We construct a machine that knows its own code, at the price of not knowing its own factivity. knowing machines; Reinhardt's strong mechanistic thesis; Lucas-Penrose argument; Kleene's recursion theorem; quantified modal logic.
Reinhardt's conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.
A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the (...) paradox. (shrink)
A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes König's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.
We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result (...) involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)
We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.
We present a reformulation of loop quantum gravity with a cosmological constant and no matter as a Fermi-liquid theory. When the topological sector is deformed and large gauge symmetry is broken, we show that the Chern–Simons state reduces to Jacobson’s degenerate sector describing 1+1 dimensional propagating fermions with nonlocal interactions. The Hamiltonian admits a dual description which we realize in the simple BCS model of superconductivity. On one hand, Cooper pairs are interpreted as wormhole correlations at the de Sitter horizon; (...) their number yields the de Sitter entropy. On the other hand, BCS is mapped into a deformed conformal field theory reproducing the structure of quantum spin networks. When area measurements are performed, Cooper-pair insertions are activated on those edges of the spin network intersecting the given area, thus providing a description of quantum measurements in terms of excitations of a Fermi sea to superconducting levels. The cosmological constant problem is naturally addressed as a nonperturbative mass-gap effect of the true Fermi-liquid vacuum. (shrink)