An uncountable countable set
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From: Randy Poe on 18 Sep 2006 14:12 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > Mike Kelly wrote: > > > > > > > [ ... snip ... ] It's not clear to me that providing finite examples then > > > > saying "obviously this holds for infinite cases too" without any > > > > justification whatsoever should be at all convincing to anyone. > > > > > > It may be not clear to any mathematician, but it is clear to any > > > scientist. The reason is that infinities do not really exist. > > > They only exist as an attempt to make the "very large" rigorous > > > in some sense. The moment you forget this, you get into trouble. > > > > But we are discussing whether there exists a uniform distribution over > > the naturals. > > Please make just the experiment. Using what distribution? > Choose at random 30 natural numbers > from the whole set N. What is the result? OK, I'll use a distribution in which the numbers divisible by 3 occur with probability 0.5. > How many of these 30 numbers > are in fact divisible by 3? About half of them, i.e., 15. - Randy
From: MoeBlee on 18 Sep 2006 14:21 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Unbounded but finite may > >> be considered potentially, but not actually, infinite. > > > > That will be jiffy, once you give axioms and/or our definitions for > > 'potentially infinite' and 'actual infinite'. Until then, it's pure > > handwaving. > > > > MoeBlee > > > > As I said, a potentially infinite set is unbounded, but will all element > indices finite. It's "countable" in standard parlance. An actually > infinite set includes elements with infinite element indices, like 1/3 > has in the decimal reals. :) Usually, 'countable' means bijectable with w or some element of w. On the other hand, sometimes, 'countable' is used to mean bijectable with w. When I use 'countable', I mean the former definition, and I use 'denumberable' to mean bijectable with w. So your 'potentially infinite' just means denumerable, I guess. As to your 'infinite element indices', I would need a definition from you. But it is no help for you to give a definition of 'infinite element indices' if you define it with yet MORE undefined verbiage. None of your definitions matter, since you never say what your primitives are, so we just get from you a treatment that is either "POTENTIALLY" INFINITE or CIRCULAR. MoeBlee
From: MoeBlee on 18 Sep 2006 14:30 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > > > >> Can a dense set like the rationals, with an infinite number of them > >> between any two naturals, really be no greater a set than the naturals, > >> which are an infinitesimal portion of the rationals? That's just poppycock. > > > > A set is not dense onto itself. A set is dense under an ordering. And > > the set of natural numbers is dense under certain orderings. > > > > MoeBlee > > > > Not in the natural quantitative order on the real line. Yes, the standard ordering on the naturals is not a dense ordering of the naturals. But you miss the point that you don't have a given standard ordering for an arbitrary set. Sure we can say that the naturals are unlike the rationals in terms of their respective standard orderings. Set theory does recognize this. But that does not give a general characterization of cardinality since we don't have a way to compare any two arbitrary sets by standard ordering since we don't have a definition of 'standard ordering' that applies to arbitrary sets. > You cannot say > that between any two naturals is another, in quantitative terms. I > meant, obviously, dense in the quantitative ordering. But, you knew that. Your 'quantitative terms' is just more undefined T-verbiage. > So, that having been said, when there are an infinite number of > rationals for every half-open unit interval, and only one natural in > every such interval, how does it make sense that there are not > infinitely many more rationals than reals? Because your definition of 'more' depends on the standard orderings, which we agreed upon for certain sets such as that of the naturals and that of the rationals, but for which there is no definition for arbitrary sets, hence not a definition of 'more' and 'less' for arbitrary sets in that respect. > Are the extra naturals that > make up the difference squashed down towards the infinite end of the > line, where there's no rationals left? Like I said, it's poppycock. No, all you need to do is define a dense ordering of the natural numbers, which is easily done. MoeBlee
From: Randy Poe on 18 Sep 2006 14:33 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Mike Kelly schrieb: > > > > > > > > > The mathematics of > > > > > the infinite can only be derived from the mathematics of the finite > > > > > (because nobody has an idea what "the infinite" is). > > > > > > > > Don't extrapolate from yourself so harshly! > > > > > > I talked to a lot of first-rate mathematicians. This in parentheses is > > > a qoute from many of them. > > > > Is that your idea of a citation? > > It is my idea of trust and honour. I do not publish names of my private > correspondents. Nor, apparently, actual quotes. - Randy
From: MoeBlee on 18 Sep 2006 15:24
Tony Orlow wrote: > I am well aware my position is "provably false", but I am > also aware that that depends on the axioms assumed and the rules > regarding logical inference. No, your "position" is not coherent enough to be addressed as either true or false. A bunch of gibberish doesn't admit of examination for truth or falsehood. > I don't know about Virgil, so maybe he IS wasting his > time. His time at least SEEMS to me to be wasted with you. Meanwhile, I KNOW that I'm wasting my time trying to reason with you. MoeBlee |