An uncountable countable set
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From: MoeBlee on 20 Sep 2006 13:53 Mike Kelly wrote: > It isn't. It *is* the set of all finite naturals. By the way, omega is > an ordinal number, not a cardinal number. With the usual definition of 'cardinal', it is a theorem that w is a cardinal as well as an ordinal, even if it is more common to emphasize that w is a cardinal by using the notation 'aleph_0' rather than 'w' when speaking of w in context of its being a cardinal. MoeBlee
From: MoeBlee on 20 Sep 2006 14:02 Tony Orlow wrote: > Omega is successor to N No, it is not not. Why do you keep repeating your error? > >> and any ordinal being > >> the set of all preceding naturals, omega is the set of all preceding > >> naturals. It's larger than all naturals. DO you disagree with that > >> simple statement? > > > > It contains terms whose meaning we do not agree on. > > > > What is the test for "largeness"? > > Number of successions, in this case, resulting in the value. A larger > number of successive increments produces a larger value. w is larger than any natural number where 'larger' is taken in the sense of the standard ordinal ordering. But w is not larger in the sense you just mentioned, since w is not any number of successions away from any natural number. > Can you really not say that an infinite count is greater than any finite > count? That seems to be the basis for omega. What does "basis for" mean? 'w' has a definition. That definition is not that of "having infinite count greater than any finite count". > There are a number of ways to do that. "Larger" can mean "farther from > 0", geometrically. So from undefined T-terminology we move on to yet more undefined T-terminology. MoeBlee
From: imaginatorium on 20 Sep 2006 14:18 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>>>> mueckenh(a)rz.fh-augsburg.de wrote: > >>>>>> Mike Kelly schrieb: > >>>>>> > >>>>>>> mueckenh(a)rz.fh-augsburg.de wrote: <snip> > >>>>>>>> Don't misunderstand me: I do not oppose the principle of induction but > >>>>>>>> the phrase "there exists" which suggests the existence of the completed > >>>>>>>> set. > >>>>>>> Why do you object to this? > >>>>>> Because of he proof above. > >>>>> It is not a proof. Division is not defined where either operand is an > >>>>> infinite cardinal number. > >>>>> > >>>> If omega is the successor to the set of all finite naturals, > >>> It isn't. What does "successor to N" even mean? > >> Ask von Neumann. > > > > I consider that a crackpot response. First the > > crackpot misconstrues some paper they're reading > > (by Cantor or Einstein, say), then when pressed > > they respond "are you arguing with Cantor/Einstein?" > > > > I'm not asking von Neumann, I'm asking you. You > > used a phrase. Did you have any idea what you > > meant by that phrase? > > > >> It is the set of all naturals, > > > > N is the successor to N? > > Omega is successor to N, and the size of N, in the von Neumann system. Uh, this is you, ignorant crank, informing us about "the von Neumann system". Well, you may consider yourself a world-class expert in "the von Neumann system", and John vH isn't around to argue with you himself. Are we supposed to take your word on this? Is there any chance you might cite a textbook so we can check? You haven't read a textbook? No, I don't suppose you have. <much foaming at the mouth removed> Really, what an idiot. Brian Chandler http://imaginatorium.org
From: Mike Kelly on 20 Sep 2006 14:31 Tony Orlow wrote: > Mike Kelly wrote: > > Han de Bruijn wrote: > >> Mike Kelly wrote: > >> > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>> Mike Kelly wrote: > >>>> > >>>>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>>>> to schoolkids. > >>>> Look who is talking. Good luck explaining alpha_0 to schoolkids. > >>> Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > >>> this is a bad analogy, because school kids don't need to know about > >>> cardinals but they do need to know how to work with natural numbers. My > >>> point, if you really missed it, was that Tony's ideas of "infinite > >>> natural numbers" don't match up to our "naive" or "intuitive" idea of > >>> what numbers should be - how we were taught to do arithmetic in school. > >>> I for one don't understand what the hell an "infinite natural number" > >>> is. And yet supposedly the advantage of his ideas are that they're more > >>> intuitive than a standard formal treatment. > >> My point is that the pot is telling the kettle that it's black (: de pot > >> verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than > >> Tony's "infinite natural number". > > > > Your analogy is terrible, as usual. > > > > My point was that Tony's "infinite natural numbers" are not compliant > > with everyday arithmetic. Aleph_0 is part of a formalisation that leads > > to an arithmetic that works exactly as we expect it to. > > > > Oh? For what finite x is x-1=x? None. This is the case in set theory, too. -- mike
From: Tony Orlow on 20 Sep 2006 14:33
MoeBlee wrote: > Tony Orlow wrote: >> >> Actually infinite(S) <-> E seS A neN index(s)>n >> >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > > So this is what you have now ('w' stands for 'N', which stands for?): Where do you see 'w'? > > S is actually infinite <-> Es(seS & An(new -> index(s)>n)) I would rather not use 'w'. Sick to N. Why not? > > S is potentially infinite <-> As(seS -> (index(s)ew & An(new -> Es(seS > & index(s) = n)))) > > Okay, now I see that your 'actually infinite' is something like what > set theory would describe as 'S has a member greater than any member of > w'; and your 'potentially infinite' is what set theory would describe > as 'S has only finite members but S has a denumerable number of finite > members'. Okay. > Neither of those correspond to the usual senses of 'actually > infinite' and 'potentially infinite'. In what way? Ask Wolfgang and Han whether they think so. Morevover, if you are going to > allow the existence of a T-potentially infinite set (I'm going to use > 'T' since your definitions don't correspond to usual senses) and just > the basic set operations, then you are still going to have sets > bijectable with proper subsets of themselves. And, you still will not > have contradicted that an unbounded set must be infinite ('infinite' > given the usual definition). No, but it will distinguish between potentially infinite sets like N and actually infinite sets like R, which can have no algebraic relation. That's because potentially infinite sets never are actually infinite, but really finite but unbounded. > >>> Next, what is the logistic system? >> First order logic > > Okay, classical first order logic with identity, I take it. I wasn't quantifying (yet) over sets or properties. :) > >> What are the non-logical primitives? >> e >> > >> index > > Good. > >>> What are the non-logical axioms? >> These are definitions of potential vs. actual infinity. Consider them >> axiomatically stated if you wish. > > They're definitions now that you fixed them. But now you still have > 'index' and 'w' to define. And you need more axioms, so that 'e' and > 'index' can work for you. Well, I define N as the set of all sizes of sets, if you will, which do not allow injections into themselves. That's a good definition of "finite", which I can accept from the standard treatment. The axiom of extensionality is one axiom I can accept from standard theory. Then there are others relating size of the set to the elements, such as subset(a,b) -> smaller(|a|,|b|), which do not work with standard theory. Yes, I have to get back to my axioms. concentration's been domestically distracted lately. This is therapy. :) > >> What is the definition of 'index'? >> >> The count of elements up to and including a given element. The number of >> steps between and element and the first element in a linear order, plus 1. > > Now you just need to define 'count', 'elements', 'up to and 'including' > or 'number of steps', 'between', 'first', 'linear order', 'plus', and > '1'. "Element" is a primitive, defined by the axioms of raw set theory. Similarly, most of those "definitions" (and I understand you can't define *everything* - just ask Virgil) follow from the rules regarding those relations. > > MoeBlee > ToeKnee |