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Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
A generalization of a theorem of Hurewicz for quasi-Polish spaces
We identify four countable topological spaces , , , and
which serve as canonical examples of topological spaces which fail to be
quasi-Polish. These four spaces respectively correspond to the , ,
, and -separation axioms. is the space of rationals, is
the natural numbers with the cofinite topology, is an infinite chain
without a top element, and is the set of finite sequences of natural
numbers with the lower topology induced by the prefix ordering. Our main result
is a generalization of Hurewicz's theorem showing that a co-analytic subset of
a quasi-Polish space is either quasi-Polish or else contains a countable
-subset homeomorphic to one of these four spaces
Quasi-Polish Spaces
We investigate some basic descriptive set theory for countably based
completely quasi-metrizable topological spaces, which we refer to as
quasi-Polish spaces. These spaces naturally generalize much of the classical
descriptive set theory of Polish spaces to the non-Hausdorff setting. We show
that a subspace of a quasi-Polish space is quasi-Polish if and only if it is
level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized
within the framework of Type-2 Theory of Effectivity as precisely the countably
based spaces that have an admissible representation with a Polish domain. They
can also be characterized domain theoretically as precisely the spaces that are
homeomorphic to the subspace of all non-compact elements of an
\omega-continuous domain. Every countably based locally compact sober space is
quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A
metrizable space is quasi-Polish if and only if it is Polish. We show that the
Borel hierarchy on an uncountable quasi-Polish space does not collapse, and
that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces
Bion Theory: an answer to the question Why is there Something rather than Nothing?
Why is there something rather than nothing? This paper explores one particular argument in favor of the answer that 'the existence of nothing' would amount to a logical contradiction. This argument consists of positing the existence of a novel entity, called a bion, of which all contingent things can be composed yet itself is non-contingent. First an overview of historical attempts to compile a systematic and exhaustive list of answers to the question is presented as context. Then follows an analysis of how the antropic principle would manifest itself in a world that consists of information and at the same time conforms to modal realism. Next, a thought experiment introduces bions as the foundation of such a world, showing how under these circumstances the ultimate origin of all existing things would be explained. The non-contingent nature of bions themselves is subsequently argued via a discussion of the principle of non-contradiction. Finally, this theory centered on the existence of bions is integrated into the worldview of Popperian metaphysics. According to the latter's criteria, I conclude that bion theory provides an integral answer to why there is something rather than nothing
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