) built up of atoms p 1,..., p n , BPC $${\vdash}$$ ? A(p 1,..., p n ) if and only if for all arithmetical sentences B 1,..., B n , BA $${\vdash}$$ ? A(B 1,..., B n ). The technique used in our proof can easily be applied to some known extensions of BA.

AXIOMS FOR GROUNDED TRUTH

AbstractWe axiomatize Leitgeb’s (2005) theory of truth and show that this theory proves all arithmetical sentences of the system of ramified analysis up to ε0. We also give alternative axiomatizations of Kripke’s (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at least as strong as the Kripke-Feferman system KF and Cantini’s VF, respectively.

Socrates did it before Gödel

We translate Socrates’ famous saying I know that I know nothing into the arithmetical sentence I prove that I prove nothing. Then it is easy to show that this translated saying is formally undecidable in formal arithmetic, using Godel’s Second Incompleteness Theorem. We investigate some variations of this Socrates-Godel sentence. In an appendix we sketch a ramified epistemic logic with propositional quantifiers in order to analyze the Socrates-Godel sentence in a more logical way, separated from the arithmetical context.

The modified Ramsey theorem is not a G_del sentence

G_delian sentences are self-referential first-order sentences in the language of arithmetics. Perhaps the most celebrated one is the sentence which asserts its own unprovability. It is well known that this sentence is neither provable nor refutable in PA (Peano Arithmetics). Some logicians and philosophers have complained that such a sentence is difficult to grasp given its ‘meta-theoretical’ content and they started to look for undecidable arithmetical statements which have a combinatorial content. One such sentence is a variant of Ramsey’s sentence: the Paris-Harrington theorem asserts its undecidability. In the present paper I shall argue that such a sentence is not first-order expressible and thereby it does not provide the desired example of a combinatorial, undecidable arithmetical sentence. Instead I shall argue that it is expressible in Independence-friendly (IF) logic.

Random reals, the rainbow Ramsey theorem, and arithmetic conservation

AbstractWe investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the Csima–Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over RCA0 + BΣ2 for -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P− + IΣ1 and no stronger than P− + BΣ2.

Reference to numbers in natural language

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.

Relative arithmetic

AbstractIn nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Modal Logic of Gödel Sentences

The modal logic of Godel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.

Injecting uniformities into Peano arithmetic

We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles (not necessarily set-theoretically true) are interpreted while maintaining unmoved the Π 2 0 -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

On Specifying Truth-Conditions

This essay is a study of ontological commitment, focused on the special case of arithmetical discourse. It tries to get clear about what would be involved in a defense of the claim that arithmetical assertions are ontologically innocent and about why ontological innocence matters. The essay proceeds by questioning traditional assumptions about the connection between the objects that are used to specify the truth-conditions of a sentence, on the one hand, and the objects whose existence is required in order for the truth-conditions thereby specified to be satisfied, on the other. This allows one to set forth an assignment of truth-conditions to arithmetical sentences whereby nothing is required of the world in order for the truth-conditions of a truth of pure arithmetic to be satisfied. The essay then argues that such an assignment can be used to account for the a priori knowability of certain arithmetical truths.

Computational complexity of sentences over fields

Hilbert’s Irreducibility Theorem is applied to find the upper bounds of the time complexities of various decision problems in arithmetical sentences and the following results are proved: 1. The decision problem of ∀ ∃ sentences over an algebraic number field is in P. 2. The decision problem of ∀ ∃ sentences over the collection of all fields with characteristic 0 is in P. 3. The decision problem of ∀ ∃ sentences over a function field with characteristic p is polynomial time reducible to the factorization of polynomials over Z p . 4. The decision problem of ∀ ∃ sentences over the collection of all fields with characteristic p is polynomial time reducible to the factorization of polynomials over Z p . 5. The decision problem of ∀ ∃ sentences over the collection of all fields is polynomial time reducible to the factorization of integers over Z and the factorization of polynomials over finite fields.

