An uncountable countable set
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From: Lester Zick on 3 Oct 2006 11:59 On 2 Oct 2006 21:28:30 -0700, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: >> On Mon, 2 Oct 2006 16:43:15 +0000 (UTC), stephen(a)nomail.com wrote: >> >> >Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >> stephen(a)nomail.com wrote: >> >> [. . .] >> >> >So one wonders what criteria you used to determine that >> >this infinity cannot be approached via limits. >> >> Which kinda takes us back to the definition of infinity wouldn't you >> say? If infinity can be approached via limits infinity would have to >> refer to the number of infinitesimals and if not infinity couldn't be >> approached via limits. > >Oh Lester, more poetry. ... pure poetry. Zen is when the truth lies beyond the ken of men and escapes reckon. Haiku anyone? >How will you respond to this post, I wonder? As to all posts with truth, justice, and the American way not to mention occasional humor. Last night I found myself in a reverie regarding the size of specific infinitesimals. They turn out to be of no particular size; some are certainly small and some even quite large. Rather it is the process of subdivision of which infinitesimals are the product that is endless. So I suspect we can more appropriately define such processes as unbound or limitless instead of their products which are themselves both bound and limited. Which ties up the problem of infinities both limited and unlimited quite nicely I think. ~v~~
From: Lester Zick on 3 Oct 2006 12:12 On Tue, 03 Oct 2006 01:51:10 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Virgil wrote: >> In article <45210227(a)news2.lightlink.com>, [. . .] >In the sense that the real world is continuous, and you don't just have >these beginnings with nothing before them. You don't? Then how is it we have both place and time? > The real line is a line, with >each point touching two others. What "real" line? There is no such thing as any one real number line and certainly no line whatsoever with each point touching two others. Points don't touch any more than zero touches one or one touches two. >Even the birth of our "universe", our spacetime bubble, had a cause >which preceded it, despite time having "begun", for all intents and >purposes as far as physics is concerned, with the Big Bang. Well, Tony, here you're just confessing your faith in causative epistemology without any probitive justification whatsoever. >Unfortunately, with all this talk, I probably just gave you fodder for >your world-view. But, the birth of the universe is not due to a >transition from finite to infinite, but is merely an intersection >between two existing entities in their own expansions. Dialectical mumbo jumbo, Tony. Whatever birth of the universe may or may not represent it certainly represents something more definite than the ken of zen. ~v~~
From: Lester Zick on 3 Oct 2006 12:24 On Tue, 03 Oct 2006 00:37:30 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <4521fa43(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: [. . .] >> In the sense that the real world is continuous > >We do not know that the real world is "continuous". Especially when we don't know what "continuous" means. >> The real line is a line, with each point touching two others. > >In a mathematical "real: line, no points can "touch" since there is >always others (uncountably many others) between them. In which case the "real" world cannot be continuous if there is no alternative basis of definition for "real" things than SOAP's. In which case SOAP math cannot be about the real world or real things which pretty much conforms to the conventional isomorphic grasp of modern math. ~v~~
From: Lester Zick on 3 Oct 2006 12:32 On Tue, 03 Oct 2006 09:42:46 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Dik T. Winter wrote: > >> In article <1159726829.184763.303470(a)i3g2000cwc.googlegroups.com> >> Han.deBruijn(a)DTO.TUDelft.NL writes: >> > stephen(a)nomail.com schreef: >> > >> > > I suppose I should clarify this. You can approach the infinite >> > > using the the limit concept, but you always have to be careful >> > > when using limits, and you have to be precise about what you >> > > mean by the limit. >> > >> > Okay. But the point is whether there exist infinities that can _not_ >> > be approached using the limit concept. >> >> Yes. In ordinals they are called the "limit ordinals". The name is not >> very appropriate, I think, because they are the ordinals that defy the >> limit concept, but that is just naming. >> >> > Obviously they exist, because >> > how can we approach e.g. the Continuum Hypothesis by employing limts? >> >> I have no idea about the meaning of this statement, but off-hand I ould >> say that there is no way. > >Now we are getting somewhere. In applicable mathematics, approaching the >infinite via the limit concept is the _only_ viable way. All the rest is >nonsense (= has no "sense": can never be measured with a sensor). But if the limit concept is the only sensible way to approach the infinite, one is left to ponder whether the limit concept is infinite and if not how a limits to the limit concept can be approached? ~v~~
From: imaginatorium on 3 Oct 2006 13:13
Lester Zick wrote: <snip lesser maundering...> > But if the limit concept is the only sensible way to approach the > infinite, one is left to ponder whether the limit concept is infinite > and if not how a limits to the limit concept can be approached? Beautifully put, Lester, but surely the only way would be by regression? (Try interplanetary travel, Lester - the sky's the limit there!) Brian Chander http://imaginatorium.org |