Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in l (or d-l) for any measure l, which is particularly appropriate for physics and in some ways more intuitive than Ko's (1991) recursive approximability (r.a.). For Lebesgue measure k, d-k implies r.a. Sets with positive kmeasure that are sufficiently "riddled" with holes are never d-k but are often r.a. This explicates Sommerer and Ott's (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-k.
Abstract. Gödel argued that Cantor's notion of cardinal number was uniquely correct. More recent work has defended alternative "Euclidean" theories of set size, in which Cantor's Principle (two sets have the same size if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). Here we see from simple examples, not that Euclidean theories of set size are wrong, nor merely that they are counterintuitive, but that they must be either very weak or in large part arbitrary and misleading. This limits their epistemic usefulness.
You are free, therefore choose-that is to say, invent. Sartre, L'existentialisme est un humanismePhilosophy, and especially metaphysics, has often been attacked on either epistemic or semantic grounds. Anything outside of experience and the laws of logic is said to be unknowable, and according to Wittgenstein and the logical positivists, there are no such things to know; metaphysical disputes are either meaningless or merely verbal. This was thought to explain philosophy's supposed lack of progress: philosophers argue endlessly and fruitlessly precisely because they are not really saying anything about matters of fact (Wittgenstein 1953, Remark 402;Carnap 1950).Since the mid-twentieth century, the tide has been against such views, and metaphysics has re-established itself within the analytic tradition. Ontology, essentialism, and de re necessity have regained credibility in many eyes and are often investigated by excavating intuitions of obscure origin. Relatedly, externalist semantic theories have claimed that meaning or reference has a secret life of its own, largely unfettered by our understanding and intentions (Kripke 1971; 1972; Putnam 1973;1975a). 'Water,' it is claimed, would denote H 2 O even if we had never discovered that particular molecular structure, and this is allied with the
A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2018) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson's (2007) Bisomorphic^events are not in fact isomorphic, but Howson is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson's physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotationinvariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances.
In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.
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