40,894 research outputs found
Is the dream solution to the continuum hypothesis attainable?
The dream solution of the continuum hypothesis (CH) would be a solution by
which we settle the continuum hypothesis on the basis of a newly discovered
fundamental principle of set theory, a missing axiom, widely regarded as true.
Such a dream solution would indeed be a solution, since we would all accept the
new axiom along with its consequences. In this article, however, I argue that
such a dream solution to CH is unattainable.
The article is adapted from and expands upon material in my article, "The
set-theoretic multiverse", to appear in the Review of Symbolic Logic (see
arXiv:1108.4223).Comment: This article is based upon an argument I gave during the course of a
three-lecture tutorial on set-theoretic geology at the summer school "Set
Theory and Higher-Order Logic: Foundational Issues and Mathematical
Developments", at the University of London, Birkbeck in August 201
Lecture Notes on Formal Program Development
This document was originally produced as lecture notes for the MSc and PG course ``Formal Program Development'' early in 1997. After some initial general considerations on this subject the paper focusses on the way one can use Extended ML (EML) for formal program development, which features EML contains and why, and which pitfalls one has to avoid when formally developing ML programs. Usage, features, and pitfalls are all presented through examples
The cost of information
We develop an axiomatic theory of information acquisition that captures the
idea of constant marginal costs in information production: the cost of
generating two independent signals is the sum of their costs, and generating a
signal with probability half costs half its original cost. Together with a
monotonicity and a continuity conditions, these axioms determine the cost of a
signal up to a vector of parameters. These parameters have a clear economic
interpretation and determine the difficulty of distinguishing states. We argue
that this cost function is a versatile modeling tool that leads to more
realistic predictions than mutual information.Comment: 52 pages, 4 figure
Rethinking the Discount Factor in Reinforcement Learning: A Decision Theoretic Approach
Reinforcement learning (RL) agents have traditionally been tasked with
maximizing the value function of a Markov decision process (MDP), either in
continuous settings, with fixed discount factor , or in episodic
settings, with . While this has proven effective for specific tasks
with well-defined objectives (e.g., games), it has never been established that
fixed discounting is suitable for general purpose use (e.g., as a model of
human preferences). This paper characterizes rationality in sequential decision
making using a set of seven axioms and arrives at a form of discounting that
generalizes traditional fixed discounting. In particular, our framework admits
a state-action dependent "discount" factor that is not constrained to be less
than 1, so long as there is eventual long run discounting. Although this
broadens the range of possible preference structures in continuous settings, we
show that there exists a unique "optimizing MDP" with fixed whose
optimal value function matches the true utility of the optimal policy, and we
quantify the difference between value and utility for suboptimal policies. Our
work can be seen as providing a normative justification for (a slight
generalization of) Martha White's RL task formalism (2017) and other recent
departures from the traditional RL, and is relevant to task specification in
RL, inverse RL and preference-based RL.Comment: 8 pages + 1 page supplement. In proceedings of AAAI 2019. Slides,
poster and bibtex available at
https://silviupitis.com/#rethinking-the-discount-factor-in-reinforcement-learning-a-decision-theoretic-approac
Forcing consequences of PFA together with the continuum large
We develop a new method for building forcing iterations with symmetric
systems of structures as side conditions. Using our method we prove that the
forcing axiom for the class of all the small finitely proper posets is
compatible with a large continuum.Comment: 35 page
Relations between some cardinals in the absence of the Axiom of Choice
If we assume the axiom of choice, then every two cardinal numbers are
comparable. In the absence of the axiom of choice, this is no longer so. For a
few cardinalities related to an arbitrary infinite set, we will give all the
possible relationships between them, where possible means that the relationship
is consistent with the axioms of set theory. Further we investigate the
relationships between some other cardinal numbers in specific permutation
models and give some results provable without using the axiom of choice
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