364 research outputs found
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
On Various Negative Translations
Several proof translations of classical mathematics into intuitionistic
mathematics have been proposed in the literature over the past century. These
are normally referred to as negative translations or double-negation
translations. Among those, the most commonly cited are translations due to
Kolmogorov, Godel, Gentzen, Kuroda and Krivine (in chronological order). In
this paper we propose a framework for explaining how these different
translations are related to each other. More precisely, we define a notion of a
(modular) simplification starting from Kolmogorov translation, which leads to a
partial order between different negative translations. In this derived
ordering, Kuroda and Krivine are minimal elements. Two new minimal translations
are introduced, with Godel and Gentzen translations sitting in between
Kolmogorov and one of these new translations.Comment: In Proceedings CL&C 2010, arXiv:1101.520
A Short and Readable Proof of Cut Elimination for Two 1st Order Modal Logics
Since 1960s, logicians, philosophers, AI people have cast eyes on modal logic. Among various modal logic systems, propositional provability logic which was established by Godel modeling provability in axiomatic Peano Arithmetic (PA) was the most striking application for mathematicians. After Godel, researchers gradually explored the predicate case in provability logic. However, the most natural application QGL for predicate provability logic is not able to admit cut elimination. Recently, a potential candidate for the predicate provability logic ML3 and its precursors BM and M3 introduced by Toulakis,Kibedi, Schwartz dedicated that A is always closed. Although ML3, BM and M3 are cut free, the cut elimination proof with the unfriendly nested induction of high multiplicity is difficult to understand. In this thesis, I will show a cut elimination proof for all (Gentzenisations) of BM, M3 and ML3, with much more readable inductions of lower multiplicity
Epistemic systems and Flagg and Friedman's translation
In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of
the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation
of Heyting arithmetic to Shapiro's epistemic arithmetic . In
\S 2, we shall prove the faithfulness of without using
stability, by introducing another translation from an epistemic system to
corresponding intuitionistic system which we shall call \it the modified
Rasiowa-Sikorski translation\rm . That is, this introduction of the new
translation simplifies the original Flagg and Friedman's proof. In \S 3, we
shall give some applications of the modified one for the disjunction property
() and the numerical existence property () of
Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule
in is proved via . So and are equivalent. In \S 5, we
shall give some relations among the translations treated in the previous
sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In
\S 7, we shall propose several(modal-)epistemic versions of Markov's rule for
Horsten's modal-epistemic arithmetic . And, as in \S 4, we shall study
some meta-implications among those versions of Markov's rules in and
one in . Friedman and Sheard gave a modal analogue (i.e.
Theorem in \cite{fs}) of Friedman's theorem (i.e. Theorem 1 in
\cite {friedman}): \it Any recursively enumerable extension of which
has also has \rm . In \S 8, we shall give a proof
of our \it Fundamental Conjecture \rm proposed in Inou\'{e}
\cite{ino90a} as follows: This is a new type of proofs. In \S 9, I
shall give discussions.Comment: 33 page
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
The Implicit Commitment of Arithmetical Theories and Its Semantic Core
According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Taitâs finitism and the role played in it by Primitive Recursive Arithmetic, Isaacsonâs thesis and Peano Arithmetic, Nelsonâs ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear (proof-theoretic) reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories
Completeness of a first-order temporal logic with time-gaps
The first-order temporal logics with ⥠and â of time structures isomorphic to Ď (discrete linear time) and trees of Ď-segments (linear time with branching gaps) and some of its fragments are compared: the first is not recursively axiomatizable. For the second, a cut-free complete sequent calculus is given, and from this, a resolution system is derived by the method of Maslov
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