Courage?
in [Logic]
|
Prev: arithmetic in ZF
Next: Derivations
From: Dennis Ritchie on 19 Apr 2005 01:09 "Will Twentyman" <wtwentyman(a)read.my.sig> wrote in message news:426461f8$1_5(a)newsfeed.slurp.net... > > > Chris Menzel wrote (quoting Twentyman): ...... > > My impression some 15 years ago when I was studying set theory more > > seriously was that most working set theorists believed (G)CH to be > > false, largely because the power set operation is too "rich" in one way > > or another only to bump you to the very next infinite cardinality. I > > don't know if this is still the general view of the matter. > > > > I was careful not to say "most" because I have no idea what the > percentages might be. 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. Dennis
From: Roligtroll on 19 Apr 2005 15:38 Hi Eckard, <You>, I < 1) Why did Cantor ignore the only reasonable notion of infinity including the pertaining rules how to handle it? > If I remember right and would like to hear oppositions, is that the definition for a infinite set includes the existence of bijection between the set and a proper subset. How to argue that notion is not reasonable?
From: Ross A. Finlayson on 20 Apr 2005 03:22 Yes, and no. Yes, because, yes, that's true, and no, because that would throw out some of the good with the bad. About the inclusion into some amorphous collection of Cantor-philes, self-proclaimed defenders of the orthodoxy, calling Cantorians hypocrites was framed strongly and to some extent unfairly, because they might be ignorant. I was concerned that you might have an opposite reaction. You might notice that there is no one here saying calculus is wrong. No one here says algebra is wrong. It's obvious that no one is proclaiming number theory wrong. Arithmetic is not wrong. What we do see is that some people, with various levels of understanding, reasoning, and justification of their arguments, think that dependence on transfinite cardinals is wrong. Why would people, in this case people with deep personal interests in mathematics and logic from varied walks, become so argumentative about an area of mathematical study? Is it because there are no real-world examples or applications of what it discusses? Is it because the transfinite cardinals attempt to address infinity, with that's varied implication? Is it because... they're wrong? The set of all sets is its own powerset. Skolemize, your model is "countable." If it's infinite, there's always one more. The powerset operation on an n-set of ordinals yields in the result exactly one new ordinal, the successor and order type. Well, this is good, I'm pleased with how this discussion was progressing. Will, I'm somewhat different than you in that I attack Cantorian and Goedelian results, and ZF itself, the foundations of the status quo's mathematical logic, quite directly, because Goedel calls you a liar. I'm glad that you agree that considering alternatives to transfinite cardinals is not wrong. The universe, or domain of discourse, is infinite. Infinite sets are equivalent. Ross
From: Ross A. Finlayson on 20 Apr 2005 09:15 No, Wolfgang, the universe has infinite numbers of things. Besides the fact that there are infinitely many theoretical theoretical particles in a non-standard model, of physical particles, of particle physics, that force vector is not just a scalar, it has infinitely many components, in a variety of ways that you can look at it. When you say, "oh, that's just a method of calculating", that's kind of ridiculous. To evaluate a solution to that system, you're evaluating those things in terms of continua, things continuous. Accept that! Confront your misperceptions. Your 10^200, or however finitely many it is this week, are domains of functions among themselves and each other ad infinitum, forever, eternally, without end, non-finitely. If you really want to believe that the set of all things is finite, why don't you go find a nice patch of sand, dig a hole in it, shove your head in that, and count the grains you can see. You would be found among various others with their heads similarly stuck, for various reasons of their own. The example of functions between physical objects actually themselves being in a way physical objects, in the physical universe defined to contain all of those things, is a facile visualization of why Cantor's paradox dooms his transfinite cardinals directly via a definition of physical reality. If you really want to address things infinite, and to some extent why our physical universe is the way it is, then that leads more or less to dual representation and the ur-element, that thing in itself about being and nothing, minimal and maximal, etcetera. Ross
From: Ross A. Finlayson on 20 Apr 2005 15:43
It's the Ouroboros, the snake eats its tail, you codefine point along with the infinite dimensional dually oriented vector space, for convenience, orthogonal. In a way, then, the null axiom theory is a set, number, physical, and geometric theory, all at once, and the same thing. Basically people point to Euclid and notice that geometric points and lines are defined in terms of each other. If you abstract that (make that abstract) in a slightly more enlightened way after thousands of years of research into geometry, the point, and line, and plane, and circle and sphere, and complex plane and hyperspheres, are all defined, at a very fundamental level of geometry and as well mathematical logic and numbers and various set relations. Keep in mind, if you want a consistent theory, it can't have any paradoxes. If you want to quantify over sets in a set theory there's a universal set. There can only be one proper class, or none. The powerset operation on an n-set of ordinals introduces only one new ordinal. There is nothing, greater than Ord. Any theory with non-logical axioms is doomed by Goedel to incompleteness. The universe is infinite, and infinite sets are equivalent. Ross Finlayson |