Self-verifying axiom systems, the incompleteness theorem and related reflection principles

2001 ◽  
Vol 66 (2) ◽  
pp. 536-596 ◽  
Author(s):  
Dan E. Willard
Keyword(s):  
Reflection Principle ◽  
Incompleteness Theorem ◽  
Semantic Tableaux ◽  
Total Function ◽  
Reflection Principles ◽  
Axiom Systems ◽  
Weak Axiom

AbstractWe will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the “Tangibility Reflection Principle”. We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further.

scholarly journals An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

2005 ◽  
Vol 70 (4) ◽  
pp. 1171-1209 ◽  
Author(s):  
Dan E. Willard
Keyword(s):  
Modus Ponens ◽  
Axiom System ◽  
Incompleteness Theorem ◽  
Deduction System ◽  
Total Function ◽  
Boundary Case ◽  
Axiom Systems

AbstractThis article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae.Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions.

Reflection principles and iterated consistency assertions

1979 ◽  
Vol 44 (1) ◽  
pp. 33-35 ◽  
Author(s):  
George Boolos
Keyword(s):  
Standard Model ◽  
Reflection Principle ◽  
Incompleteness Theorem ◽  
First Order ◽  
Starting Point ◽  
The Standard Model ◽  
Reflection Principles ◽  
Order Arithmetic

This paper compares the strength of two sorts of sentences of PA (classical first-order arithmetic with induction): reflection principles and sentences that may be called iterated consistency assertions.Let Bew(x) be the standard provability predicate for PA, and for any sentence S of PA, let ⌈S⌉ be the numeral for the Gödel number of S. The reflection principle for S is the sentence Bew(⌈S⌉) → S, and a reflection principle is simply the reflection principle for some sentence. Nothing false (in the standard model for PA) is provable in PA, and therefore every reflection principle is true. Löb's theorem asserts that S is provable (in PA) if the reflection principle for S is provable.We shall suppose that the 0-ary propositional connectives ⊤ and ⊥ are taken as primitives in the formulation of PA. We define the iterated consistency assertions Conm by: Con0 = ⊤; Conm−1 = − Bew(⌈ − Conm⌉). Con1 may be taken to be the sentence of PA that expresses the consistency of PA; Conn−1, the sentence that expresses the consistency of PA ⋃ {Conn}.Our starting point is the observation that Con1 is equivalent (in PA) to the reflection principle for ⊥. (The second incompleteness theorem thus follows in a well-known way from Löb's theorem: if PA is consistent, then ⊥ is not provable, the reflection principle for ⊥ is not provable, and the consistency of PA is not provable either.)

How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic q

2002 ◽  
Vol 67 (1) ◽  
pp. 465-496 ◽  
Author(s):  
Dan E. Willard
Keyword(s):  
Axiom System ◽  
Incompleteness Theorem ◽  
Semantic Tableaux ◽  
The Standard Model ◽  
Natural Extensions ◽  
Consistent Extension ◽  
Infinite Cardinality ◽  
Axiom Systems ◽  
A Minor ◽  
Open Question

AbstractLet us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers and denoted as V. such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property.Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)

scholarly journals MODAL STRUCTURALISM AND REFLECTION

2018 ◽  
Vol 12 (4) ◽  
pp. 823-860 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Reflection Principles

AbstractModal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency

2006 ◽  
Vol 71 (4) ◽  
pp. 1189-1199 ◽  
Author(s):  
Dan E. Willard
Keyword(s):  
Incompleteness Theorem ◽  
Floating Point ◽  
Floating Point Arithmetic ◽  
Axiom Systems ◽  
Sufficient Strength ◽  
Point Arithmetic

AbstractGödel's Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer's floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.

Iterated reflection principles and the ω-rule

1982 ◽  
Vol 47 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Equivalent System ◽  
Formal Similarity ◽  
Sound System ◽  
Formal Systems ◽  
Ordinal Notations ◽  
Axiomatizable Theory ◽  
Reflection Principles ◽  
Image Position

The ω-rule,with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

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