Deflationism and the Godel Phenomena: Reply to Ketland

I am not a deflationist. I believe that truth and falsity are substantial. The truth of a proposition consists in its having a constructive proof, or truthmaker. The falsity of a proposition consists in its having a constructive disproof, or falsitymaker. Such proofs and disproofs will need to be given modulo acceptable premisses. The choice of these premisses will depend on the discourse in question. In the case of an arithmetical proposition, a truthmaker would be an intuitionistic proof from acceptable first principles, or axioms; and a falsitymaker would be an intuitionistic disproof from the same. For many an arithmetical proposition (p, one is not at present in possession of either a proof or a disproof of (p-indeed, one is not possessed of any effective method that one knows will eventually produce either a proof or a disproof of cp.1 That, says the intuitionist, is why one should refrain from asserting (p V -ip. In my paper 'Deflationism and the Godel Phenomena', I was not wearing the hat of an anti-realist committed to a substantial notion of truth. Rather, I was playing devil's advocate. The devil-an unwitting client, for whom I was appearing pro bono was the deflationist. I was seeking to show that a certain argument appealing to the orthodox treatment of Godel sentences in arithmetic did not count against deflationism about truth. Other writers, such as Shapiro and Ketland, had claimed that deflationism about truth could be discredited by consideration of the special status of independent Godel sentences. Ketland, in particular, had written (1999, p. 88)

Deflationism and the Godel Phenomena: Reply to Tennant

I am not a deflationist. I believe that truth and falsity are substantial. The truth of a proposition consists in its having a constructive proof, or truthmaker. The falsity of a proposition consists in its having a constructive disproof, or falsitymaker. Such proofs and disproofs will need to be given modulo acceptable premisses. The choice of these premisses will depend on the discourse in question. In the case of an arithmetical proposition, a truthmaker would be an intuitionistic proof from acceptable first principles, or axioms; and a falsitymaker would be an intuitionistic disproof from the same. For many an arithmetical proposition (p, one is not at present in possession of either a proof or a disproof of (p-indeed, one is not possessed of any effective method that one knows will eventually produce either a proof or a disproof of cp.1 That, says the intuitionist, is why one should refrain from asserting (p V -ip. In my paper 'Deflationism and the Godel Phenomena', I was not wearing the hat of an anti-realist committed to a substantial notion of truth. Rather, I was playing devil's advocate. The devil-an unwitting client, for whom I was appearing pro bono was the deflationist. I was seeking to show that a certain argument appealing to the orthodox treatment of Godel sentences in arithmetic did not count against deflationism about truth. Other writers, such as Shapiro and Ketland, had claimed that deflationism about truth could be discredited by consideration of the special status of independent Godel sentences. Ketland, in particular, had written (1999, p. 88)

Skolem functions of arithmetical sentences

Arithmetical formulas are the formulas containing the usual logical and arithmetical symbols +, ·, and constants of Z . If an arithmetical sentence ∀x∃y ψ(x,y) is true in a model M , then there is function f ( x ) defined on M such that ∀x ψ(x,f(x)) is true in M . Such a function is called a Skolem function of the arithmetical sentence ∀x∃y ψ(x,y) . In this paper, we study the bounds of the Skolem functions when the model M is the set of all natural numbers N or the ring of integers Z . We define the Skolem function f ( x ) for ∀x∃y ψ(x,y) as follows. For any a in N (or Z ) let f ( a ) be the least (or least absolute value of) b such that ψ ( a , b ) is true in N (or Z ). For every arithmetical sentence ∀x∃y∀z ψ(x,y,z) true in N (or Z ) there is a polynomial g ( x ) over Z such that the corresponding Skolem function f ( x )< g (| x |) for any x in N (or Z ). An application of considering the bounds of these Skolem function is the following: If the Generalized Riemann Hypothesis holds, then for every d there is a polynomial time algorithm for the following problem: given a quantifier-free arithmetical formula φ ( x , y ) of degree at most d , does ∀x∃y φ(x,y) hold in Z . Moreover, if the sentence is false in Z , then the algorithm outputs an a ∈ Z such that ∀y ¬φ(a,y) .

