42,992 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Explicit Evidence Systems with Common Knowledge
Justification logics are epistemic logics that explicitly include
justifications for the agents' knowledge. We develop a multi-agent
justification logic with evidence terms for individual agents as well as for
common knowledge. We define a Kripke-style semantics that is similar to
Fitting's semantics for the Logic of Proofs LP. We show the soundness,
completeness, and finite model property of our multi-agent justification logic
with respect to this Kripke-style semantics. We demonstrate that our logic is a
conservative extension of Yavorskaya's minimal bimodal explicit evidence logic,
which is a two-agent version of LP. We discuss the relationship of our logic to
the multi-agent modal logic S4 with common knowledge. Finally, we give a brief
analysis of the coordinated attack problem in the newly developed language of
our logic
A bound for Dickson's lemma
We consider a special case of Dickson's lemma: for any two functions on
the natural numbers there are two numbers such that both and
weakly increase on them, i.e., and . By a
combinatorial argument (due to the first author) a simple bound for such
is constructed. The combinatorics is based on the finite pigeon hole principle
and results in a descent lemma. From the descent lemma one can prove Dickson's
lemma, then guess what the bound might be, and verify it by an appropriate
proof. We also extract (via realizability) a bound from (a formalization of)
our proof of the descent lemma.
Keywords: Dickson's lemma, finite pigeon hole principle, program extraction
from proofs, non-computational quantifiers
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is âpureâ if it draws only on what is âcloseâ or âintrinsicâ to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is âextrinsic,â âextraneous,â âdistant,â âremote,â âalien,â or âforeignâ to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
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