In the early twentieth century two extremely influential research programs aimed to establish solid foundations for mathematics with the help of new formal logic. The logicism of Gottlob Frege and Bertrand Russell claimed that all mathematics can be shown to be reducible to logic. David Hilbert and his school in turn intended to demonstrate, using logical formalization, that the use of infinistic, set-theoretical methods in mathematics—viewed with suspicion by many—can never lead to finitistically meaningful but false statements and is thus safe. This came to be known as Hilbert’s program. These grand aims were shown to be impossible by applying the exact methods of logic to itself: the limitative results of Kurt Gödel, Alonzo Church, and Alan Turing in the 1930s revolutionized the whole understanding of logic and mathematics (the key papers are reprinted in [5]). Panu Raatikainen is a fellow in the Helsinki Collegium for Advanced Study and a docent of theoretical philosophy at the University of Helsinki. His e-mail address is

Abstract not found

Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a PA-degree implies strong f-randomness, but f-randomness does not imply f-randomness relative to a PA-degree. Keywords: partial randomness, effective Hausdorff dimension, Martin-Löf randomness, Kolmogorov complexity, models of arithmetic.