Incompleteness and Inconsistency

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or 'naive' notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Godel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent.

Two Proof-Theoretic Remarks on EA + ECT

In this note two propositions about the epistemic formalization of Church's Thesis (ECT) are proved. First it is shown that all arithmetical sentences deducible in Shapiro's system EA of Epistemic Arithmetic from ECT are derivable from Peano Arithmetic PA + uniform reflection for PA. Second it is shown that the system EA + ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas.

Is there a simple, pedestrian arithmetic sentence which is independent of ZFC?

We show that the P < NP conjecture can beformulated as a Pi20 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. da Costa on the P<NPconjecture and on the possibility that this sentence isindependent of ZFC supposed consistent.

Sentences over Integral Domains and Their Computational Complexities

LetRbe a Hilbertian domain and letKbe its fraction field. Letψ(x1, …, xn, y) be a quantifier free arithmetical formula overR. We may also takeψ(x1, …, xn, y) to be an arithmetical formula overK[x1, …, xn] and write it asψ(y). In this paper we show that ifRhas enough non-units and ∀x1…∀xn ∃y ψ(x1, …, xn, y), called an ∀n ∃ sentence, is true inR, then ∃y ψ(y) is true inK[x1, …, xn]. Also, ifR=K[T], whereKis an infinite integral domain and∀x1…∀xn∃yψ(x1,…,xn,y)is true inR, then ∃y ψ(y) is true inR[x1, …, xn]. These results are applied to find the upper and lower bounds of the time complexities of various decision problems on diophantine equations with parameters and arithmetical sentences. Some of the results are: 1. The decision problem of ∀∃ sentences and diophantine equations with parameters over the ring of integers of a global field are co-NP-complete. 2. The decision problem of ∀∃ sentences over the ring of integers of a global field is NP-complete. 3. LetKbe an infinite domain, the time complexities of the decision problems of equations with parameters and ∀∃ sentences over the polynomial ringK[t] are polynomial time reducible to factoring polynomials overK. 4. The decision problem of ∀∃ sentences over all algebraic integer rings is in P. 5. The decision problem of ∀∃ sentences over all integral domains with characteristic 0 is in P. 6. The time complexity of the decision problem of ∀∃ sentences over all integral domains is polynomial time reducible to factoring integers overZand factoring polynomials over finite fields.

A Substitutional Framework for Arithmetical Validity

A platonist in mathematics believes that arithmetic has a subject matter, i.e., that the statements of arithmetic are about certain objects – the natural numbers. For a platonist, the language of (first-order) arithmetic La is referential and he is licensed to speak of true and false sentences of La and to endorse Tarski’s analysis of truth. It follows from this Tarskian analysis plus the fact that every natural number is denoted by some closed term of La (a numeral, if one insists on canonicity) that the truth values of arithmetical sentences are determined by the truth values of its atomic sentences. Consider now a philosopher who, while not a platonist in any sense, broadly accepts the results of mathematics – however tentatively – and is persuaded that the truths of arithmetic are determined by the truth values of its atomic sentences (whatever may be his reformulation of the notion of arithmetical truth). This article may be viewed as an attempt to frame a position for such a non-platonist philosopher of a non-revisionist bent. It is well known that certain atomic sentences of arithmetic have a persuasive rendering in terms of schemata of formulas of first-order languages with equality. This rendering is specially persuasive insofar as we focus on the cardinal role of numbers (and leave their ordinal role aside). For instance, the sentence 7+5 =12 can be rendered as

Inconsistent Models of Arithmetic Part I: Finite Models

The paper concerns interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first order arithmetic. The paper demonstrates the existence of such models and provides a complete taxonomy of the finite ones.