Courage?
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From: Ross A. Finlayson on 24 Apr 2005 14:30 I wrote that, and now I have to justify to myself why I think that's true, or disavow it. Perhaps it has to do with the fact that rationality, in terms of logic, is what I use to explain my chosen responses to life's little stimuli, in an overall sense. For example, when I'm driving down the road, I pick a lane and use my turn signals and don't go plowing through mailboxes, because that would be much more trouble than it's worth. Where thousands of years ago pretty much all natural phenomena were ascribed to spirits, a higher power, or the Godhead, today almost none of those things are. While that is so, you might be an atheist, in developing a logical theory, one aspect of its robustness is to imagine if a higher power came manifest, what would happen. If God made light dark and gravity repulsion, basically in the paraconsistent sense, could he alter rules of inference? I guess one notion of the null axiom theory is that it's resistant to meddling, it goes around and around, because at its root is the inconsistency and hopefully the root probabilistic flaw, its opposite is there true. Its opposite, the opposite of rationale and inference, is the same thing, rationale and inference. So, to get to talking basically about the consequences of divinity or a higher power, and basically a higher power on all high, a highest power, we get to consequences of the higher power being neither provable, nor disprovable, in terms of mathematical logic. That is to be expected, theologians have been trying to explain divinity's impact on reason for a long time. In an otherwise very sterile or clinical sense, it's convenient to assume atheism in developing mathematical logic, for God not to put mathematical rocks he (neutral) can't move in front of you. Many religions have a universalist notion where basically the divine is represented in all things, it's the Alpha, and the Omega, those are just the first and last letters of the Greek alphabet, and trivially is a statement that zero equals infinity, that's a true statement of sorts in the null axiom theory. That's different from some of the other more technical notions that in terms of a mathematical logic are used to both explain things and provide an avenue of discourse free from inconsistency. It's counterproductive, often, to introduce divinity into conversation about mathematical logic. Calling each of us morons is basically wrong. With the conceit and basically pride in ability to explain some trivia of the foundations of mathematical logic, I introduce divinity into the conversation to reduce the value of others' participation, and perhaps as well my own, because I'm greedy and think I'm on to something. The irrational is thrown in as a monkeywrench, a spanner, in that way my sabotage, which might be wrong, incorrect, or a bad idea, prevents others from more immediately sharing. That's damn rude! I actually believe that. While in terms of mathematical logic there are some technical concepts of, for example, a null axiom or axiom-free theory, that are readily communicable, and that I can readily rationalize to myself, something that by its very definition is irrational, entertained and yes, entertaining, should be guarded against for it's delusional. There's some power in that. There can be, only one, theory. Now, that's said for reasons of these technical philosophy notions. If you quantify over sets, there's an implicit universal set, that's basically the reason why. Also it's a nod to the guy from Highlander going "There can be, only one." There can be theory. There can be, only one, theory. Ross
From: Keith Ramsay on 28 Apr 2005 02:04 Dennis Ritchie wrote: [...] |In his 1966 monograph about his independence result for CH, |Cohen remarked in the conclusion: | | "A point of view which the author feels may eventually come to | be accepted is that CH is *obviously* false. The main reason one | accepts the Axiom of Infinity is that we feel it absurd to think | that the process of adding only one set at a time can exhaust | the universe.... The set C is is, in contrast, generated by a totally | new and more powerful principle, namely the Power Set Axiom.... | Thus C is greater than aleph_n, aleph_omega, aleph_alpha where | alpha = aleph_omega, etc. This point of view regards C as | an incredibly rich set given us by one bold new axiom, which | can never be reached by any piecemeal process of construction...." | |I don't know the consensus about this, nor indeed whether Cohen |still believes it. No. I've seen a more recent paper in which he remarked that he had come to regard it as not having a determinate answer. It still doesn't seem like a consensus has emerged. Woodin has some recent arguments that apparently "motivate" taking CH as false. Whether one agrees that this is a good way to go, and whether one believes that CH is objectively false, are not exactly the same issue. Probably most of us who think that the cumulative hierarchy is well enough defined to preclude CH being simply neither true nor false also think that if we have a good idea of what the truth of the matter is, it's also a good idea to develop the correct theory of the cumulative hierarchy. But someone who doesn't think there's an objective answer still might conclude based on this kind of argumentation that mathematics will be better furthered by spending time studying the kind of model Woodin has in mind than studying models satisfying CH. Keith Ramsay
From: Ross A. Finlayson on 28 Apr 2005 06:05
Where does that lead you, Keith? Is that, how shall we say, handwaving? Basically I'm looking at the cumulative hierarchy as zero to omega. Correspondingly, it's also zero to omega and omega to omega plus omega, the integers. It is the natural integers, and the integers. Where that is so, what you may call the transfinite cardinal aleph_n is omega + n. This is where the satisfactory function f(x) = x+1 leads to that for the natural integers as ordinals that the S of Cantor's powerset result is empty, and the null set, and dually represented, as a resolution to a variety of paradoxes including Cantor's and Burali-Forti's. Consider for example the unary representation of numbers: base one. For the integers, the base one representation is a sequence of tally marks, for which the Romans instigated a shorthand, eg V for five, X for ten, etcetera. On the left side of the radix, in conventional Arabic notation, the integer is simply that many tlly marks, for however many the integer is. On the right side of the radix, there is a somewhat different representation than you might expect. There can only be a sequence of tally marks, and they represent the integral iota multiples between zero and one. Another notion is to consider the infinite base representation of a (real) number. On the left of the radix, there is only one digit, the infinite base implies an infinite alphabet of symbols. It goes from zero to infinity in one place. On the right, again, there is an expression of integral iota-multiples, as might be expected where on the left of the radix, or in decimal the decimal point, there is the integer part of the real number, and on the right, the unit "fraction" part, where the denominator in this case is the scalar infinity. There are only real numbers between zero and one. In currently non-standard models of the real numbers, there are infinitesimals between zero and one. What does that directly imply? Ross |