2,530 research outputs found
Pincherle's theorem in Reverse Mathematics and computability theory
We study the logical and computational properties of basic theorems of
uncountable mathematics, in particular Pincherle's theorem, published in 1882.
This theorem states that a locally bounded function is bounded on certain
domains, i.e. one of the first 'local-to-global' principles. It is well-known
that such principles in analysis are intimately connected to (open-cover)
compactness, but we nonetheless exhibit fundamental differences between
compactness and Pincherle's theorem. For instance, the main question of Reverse
Mathematics, namely which set existence axioms are necessary to prove
Pincherle's theorem, does not have an unique or unambiguous answer, in contrast
to compactness. We establish similar differences for the computational
properties of compactness and Pincherle's theorem. We establish the same
differences for other local-to-global principles, even going back to
Weierstrass. We also greatly sharpen the known computational power of
compactness, for the most shared with Pincherle's theorem however. Finally,
countable choice plays an important role in the previous, we therefore study
this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi
The origin of the difference between space and time
All differences between the role of space and time in nature are explained by
proposing the principles in which none of the space-time coordinates has an
{\it a priori} special role. Spacetime is treated as a nondynamical manifold,
with a fixed global topology. Dynamical theory of gravity
determines only the metric tensor on a fixed manifold. All dynamics is treated
as a Cauchy problem, so it {\em follows} that one coordinate takes a special
role. It is proposed that {\em any} boundary condition that is finite
everywhere leads to a solution which is also finite everywhere. This explains
the signature of the metric, the boundedness of energy from below,
the absence of tachyons, and other related properties of nature. The time arrow
is explained by proposing that the boundary condition should be ordered. The
quantization is considered as a boundary condition for field operators. Only
the physical degrees of freedom are quantized.Comment: 22 pages, late
Progressive and merging-proof taxation
We investigate the implications and logical relations between progressivity (a principle of distributive justice) and merging-proofness (a strategic principle) in taxation. By means of two characterization results, we show that these two principles are intimately related, despite their different nature. In particular, we show that, in the presence of continuity and consistency (a widely accepted framework for taxation) progressivity implies merging-proofness and that the converse implication holds if we add an additional strategic principle extending the scope of merging-proofness to a multilateral setting. By considering operators on the space of taxation rules, we also show that progressivity is slightly more robust than merging-proofness.taxation, progressivity, merging-proofness, consistency, operators
Progressive and merging-proof taxation
We investigate the implications and logical relations between progressivity (a principle of distributive justice) and merging-proofness (a strategic principle) in taxation. By means of two characterization results, we show that these two principles are intimately related, despite their different nature. In particular, we show that, in the presence of continuity and consistency (a widely accepted framework for taxation) progressivity implies merging-proofness and that the converse implication holds if we add an additional strategic principle extending the scope of merging-proofness to a multilateral setting. By considering operators on the space of taxation rules, we also show that progressivity is slightly more robust than merging-proofness.taxation, progressivity, merging-proofness, consistency, operators.
Axiomatic approach to the cosmological constant
A theory of the cosmological constant Lambda is currently out of reach.
Still, one can start from a set of axioms that describe the most desirable
properties a cosmological constant should have. This can be seen in certain
analogy to the Khinchin axioms in information theory, which fix the most
desirable properties an information measure should have and that ultimately
lead to the Shannon entropy as the fundamental information measure on which
statistical mechanics is based. Here we formulate a set of axioms for the
cosmological constant in close analogy to the Khinchin axioms, formally
replacing the dependency of the information measure on probabilities of events
by a dependency of the cosmological constant on the fundamental constants of
nature. Evaluating this set of axioms one finally arrives at a formula for the
cosmological constant that is given by Lambda = (G^2/hbar^4) (m_e/alpha_el)^6,
where G is the gravitational constant, m_e is the electron mass, and alpha_el
is the low energy limit of the fine structure constant. This formula is in
perfect agreement with current WMAP data. Our approach gives physical meaning
to the Eddington-Dirac large number hypothesis and suggests that the observed
value of the cosmological constant is not at all unnatural.Comment: 7 pages, no figures. Some further references adde
Bounded Situation Calculus Action Theories
In this paper, we investigate bounded action theories in the situation
calculus. A bounded action theory is one which entails that, in every
situation, the number of object tuples in the extension of fluents is bounded
by a given constant, although such extensions are in general different across
the infinitely many situations. We argue that such theories are common in
applications, either because facts do not persist indefinitely or because the
agent eventually forgets some facts, as new ones are learnt. We discuss various
classes of bounded action theories. Then we show that verification of a
powerful first-order variant of the mu-calculus is decidable for such theories.
