2,713 research outputs found
A Modern Look at Social Trinitarianism
This paper attempts to show through the modern literature that Social Trinitarianism (ST) is a more plausible explanation of the Trinity than Latin Trinitarianism (LT). It will look at ST\u27s solution to Trinitarian procession and LT\u27s likeness to modalism. It will focus on essays written in response to Keith Ward’s Christ and the Cosmos and shall offer a new way to speak of the Trinity through the combining of the methodology proposed by H. E. Barber and Richard Swinburne’s view of necessity and procession
The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of , algorithmically
random continuous functions from to , and algorithmically
random Borel probability measures on , especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
is a random closed set if and only if it is the
set of zeros of a random continuous function on . As a corollary we
obtain the result that the collection of random continuous functions on
is not closed under composition. Next, we construct a computable
measure on the space of measures on such that
is a random closed set if and only if
is the support of a -random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in
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