142 research outputs found
From Tarski to G\"odel. Or, how to derive the Second Incompleteness Theorem from the Undefinability of Truth without Self-reference
In this paper, we provide a fairly general self-reference-free proof of the
Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of
Truth.Comment: 7 page
Sharpened lower bounds for cut elimination
We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our new lower bounds remove the constant of proportionality, giving an exponential stack of height equal to d − O(1). The proof method is based on more efficiently expressing the Gentzen-Solovay cut formulas as low depth formulas
The Small-Is-Very-Small Principle
The central result of this paper is the small-is-very-small principle for
restricted sequential theories. The principle says roughly that whenever the
given theory shows that a property has a small witness, i.e. a witness in every
definable cut, then it shows that the property has a very small witness: i.e. a
witness below a given standard number.
We draw various consequences from the central result. For example (in rough
formulations): (i) Every restricted, recursively enumerable sequential theory
has a finitely axiomatized extension that is conservative w.r.t. formulas of
complexity . (ii) Every sequential model has, for any , an extension
that is elementary for formulas of complexity , in which the
intersection of all definable cuts is the natural numbers. (iii) We have
reflection for -sentences with sufficiently small witness in any
consistent restricted theory . (iv) Suppose is recursively enumerable
and sequential. Suppose further that every recursively enumerable and
sequential that locally inteprets , globally interprets . Then,
is mutually globally interpretable with a finitely axiomatized sequential
theory.
The paper contains some careful groundwork developing partial satisfaction
predicates in sequential theories for the complexity measure depth of
quantifier alternations
Curious satisfaction classes
We present two new constructions of satisfaction/truth classes over models of
PA (Peano Arithmetic) that provide a foil to the fact that the existence of a
disjunctively correct full truth class over a model M of PA implies that
Con(PA) holds in M.Comment: 12 page
A Definability Dichotomy for Finite Valued CSPs
Finite valued constraint satisfaction problems are a formalism for describing many natural optimisation problems, where constraints on the values that variables can take come with rational weights and the aim is to find an assignment of minimal cost. Thapper and Zivny have recently established a complexity dichotomy for valued constraint languages. They show that each such languages either gives rise to a polynomial-time solvable optimisation problem, or to an NP-hard one, and establish a criterion to distinguish the two cases. We refine the dichotomy by showing that all optimisation problems in the first class are definable in fixed-point language with counting, while all languages in the second class are not definable, even in infinitary logic with counting. Our definability dichotomy is not conditional on any complexity-theoretic assumption
Mathematical constraints and their philosophical impact
This paper investigates some of the constraints encountered in mathematics. It proves
the incompleteness theorems. It investigates or discusses theorem proving procedures,
complexity, undecidability, undefinability, and covers some examples of independent
formulas. Some dialogue on the influence of these concepts on the development
of mathematics, and on mathematical philosophy since the 20th century is
included. This paper should give any person with an interest in philosophy sufficient
understanding to discuss mathematical philosophy with mathematicians
Proof Theory for Lax Logic
In this paper some proof theory for propositional Lax Logic is developed. A
cut free terminating sequent calculus is introduced for the logic, and based on
that calculus it is shown that the logic has uniform interpolation.
Furthermore, a separate, simple proof of interpolation is provided that also
uses the sequent calculus. From the literature it is known that Lax Logic has
interpolation, but all known proofs use models rather than proof systems
Varieties of truth definitions
We study the structure of the partial order induced by the definability
relation on definitions of truth for the language of arithmetic. Formally, a
definition of truth is any sentence which extends a weak arithmetical
theory (which we take to be EA) such that for some formula and any
arithmetical sentence , is provable in . We say that a sentence is definable
in a sentence , if there exists an unrelativized translation from the
language of to the language of which is identity on the
arithmetical symbols and such that the translation of is provable in
. Our main result is that the structure consisting of truth definitions
which are conservative over the basic arithmetical theory forms a countable
universal distributive lattice. Additionally, we generalize the result of
Pakhomov and Visser showing that the set of (G\"odel codes of) definitions of
truth is not -definable in the standard model of arithmetic. We
conclude by remarking that no -sentence, satisfying certain further
natural conditions, can be a definition of truth for the language of
arithmetic
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