Essential Incompleteness of Arithmetic Verified by Coq

A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq's type theory and therefore is incomplete.Comment: This paper is part of the proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005). For the associated Coq source files see the TeX sources, or see <http://r6.ca/Goedel20050512.tar.gz

On closed rotating worlds

A new solution for the stationary closed world with rigid rotation is obtained for the spinning fluid source. It is found that the spin and vorticity are locally balanced. This model qualitatively shows that the local rotation of the cosmological matter can be indeed related to the global cosmic vorticity, provided the total angular momentum of the closed world is vanishing.Comment: 10 pages, Revtex, to appear in Phys. Rev. D6

An exterior for the G\"{o}del spacetime

We match the vacuum, stationary, cylindrically symmetric solution of Einstein's field equations with $\Lambda$, in a form recently given by Santos, as an exterior to an infinite cylinder of dust cut out of a G\"{o}del universe. There are three cases, depending on the radius of the cylinder. Closed timelike curves are present in the exteriors of some of the solutions. There is a considerable similarity between the spacetimes investigated here and those of van Stockum referring to an infinite cylinder of rotating dust matched to vacuum, with $\Lambda=0$.Comment: 11 pages, LaTeX 2.09, no figures. Submitted to Classical and Quantum Gravit

On certain relationships between cosmological observables in the Einstein-Cartan gravity

We show that in the Einstein-Cartan gravity it is possible to obtain a relation between Hubble's expansion and the global rotation (vorticity) of the Universe. Gravitational coupling can be reduced to dimensionless quantity of order unity, fixing the scalar mass density and the resulting negative cosmological constant at spacelike infinity. Current estimates of the expansion and rotation (see also astro-ph/9703082) of the Universe favour the massive spinning particles as candidate particles for cold and hot dark matter. Nodland and Ralston vorticity (Phys. Rev. Lett. 78 (1997) 3043) overestimates the value favoured by the Einstein-Cartan gravity for three orders of magnitude.Comment: 7 pages, LaTeX styl

Bogomolnyi Bound with a Cosmological Constant

Bogomolnyi-type bound is constructed for the topological solitons in O(3) nonlinear $\sigma$ model coupled to gravity with a negative cosmological constant. Spacetimes made by self-dual solutions form a class of G\"{o}del-type universe. In the limit of a spinless massive point particle, the obtained stationary metric does not violate the causality and it is a new point particle solution different from the known static hyperboloid and black hole. We also showed that static Nielsen-Olesen vortices saturate Bogomolnyi-type bound only when the cosmological constant vanishes.Comment: 11 pages, RevTe

Boundary States for Supertubes in Flat Spacetime and Godel Universe

We construct boundary states for supertubes in the flat spacetime. The T-dual objects of supertubes are moving spiral D1-branes (D-helices). Since we can obtain these D-helices from the usual D1-branes via null deformation, we can construct the boundary states for these moving D-helices in the covariant formalism. Using these boundary states, we calculate the vacuum amplitude between two supertubes in the closed string channel and read the open string spectrum via the open closed duality. We find there are critical values of the energy for on-shell open strings on the supertubes due to the non-trivial stringy correction. We also consider supertubes in the type IIA Godel universe in order to use them as probes of closed timelike curves. This universe is the T-dual of the maximally supersymmetric type IIB PP-wave background. Since the null deformations of D-branes are also allowed in this PP-wave, we can construct the boundary states for supertubes in the type IIA Godel universe in the same way. We obtain the open string spectrum on the supertube from the vacuum amplitude between supertubes. As a consequence, we find that the tachyonic instability of open strings on the supertube, which is the signal of closed time like curves, disappears due to the stringy correction.Comment: 26 pages, 3 figures, v2: explanations added, references added, v3: explanations adde

Revisiting consistency conditions for quantum states of systems on closed timelike curves: an epistemic perspective

There has been considerable recent interest in the consequences of closed timelike curves (CTCs) for the dynamics of quantum mechanical systems. A vast majority of research into this area makes use of the dynamical equations developed by Deutsch, which were developed from a consistency condition that assumes that mixed quantum states uniquely describe the physical state of a system. We criticise this choice of consistency condition from an epistemic perspective, i.e., a perspective in which the quantum state represents a state of knowledge about a system. We demonstrate that directly applying Deutsch's condition when mixed states are treated as representing an observer's knowledge of a system can conceal time travel paradoxes from the observer, rather than resolving them. To shed further light on the appropriate dynamics for quantum systems traversing CTCs, we make use of a toy epistemic theory with a strictly classical ontology due to Spekkens and show that, in contrast to the results of Deutsch, many of the traditional paradoxical effects of time travel are present.Comment: 10 pages, 6 figures, comments welcome; v2 added references and clarified some points; v3 published versio

Shear-free rotating inflation

We demonstrate the existence of shear-free cosmological models with rotation and expansion which support the inflationary scenarios. The corresponding metrics belong to the family of spatially homogeneous models with the geometry of the closed universe (Bianchi type IX). We show that the global vorticity does not prevent the inflation and even can accelerate it.Comment: Revtex, 12 pages; to appear in Phys. Rev.

Exact Solution of Photon Equation in Stationary G\"{o}del-type and G\"{o}del Space-Times

In this work the photon equation (massless Duffin-Kemmer-Petiau equation) is written expilicitly for general type of stationary G\"{o}del space-times and is solved exactly for G\"{o}del-type and G\"{o}del space-times. Harmonic oscillator behaviour of the solutions is discussed and energy spectrum of photon is obtained.Comment: 9 pages,RevTeX, no figure, revised for publicatio