Quantum Randomness and Underdetermination

We consider the nature of quantum randomness and how one might have empirical evidence for it. We will see why, depending on one's computational resources, it may be impossible to determine whether a particular notion of randomness properly characterizes one's empirical data. Indeed, we will see why even an ideal observer under ideal epistemic conditions may never have any empirical evidence whatsoever for believing that the results of one's quantum-mechanical experiments are randomly determined. This illustrates a radical sort of empirical underdetermination faced by fundamentally stochastic theories like quantum mechanics

On Martin's Pointed Tree Theorem

We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT)

Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega

Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)

Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science