2,356 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
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
A quasi-isometric embedding theorem for groups
We show that every group of at most exponential growth with respect to
some left invariant metric admits a bi-Lipschitz embedding into a finitely
generated group such that is amenable (respectively, solvable,
satisfies a non-trivial identity, elementary amenable, of finite decomposition
complexity, etc.) whenever is. We also discuss some applications to
compression functions of Lipschitz embeddings into uniformly convex Banach
spaces, F{\o}lner functions, and elementary classes of amenable groups
Intersection theory and the Horn inequalities for invariant subspaces
We provide a direct, intersection theoretic, argument that the Jordan models
of an operator of class C_{0}, of its restriction to an invariant subspace, and
of its compression to the orthogonal complement, satisfy a multiplicative form
of the Horn inequalities, where `inequality' is replaced by `divisibility'.
When one of these inequalities is saturated, we show that there exists a
splitting of the operator into quasidirect summands which induces similar
splittings for the restriction of the operator to the given invariant subspace
and its compression to the orthogonal complement. The result is true even for
operators acting on nonseparable Hilbert spaces. For such operators the usual
Horn inequalities are supplemented so as to apply to all the Jordan blocks in
the model
Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces
Assuming the existence of certain large cardinal numbers, we prove that for
every projective filter over the set of natural numbers,
-bases in Banach spaces have continuous coordinate functionals. In
particular, this applies to the filter of statistical convergence, thereby we
solve a problem by V. Kadets (at least under the presence of certain large
cardinals). In this setting, we recover also a result of Kochanek who proved
continuity of coordinate functionals for countably generated filters (Studia
Math., 2012).Comment: 10 p
Matter as Information. Quantum Information as Matter
Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latter. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of âHumeâs principleâ to be interpreted in terms of quantum mechanics as equivalence of âmanyâ and âmuchâ underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis
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