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.

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 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 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 GENERALIZATIONS OF GÖDEL’S INCOMPLETENESS THEOREMS FOR ∑n-DEFINABLE THEORIES OF ARITHMETIC

2017 ◽  
Vol 10 (4) ◽  
pp. 603-616 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Consistent Theory ◽  
Sound Theory ◽  
Incompleteness Theorems ◽  
Gödel’S Incompleteness Theorems

AbstractIt is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1 set of theorems has a true but unprovable ∏n sentence. Lastly, we prove that no ∑n+1-definable ∑n -sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively.

Recursion Theory for Metamathematics

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Incompleteness Theorem ◽  
Incompleteness Theorems ◽  
Gödel’S Incompleteness Theorems

This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

Where’s Waldo: Revisiting the Administrative Novel

Public Voices ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 1
Author(s):  
George Connor
Keyword(s):  
Organization And Administration ◽  
Player Piano ◽  
Simple Observation ◽  
Novel Approach ◽  
Contemporary Work

In 1968 Dwight Waldo published The Novelist on Organization and Administration: An Inquiry into the Relationships Between the Two Worlds.  His very simple observation was that “one can learn much about administration from novels” (1968, 4). After thirty years, it may be a time to re-examine both the literary and academic side of Waldo’s novel approach. The article offers Kurt Vonnegut’s Player Piano and David Foster Wallace’s The Pale King, one classic and one contemporary work, as vehicles for revisiting Waldo’s pedagogy and reestablishing the linkage between administration and administrative novels.

Coinage and the Tributary Mode of Production

2018 ◽  
Author(s):  
Eduardo Manzano Moreno
Keyword(s):  
Roman Empire ◽  
Middle Ages ◽  
Social Role ◽  
State Of The Art ◽  
Early Middle Ages ◽  
The Social ◽  
Simple Question ◽  
Mode Of Production

This chapter addresses a very simple question: is it possible to frame coinage in the Early Middle Ages? The answer will be certainly yes, but will also acknowledge that we lack considerable amounts of relevant data potentially available through state-of-the-art methodologies. One problem is, though, that many times we do not really know the relevant questions we can pose on coins; another is that we still have not figured out the social role of coinage in the aftermath of the Roman Empire. This chapter shows a number of things that could only be known thanks to the analysis of coins. And as its title suggests it will also include some reflections on greed and generosity.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals Modal Structuralism with Theoretical Terms

Erkenntnis ◽  
2021 ◽  
Author(s):  
Holger Andreas ◽  
Georg Schiemer
Keyword(s):  
Philosophy Of Mathematics ◽  
Formal Semantics ◽  
Peano Arithmetic ◽  
Theoretical Terms ◽  
Structural Interpretation ◽  
Oxford University ◽  
Structuralist Approach ◽  
Modal Semantics ◽  
Oxford University Press

AbstractIn this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic.

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