Uncountability of the Rationals?
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From: MoeBlee on 12 Jun 2010 13:50 On Jun 11, 5:17 pm, Virgil <Vir...(a)home.esc> wrote: > John Von Neumann, whom I once met Please tell us more about that. MoeBlee
From: David R Tribble on 12 Jun 2010 14:03 David R Tribble writes: >> The funny thing about this is that this does not seem to matter >> in the least. Tony keeps saying that IFR solves all sorts of >> problems and "obfuscations" with Cantor's cardinalities, yet >> he hasn't given a single concrete example of it. > Tony Orlow wrote: > So, you can't even remember what it stands for or what it means, but > you are sure it's incosistent. Well, I've never tried eating hippo > meat, but I don't like it. I just know I don't like it because it's > not a hamburger. I mean, come on. If you don't understand it how can > you criticize it? I do understand it; it is simply not that complicated. I don't recall exactly what the acronym stands for, but so what? A bad idea is still a bad idea, no matter what you call it. All that your IFR really accomplishes is to establish the limiting ratio between two real number series. That's all it is. That ratio can be applied to ordered sets built from those series, but ultimately, IFR is only about the limit of the ratio between finite series (plain old sequences included). By nothing resembling coherent logic have you been able to demonstrate that this limit of real functions applies to infinite sets, or even to function on non-finite domains. Sadly, it's just that simple.
From: David R Tribble on 12 Jun 2010 14:05 Tony Orlow wrote: > By the way, the statetment that "he hasn't given a > single concrete example of it" is not merely disingenuous, but an > outright lie, and you know it. "sqrt(BigUn)" is not a concrete example. You have yet, after all these years, to provide a workably consistent definition of BigUn arithmetic. You can keep claiming you have, and that we're simply too stupid to understand it, but it doesn't change the reality that you're just trying to make a "unit infinity" act like a real number, and that you have simply failed to contrive a coherent self-consistent system to do it. Sorry, but that's the truth.
From: David R Tribble on 12 Jun 2010 15:35 Tony Orlow wrote: >> I would like you consider a slightly modified goal: To exactly order >> sets according to size, and to distinguish between those that have an >> absolute size and those that do not. For me, no countably infinite set >> has an absolute size, but only one relative to other countably >> infinite sets, especially the standard N, starting at 0. > David R Tribble wrote: >> Wouldn't it be easier to say that some chosen countably infinite >> set has a standard "absolute" size, and then that all other such >> sets have a size that is some fraction of that size? Thus every >> set then has an "absolute" size, based on the standard set size. > Tony Orlow wrote: > That might be easier, and would lead, as suggested, to natural > density. However, that doesn't address the relations between infinite > sets that have other ratios or offsets besides the finite. It is much > more fruitful to look for a unit infinity based on uncountable > infinity, which can be much more easily combined with measure and > topology. So having an "absolute" set size for countable sets is not a good idea? You'd rather just say that non-finite countable sets don't have an absolute size at all and leave it at that? Why go through the trouble of dealing with uncountable sets but just write off the simpler domain of countable sets? David R Tribble wrote: >> For convenience, you'd want to choose the "largest" countably >> infinite set for the standard "absolute" set size. This would >> probably be N, since it contains all of the countable naturals. >> (Another logical choice would be Z.) Given this, you could then >> say that every countably infinite set does indeed have an >> "absolute" set size. > Tony Orlow wrote: > But, for me, they do not. If N starts at the first location with 1, > and then the second with 2, etc, then the nth natural is n, and unless > a set has n elements, there is no nth one. If there are omega > naturals, then there is an omegath, which by the order=value > definition of that set must equal omega. What makes you think that? Statements like that make it look like you don't really understand what terms like "omega" mean. > However omega cannot exist within the set. Consider this. > Are the following two statements logically equivalent? > > AneN n<omega > > The first says there is no ntaural that can be the size of the entire > set of naturals. That's true. Yes. > ~EneN n>=omega > > The second says that there is no n in N that is greater than or equal > to this "size". Yes. > Does either imply, actually, that such a size exists? No. Both assert > that no natural number will suffice. However, this is not to say that > there exists ANY number which does. Logically, you must concede, > neither statement proves that omega exists as a number. Yes, that's correct. Neither of your two statements implies that there exists any number equal to card(N). Neither statement implies card(N) does not exist, either. > I am not > obligated to imagine such a number, or give it any credibility. So you're saying that if the two statements you provided do not imply that card(N) exists, that proves that card(N) does not exist? That's an extremely weak argument to make. Have you considered any of the other statements you *could* have made but didn't, that maybe one of them might imply the existence of card(N)? After all, your statements simply state what does *not* exist, but nothing about what *could* exist. How is you argument any different than: 1. For all x, f(x) < z; 2. There does not exist x such that f(x) >= z; Therefore z does not exist. > When > it comes to the countably infinite, we can only look at this as some > kind of limit, and for most discriminating results, consider them as > Big-O measures or something similar. But then that's sort of what omega (or Aleph_0) is, it being a kind of (lower) limit on infinite set sizes.
From: David R Tribble on 12 Jun 2010 15:42
Jesse F. Hughes wrote: >> You [Tony] sometimes say that you >> don't think omega is the smallest infinite ordinal, but I'm not sure >> what you mean when you say that. > Tony Orlow wrote: > I don't use the word "ordinal" in my arguments (sure, go find one > mention from 1996 or whatever). I say there are a wide spectrum of > countable and uncountable infiniites, given the right techniques. What Tony means is that size(E) < size(N) for the sets N and E = {0,2,4,6,...}. Specifically, he means that size(E) = size(N)/2, where "size(S)" is his personal definition of "set size". He also allows for things like sqrt(size(N)) and log(size(R)). These "values" of his are "different infinities", all based on his idea of a "unit infinity". |