Hi everyone! My name is Abram Demski. My interest, when it comes to this list, is: what is the correct logic, the logic that can refer to (and reason about) any mathematical structure? The logic that can define everything definable? If every possible universe exists, then of course we've got to ask which universes are possible. As someone mentioned recently, a sensible approach is to take the logically consistent ones. So, I'm asking: in what logic?
I am also interested in issues of specifying a probability distribution over these probabilities, and what such a probability distribution really means. Again there was some recent discussion on this... I was very tempted to comment, but I wanted to lurk a while to get the idea of the group before posting my join post. Following is my view on what the big questions are when it comes to specifying the correct logic. The first two big puzzles are presented to us by Godel's incompleteness theorem and Tarski's undefinability theorem. The way I see it, Godel's theorem presents a "little" puzzle, which points us in the direction of the "big" puzzle presented by Tarski's theorem. http://en.wikipedia.org/wiki/Godel%27s_theorem http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem The little puzzle is this: Godel's theorem tells us that any sufficiently strong logic does not have a complete set of deduction rules; the axioms will fail to capture all truths about the logical entities we're trying to talk about. But if these entities cannot be completely axiomized, then in what sense are they well-defined? How is logic logical, if it is doomed to be incompletely specified? One way out here is to say that numbers (which happen to be the logical entities that Godel showed were doomed to incompleteness, though of course the incompleteness theorem has since been generalized to other domains) really are incompletely specified: the axioms are incomplete in that they fail to prove every sentence about numbers either true or false, but they are complete in that the ones they miss are in some sense actually not specified by our notion of number. I don't like this answer, because it is equivalent to saying that the halting problem really has no answer in the cases where it is undecidable. http://en.wikipedia.org/wiki/Halting_problem Instead, I prefer to say that while decidable facts correspond to finite computations, undecidable facts simply correspond to infinite computations; so, there is still a well-defined procedure for deciding them, it simply takes too long for us to complete. For the case of number theory, this can be formalized with the arithmetical hierarchy: http://en.wikipedia.org/wiki/Arithmetical_hierarchy Essentially, each new quantifier amounts to a potentially infinite number of cases we need to check. There are similar hierarchies for broader domains: http://en.wikipedia.org/wiki/Hyperarithmetical_hierarchy http://en.wikipedia.org/wiki/Analytical_hierarchy http://en.wikipedia.org/wiki/Projective_hierarchy This brings us to the "big" puzzle. To specify the logic an refer to any structure I want, I need to define the largest of these hierarchies: a hierarchy that includes all truths of mathematics. Unfortunately, Tarski's undefinability theorem presents a roadblock to this project: If I can use logic L to define a hierarchy H, then H will necessarily fail to include all truths of L. To describe the hierarchy of truths for L, I will always need a more powerful language L+1. Tarski proved this under some broad assumptions; since Tarski's theorem completely blocks my project, it appears I need to examine these assumptions and reject some of them. I am, of course, not the first to pursue such a goal. There is an abundant literature on theories of truth. From what I've seen, the important potential solutions are Kripke's fixed-points, revision theories, and paraconsistent theories: http://en.wikipedia.org/wiki/Saul_Kripke#Truth http://plato.stanford.edu/entries/truth-revision/ http://en.wikipedia.org/wiki/Paraconsistent_logic All of these solutions create reference gaps: they define a language L that can talk about all of its truths, and therefore could construct its own hierarchy in one sense, but in addition to simple true and false more complicated truth-states are admitted that the language cannot properly refer to. For Kripke's theory, we are unable to talk about the sentences that are neither-true-nor-false. For revision theories, we are unable to talk about which sentences have unstable truth values or multiple stable truth values. In paraconsistent logic, we are able to refer to sentences that are both-true-and-false, but we can't state within the language that a statement is *only* true or *only* false (to my knowledge; paraconsistent theory is not my strong suit). So using these three theories, if we want a hierarchy that defines all the truth value *combinations* within L, we're still out of luck. As I said, I'm also interested in the notion of probability. I disagree with Solomonoff's universal distribution (http://en.wikipedia.org/wiki/Ray_Solomonoff), because it assumes that the universe is computable. I cannot say whether the universe we actually live in is computable or not; however, I argue that, regardless, an uncomputable universe is at least conceivable, even if it has a low credibility. So, a universal probability distribution should include that possibility. I also want to know exactly what it means to measure a probability. I think use of subjective probabilities is OK; a probability can reflect a state of belief. But, I think the reason that this is an effective way of reasoning is because these subjective probabilities tend to converge to the "true" probabilities as we gain experience. It seems to me that this "true probability" needs to be a frequency. It also seems to me that this would be meaningful even in universes that actually happened to have totally deterministic physics-- so by a "true probability" I don't mean to imply a physically random outcome, though I don't mean to rule it out either (like uncomputable universes, I think it should be admitted as possible). Well, I think that is about it. For now. --Abram Demski --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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