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13
Undecidability and incompleteness in classical mechanics
 Internat. J. Theoret. Physics
, 1991
"... We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian ..."
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We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of G6del's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained. 1.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
The Mathematician as a Formalist
 in Truth in Mathematics (H.G. Dales and
, 1998
"... Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millenni ..."
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Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)
Summable gaps
 Ann. Pure Appl. Logic
, 2003
"... Abstract. It is proved, under Martin’s Axiom, that all (ω1, ω1) gaps in P(N) / fin are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. ..."
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Abstract. It is proved, under Martin’s Axiom, that all (ω1, ω1) gaps in P(N) / fin are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin’s Axiom the Baire categoricity of all ∆ 1 3 non ∆ 1 3complete sets of reals requires a weakly compact cardinal. One of the most striking early discoveries in Set Theory was Hausdorff’s construction of a gap inside the Boolean algebra P(N) / fin. And one of the more interesting phenomenon associated with these gaps is the phenomenon of destructibility, that is the possibility of interpolating a gap (and
LUZIN GAPS
"... Abstract. We isolate a class of Fσδ ideals on N that includes all analytic Pideals and all Fσ ideals, and introduce ‘Luzin gaps ’ in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between qu ..."
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Abstract. We isolate a class of Fσδ ideals on N that includes all analytic Pideals and all Fσ ideals, and introduce ‘Luzin gaps ’ in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author’s Rigidity Conjecture for quotients P(N)/I. We also prove that the ideals NWD(Q) and NULL(Q) have the Radon–Nikodym property, and (using OCA∞) an uniformization result for Kcoherent families of continuous partial functions. One of the most fascinating facts about the Boolean algebra P(N) / Fin was discovered by Hausdorff in 1908. In [20], he constructed two families A and B of sets of integers such that (a) A ∩ B is finite for all A ∈ A and all B ∈ B, (b) for every C ⊆ N either A \ C is infinite for some A ∈ A or B ∩ C is infinite for some B ∈ B, and
THE RELATIVE COMMUTANT OF SEPARABLE C*ALGEBRAS OF REAL RANK ZERO
, 2008
"... Abstract. We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*algebras and C*subalgebras, respectively, and all ultrafilter ..."
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Abstract. We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*algebras and C*subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C*terminology is standard (see e.g., [2]). In the following U ranges over nonprincipal ultrafilters on N. With A U denoting the (norm, also called C*) ultrapower of a C*algebra A associated with U we have FU(A) = A ′ ∩ A U, the relative commutant of A in its ultrapower. This invariant plays an important role in [8] and [7]. Theorem 1. For every separable infinitedimensional C*algebra A of real rank zero the following are equivalent. (1) FU(A) ∼ = FV(A) for any two nonprincipal ultrafilters U and V on N. (2) A U ∼ = A V for any two nonprincipal ultrafilters U and V on N. (3) The Continuum Hypothesis. The equivalence of (3) and (2) in Theorem 1 for every infinitedimensional C*algebra A of cardinality 2 ℵ0 that has arbitrarily long finite chains in the Murrayvon Neumann ordering of projections was proved in [6, Corollary 3.8], using the same Dow’s result from [4] used here. We shall prove (1) implies (3) and (2) implies (3) in Corollary 10 below. The reverse implications are wellknown consequences of countable saturatedness of ultrapowers associated with nonprincipal ultrafilters on N (see [1, Proposition 7.6]). The implication from (3) to (1) holds for every separable C*algebra A and the implication from (3) to (2) holds for every C*algebra A of size 2 ℵ0. The point is that if A is separable then the isomorphism between diagonal copies of A extends to an isomorphism between
Questions on Automatic Continuity
"... . We present a variety of open questions in automatic continuity theory, concentrating on homomorphisms between Banach algebras and derivations from a Banach algebra A into a Banach Abimodule. 1991 Mathematics Subject Classification: 46H40. 1. Introduction In automatic continuity theory, we are c ..."
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. We present a variety of open questions in automatic continuity theory, concentrating on homomorphisms between Banach algebras and derivations from a Banach algebra A into a Banach Abimodule. 1991 Mathematics Subject Classification: 46H40. 1. Introduction In automatic continuity theory, we are concerned with conditions that imply that a linear map between Banach spaces (or more general topological linear spaces) is necessarily continuous. A rather general context in which questions are posed is the following. Let (A; k \Delta k) be a Banach algebra, and let (E; k \Delta k) be a Banach space which is an Abimodule for the maps (a; x) 7! a \Delta x and (a; x) 7! x \Delta a. Then E is a weak Banach Abimodule if the maps x 7! a \Delta x and a 7! x \Delta a on E are continuous for each a 2 A. For example, suppose that A and B are Banach algebras and that ` : A ! B is a homomorphism. Then B is a weak Banach Abimodule for the module operations given by a \Delta b = `(a)b and b \Delta a...
Automatic continuity for Banach algebras
"... ollows easily from the continuity of characters that every homomorphism ` : A ! B from a Banach algebra A into a commutative, semisimple Banach algebra B is continuous. A closely related result is Johnson's uniquenessofnormtheorem: every semisimple Banach algebra has a unique complete algebra n ..."
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ollows easily from the continuity of characters that every homomorphism ` : A ! B from a Banach algebra A into a commutative, semisimple Banach algebra B is continuous. A closely related result is Johnson's uniquenessofnormtheorem: every semisimple Banach algebra has a unique complete algebra norm. For lovely alternative proofs of this theorem, see [1] and [13]. There are nonsemsimple, commutative Banach algebras which have a unique complete algebra norm. For example, this is true of the convolution algebras L (R ; !), where ! is a weight function on R (see [4], x5.2). On the other hand, there are even commutative Banach algebras with a onedimensional (Jacobson) radical which do not have a unique complete algebra norm (see [4], x5.1). Nevertheless there are striking open questions in this area: we do not know whether a commutative Banach algebra which is an integral domain necessarily has a unique complete algebra norm; the question is also open for Banach algebras with
and
, 2000
"... Abstract. This is a survey of the author’s recent results on the Kadison and Halmos similarity problems and the closely connected notion of “length ” of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison’s similarity problem. Let A be a uni ..."
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Abstract. This is a survey of the author’s recent results on the Kadison and Halmos similarity problems and the closely connected notion of “length ” of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison’s similarity problem. Let A be a unital C ∗algebra and let u: A → B(H) (H Hilbert) be a unital homomorphism (i.e. we have u(1) = 1 and u(ab) = u(a)u(b) ∀ a, b ∈ A). Show that if u is bounded, then u is similar to a ∗homomorphism, i.e. ∃ ξ: H → H invertible such that uξ: a → ξ −1 u(a)ξ is a ∗homomorphism ( = C ∗representation). Explicitly, the conclusion means that ∀ a ∈ A ξ −1 u(a ∗)ξ = (ξ −1 u(a)ξ) ∗, when this holds, Kadison calls u “orthogonalizable”. Many partial results are known, mainly due to Erik Christensen ([C1–C4]) and Uffe Haagerup ([H1]). In particular, they established (see [C3] and [H1]) this conjecture for cyclic homomorphisms, i.e. when u admits a cyclic vector h in H ( = a vector h such that u(A)h = H) or more generally when u admits a finite cyclic set h1,..., hn (so that we have u(A)h1 + · · · + u(A)hn = H). In addition, the Kadison conjecture is known in the following cases: (i) A is commutative.