Infinite Regresses, Infinite Beliefs

One way of mapping part of the domain of epistemology is to represent various theories as responses to the following argument: (I) A belief (strictly, a token state of belief) is justified only if a justified belief is a reason for it. (Premiss.) (II) There are justified beliefs. (Premiss.) (III) The proper ancestral of the reason-relation is irreflexive. (Premiss.) (IV) There is an infinite sequence (strictly, a sequence with infinite range) of justified beliefs each of which is a reason for its predecessor, if any. (From (I) to (III).) (V) There is no such sequence. (Premiss.) (VI) There both is and is not such a sequence. (From (IV) and (V).) (VII) Not-(I) / not-(II) / not-(III) / not-(V). (Reductio.

Entanglement convertibility for infinite dimensional pure bipartite states

It is shown that the order property of pure bipartite states under SLOCC (stochastic local operations and classical communications) changes radically when dimensionality shifts from finite to infinite. In contrast to finite dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the ``infinite workspace'' property, remains even in physically relevant infinite dimensional systems

A trifurcated waveguide problem

We consider the diffraction of the dominant wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. A closed form solution of the resulting matrix Wiener-Hopf equation is obtained

Structures of boson and fermion Fock spaces in the space of symmetric functions

We realize the Weil representation of infinite dimensional symplectic group and spinor representation of infinite-dimensional group $GL$ by linear operators in the space of symmetric functions in infinite number of variables.Comment: 32 page

Cyclic Universe and Infinite Past

We address two questions about the past for infinitely cyclic cosmology. The first is whether it can contain an infinite length null geodesic into the past in view of the Borde-Guth-Vilenkin (BGV) "no-go" theorem, The second is whether, given that a small fraction of spawned universes fail to cycle, there is an adequate probability for a successful universe after an infinite time. We give positive answers to both questions then show that in infinite cyclicity the total number of universes has been infinite for an arbitrarily long time.Comment: 7 pages. Clarification in discussion of infinite pas

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200