Notably, this variant supports a controlled form of quantification across
situations. We also show that through verification, we can actually check
whether an arbitrary action theory maintains boundedness.Comment: 51 page
Homotopy theory of diagrams
In this paper we develop homotopy theoretical methods for studying diagrams.
In particular we explain how to construct homotopy colimits and limits in an
arbitrary model category. The key concept we introduce is that of a model
approximation. Our key result says that if a category admits a model
approximation then so does any diagram category with values in this category.
From the homotopy theoretical point of view categories with model
approximations have similar properties to those of model categories. They admit
homotopy categories (localizations with respect to weak equivalences). They
also can be used to construct derived functors by taking the analogs of fibrant
and cofibrant replacements.
A category with weak equivalences can have several useful model
approximations. We take advantage of this possibility and in each situation
choose one that suits our needs. In this way we prove all the fundamental
properties of the homotopy colimit and limit: Fubini Theorem (the homotopy
colimit -respectively limit- commutes with itself), Thomason's theorem about
diagrams indexed by Grothendieck constructions, and cofinality statements.
Since the model approximations we present here consist of certain functors
"indexed by spaces", the key role in all our arguments is played by the
geometric nature of the indexing categories.Comment: 95 pages with inde
Bounded Arithmetic in Free Logic
One of the central open questions in bounded arithmetic is whether Buss'
hierarchy of theories of bounded arithmetic collapses or not. In this paper, we
reformulate Buss' theories using free logic and conjecture that such theories
are easier to handle. To show this, we first prove that Buss' theories prove
consistencies of induction-free fragments of our theories whose formulae have
bounded complexity. Next, we prove that although our theories are based on an
apparently weaker logic, we can interpret theories in Buss' hierarchy by our
theories using a simple translation. Finally, we investigate finitistic G\"odel
sentences in our systems in the hope of proving that a theory in a lower level
of Buss' hierarchy cannot prove consistency of induction-free fragments of our
theories whose formulae have higher complexity
On the equivalence between progressive taxation and inequality reduction
We establish the precise connections between progressive taxation and inequality reduction, in a setting where the level of tax revenue to be raised is endogenously fixed and tax schemes are balanced. We show that, in contrast with the traditional literature on taxation, the equivalence between inequality reduction and the combination of progressivity and income order preservation does not always hold in this setting. However, we show that, among rules satisfying consistency and, either revenue continuity, or revenue monotonicity, the equivalence remains intact.progressivity, inequality reduction, income order preservation, consistency, taxation
Lower bounds rule!
We propose two axioms that introduce lower bounds into resource monotonicity requirements for rules for the problem of adjudicating conflicting claims. Suppose the amount to divide increases. The first axiom requires that two claimants whose lower bound changes equally experience an equal change in awards. The second axiom requires that extra resources are divided only among those claimants who experience a strictly positive change in their lower bound. We show that, in the two-claimant case, Concede-and-Divide is the only rule that satisfies both axioms when the axioms are defined over a large set of lower bounds that include the minimal rights lower bound and the secured lower bound. We also show that, in the n-claimant case where at least one claimant claims the total amount, the Minimal Overlap rule is the only rule that satisfies both axioms when the axioms are defined over the secured lower bound.claims problems, lower bounds, concede-and-divide, minimal overlap rule
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