7,368 research outputs found
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
Singly generated quasivarieties and residuated structures
A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K is
PSC then it still has the JEP, and if it has the JEP and its nontrivial members
lack trivial subalgebras, then its relatively simple members all belong to the
universal class generated by one of them. Under these conditions, if K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff its nontrivial members have a common retract. The
theory is then applied to the variety of De Morgan monoids, where we isolate
the sub(quasi)varieties that are PSC and those that have the JEP, while
throwing fresh light on those that are structurally complete. The results
illuminate the extension lattices of intuitionistic and relevance logics
Compactness of powers of \omega
We characterize exactly the compactness properties of the product of \kappa\
copies of the space \omega\ with the discrete topology. The characterization
involves uniform ultrafilters, infinitary languages, and the existence of
nonstandard elements in elementary estensions. We also have results involving
products of possibly uncountable regular cardinals.Comment: v2 slightly improve
Structural Relativity and Informal Rigour
Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures
Degrees of the finite model property: the antidichotomy theorem
A classic result in modal logic, known as the Blok Dichotomy Theorem, states
that the degree of incompleteness of a normal extension of the basic modal
logic is or . It is a long-standing open problem
whether Blok Dichotomy holds for normal extensions of other prominent modal
logics (such as or ) or for extensions of the intuitionistic
propositional calculus . In this paper, we introduce the notion
of the degree of finite model property (fmp), which is a natural variation of
the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem
that the degree of fmp of a normal extension of remains or
. In contrast, our main result establishes the following
Antidichotomy Theorem for the degree of fmp for extensions of :
each nonzero cardinal such that or is realized as the degree of fmp of some extension of
. We then use the Blok-Esakia theorem to establish the same
Antidichotomy Theorem for normal extensions of and
Proceedings of International Workshop "Global Computing: Programming Environments, Languages, Security and Analysis of Systems"
According to the IST/ FET proactive initiative on GLOBAL COMPUTING, the goal is to obtain techniques (models, frameworks, methods, algorithms) for constructing systems that are flexible, dependable, secure, robust and efficient.
The dominant concerns are not those of representing and manipulating data efficiently but rather those of handling the co-ordination and interaction, security, reliability, robustness, failure modes, and control of risk of the entities in the system and the overall design, description and performance of the system itself.
Completely different paradigms of computer science may have to be developed to tackle these issues effectively. The research should concentrate on systems having the following characteristics: • The systems are composed of autonomous computational entities where activity is not centrally controlled, either because global control is impossible or impractical, or because the entities are created or controlled by different owners.
• The computational entities are mobile, due to the movement of the physical platforms or by movement of the entity from one platform to another.
• The configuration varies over time. For instance, the system is open to the introduction of new computational entities and likewise their deletion.
The behaviour of the entities may vary over time.
• The systems operate with incomplete information about the environment.
For instance, information becomes rapidly out of date and mobility requires information about the environment to be discovered.
The ultimate goal of the research action is to provide a solid scientific foundation for the design of such systems, and to lay the groundwork for achieving effective principles for building and analysing such systems.
This workshop covers the aspects related to languages and programming environments as well as analysis of systems and resources involving 9 projects (AGILE , DART, DEGAS , MIKADO, MRG, MYTHS, PEPITO, PROFUNDIS, SECURE) out of the 13 founded under the initiative. After an year from the start of the projects, the goal of the workshop is to fix the state of the art on the topics covered by the two clusters related to programming environments and analysis of systems as well as to devise strategies and new ideas to profitably continue the research effort towards the overall objective of the initiative.
We acknowledge the Dipartimento di Informatica and Tlc of the University of Trento, the Comune di Rovereto, the project DEGAS for partially funding the event and the Events and Meetings Office of the University of Trento for the valuable collaboration
Heating of blue compact dwarf galaxies: gas distribution and photoionization by stars in I Zw 18
Photoionization models so far are unable to account for the high electron
temperature Te([O III]) implied by the line ratio [O III]4363A/[O III]5007A in
low-metallicity blue compact dwarf galaxies, casting doubts on the assumption
of photoionization by hot stars as the dominant source of heating of the gas in
these objects. Combinations of runs of the 1-D photoionization code NEBU are
used to explore alternative models for the giant H II region shell I Zw 18 NW.
Acceptable models are obtained, which represent schematically an incomplete
shell comprising radiation-bounded condensations embedded in a low-density
matter-bounded diffuse medium. The thermal pressure contrast between gas
components is about a factor 7. The diffuse phase can be in pressure balance
with the hot superbubble fed by mechanical energy from the inner massive star
cluster. The failure of previous modellings is ascribed to (1) the adoption of
an inadequate small-scale gas density distribution, which proves critical when
the collisional excitation of hydrogen contributes significantly to the cooling
of the gas, and possibly (2) a too restrictive implementation of Wolf-Rayet
stars in synthetic stellar cluster spectral energy distributions. A neutral gas
component heated by soft X-rays, whose power is less than 1% of the star
cluster luminosity and consistent with CHANDRA data, can explain the
low-ionization fine-structure lines detected by SPITZER. [O/Fe] is slightly
smaller in I Zw 18 NW than in Galactic Halo stars of similar metallicity and
[C/O] is correlatively large. Extra heating by, e.g., dissipation of mechanical
energy is not required to explain Te([O III]) in I Zw 18. Important
astrophysical developments are at stakes in the 5% uncertainty attached to [O
III] collision strengths.Comment: 20 pages, 8 figures, to be published in A&
Brentanian Continua
Brentano’s theory of continuity is based on his account of boundaries. The core idea of the theory is that boundaries and coincidences thereof belong to the essence of continua. Brentano is confident that he developed a full-fledged, boundary-based, theory of continuity1; and scholars often concur: whether or not they accept Brentano’s take on continua they consider it a clear contender. My impression, on the contrary, is that, although it is infused with invaluable insights, several aspects of Brentano’s account of continuity remain inchoate. To be clear, the theory of boundaries on which it relies, as well as the account of ontological dependence that Brentano develops alongside his theory of boundaries, constitute splendid achievements. However, the passage from the theory of boundaries to the account of continuity is rather sketchy. This paper pinpoints some chief problems raised by this transition, and proposes some solutions to them which, if not always faithful to the letter of Brentano’s account of continua, are I believe faithful to its spirit.
§1 presents Brentano’s critique of the mathematical account of the continuous. §2 introduces Brentano’s positive account of continua. §3 raises three worries about Brentano’s account of continuity. §4 proposes a Neo-Brentanian approach to continua that handles these worries
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