scholarly journals Inconsistent Models for Relevant Arithmetics

2021 ◽  
Vol 18 (5) ◽  
pp. 380-400
Author(s):  
Robert Meyer ◽  
Chris Mortensen
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Intrinsic Interest ◽  
Recursively Enumerable ◽  
Classical Quantum ◽  
The Absolute ◽  
Quantum Arithmetic ◽  
Inconsistent Theories

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest.

scholarly journals Inconsistent models for relevant arithmetics

1984 ◽  
Vol 49 (3) ◽  
pp. 917-929 ◽  
Author(s):  
Robert K. Meyer ◽  
Chris Mortensen
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
The Absolute

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano's axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R#, was absolutely consistent. It was pointed out that such a result escapes incautious formulations of Gödel's second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not ordinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein constitutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R#. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P#. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here).

Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic.

1991 ◽  
Vol 56 (2) ◽  
pp. 643-660 ◽  
Author(s):  
Robert A. Di Paola ◽  
Franco Montagna
Keyword(s):  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Easy Consequence ◽  
Recursively Enumerable ◽  
Bona Fide ◽  
Precise Relationship

The dominical categories were introduced by Di Paola and Heller, as a first step toward a category-theoretic treatment of the generalized first Godel incompleteness theorem [1]. In his Ph.D. dissertation [7], Rosolini subsequently defined the closely related p-categories, which should prove pertinent to category-theoretic representations of incompleteness for intuitionistic systems. The precise relationship between these two concepts is as follows: every dominical category is a pointed p-category, but there are p-categories, indeed pointed p-isotypes (all pairs of objects being isomorphic) with a Turing morphism that are not dominical. The first of these assertions is an easy consequence of the fact that in a dominical category C by definition the near product functor when restricted to the subcategory Ct, of total morphisms of C (as “total” is defined in [1]) constitutes a bona fide product such that the derived associativity and commutativity isomorphisms are natural on C × C × C and C × C, respectively, as noted in [7]. As to the second, p-recursion categories (that is, pointed p-isotypes having a Turing morphism) that are not dominical were defined and studied by Montagna in [6], the syntactic p-categories ST and S′T associated with consistent, recursively enumerable extensions of Peano arithmetic, PA. These merit detailed investigation on several counts.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Whither relevant arithmetic?

1992 ◽  
Vol 57 (3) ◽  
pp. 824-831 ◽  
Author(s):  
Harvey Friedman ◽  
Robert K. Meyer
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Central Question ◽  
Proof Methods ◽  
Classical Proof ◽  
Natural Way

AbstractBased on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R#, which does formalize arithmetic on relevant principles, but also admits γ in a natural way?

The Unprovability of Consistency

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Peano Arithmetic ◽  
Incompleteness Theorem

Gödel’s second incompleteness theorem, roughly stated, is that if Peano Arithmetic is consistent, then it cannot prove its own consistency. The theorem has been generalized and abstracted in various ways and this has led to the notion of a provability predicate, which plays a fundamental role in much modern metamathematical research. To this notion we now turn. A formula P(v1) is called a provability predicate for S if for all sentences X and Y the following three conditions hold: Suppose now P(v1) is a Σ1-formula that expresses the set P of the system P.A. Under the assumption of ω-consistency, P(v1) represents P in P.A. Under the weaker assumption of simple consistency, all that follows is that P(v1) represents some superset of P, but that is enough to imply that if X is provable in P.A., then so is P (x̄).

scholarly journals Boolean negation and non-conservativity II: The variable-sharing property

2020 ◽  
Author(s):  
Tore Fjetland Øgaard
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Relevant Logics ◽  
Acute Form ◽  
Vast Range ◽  
Boolean Extension

Abstract Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended logic.

Π10 classes and Boolean combinations of recursively enumerable sets

1974 ◽  
Vol 39 (1) ◽  
pp. 95-96 ◽  
Author(s):  
Carl G. Jockusch
Keyword(s):  
Turing Machine ◽  
Peano Arithmetic ◽  
Binary Sequences ◽  
First Order ◽  
Recursive Tree ◽  
Recursively Enumerable ◽  
Private Correspondence ◽  
Recursively Enumerable Sets ◽  
Binary Input

Let be the collection of all sets which are finite Boolean combinations of recursively enumerable (r.e.) sets of numbers. Dale Myers asked [private correspondence] whether there exists a nonempty class of sets containing no member of . In this note we construct such a class. The motivation for Myers' question was his observation (reported in [1]) that the existence of such a class is equivalent to the assertion that there is a finite consistent set of tiles which has no m-trial tiling of the plane for any m (obeying the “origin constraint”). (For explanations of these terms and further results on tilings of the plane, cf. [1] and [5].) In addition to the application to tilings, the proof of our results gives some information on bi-immune sets and on complete extensions of first-order Peano arithmetic.A class of sets may be roughly described as the class of infinite binary input tapes for which a fixed Turing machine fails to halt, or alternatively as the class of infinite branches of a recursive tree of finite binary sequences. (In these definitions, sets of numbers are identified with the corresponding binary sequences.) Precise definitions, as well as many results concerning such classes, may be found in [3] and [4].

Sentences implying their own provability

1983 ◽  
Vol 48 (3) ◽  
pp. 777-789 ◽  
Author(s):  
David Guaspari
Keyword(s):  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Simple Observation ◽  
Simple Question

We begin with a simple observation and a simple question. If we fix Th(x), some reasonable formulation of “x is the Gödel number of a theorem of Peano Arithmetic”, then for any sentence σ, Peano Arithmetic proves σ → Th(⌈σ⌉). (Here ⌈σ⌉ is the canonical term denoting the Gödel number of σ.) This observation is crucial to the proof of the Second Incompleteness Theorem. Call ψ a self-prover (with respect to Th(x)) if ψ → Th(⌈ψ⌉) is a theorem of Peano Arithmetic (from now on, PA). Our simple question is this: Does the observation have a converse—must every self-prover be provably equivalent to a sentence? Whatever φ happens to be, the formula φ ∧ Th(⌈φ⌉) is a self-prover. This makes it seem clear that there must exist self-provers which are not provably : We need only use a suitably complicated φ.Deciding what sort of complication is “suitable” and finding such a φ is surprisingly annoying. Here is a quick example: One might guess that any φ which is unprovable and would work. In that case we could take φ to be CON—that is, ¬Th(⌈0 = 1⌉); but CON ∧ Th(⌈CON⌉) is refutable in PA, so is provably equivalent to the quantifier-free formula 0 = 1.

scholarly journals A non-transitive relevant implication corresponding to classical logic consequence

2019 ◽  
Vol 16 (2) ◽  
pp. 10
Author(s):  
Peter Verdée ◽  
Inge De Bal ◽  
Aleksandra Samonek
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Modus Ponens ◽  
Relevant Logic ◽  
Object Language ◽  
Cut Rule ◽  
Consequence Relation ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Traditional Sense

In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence.          Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to thesequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word.       By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible.

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