An uncountable countable set
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From: David Marcus on 27 Oct 2006 20:00 Virgil wrote: > In article <4542888c(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > David Marcus wrote: > > > Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > > > (The vertical lines denote "cardinality".) > > > > Um, before I answer that question, I think you need to define what you > > mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > > you define '-' on these "numbers"? > > I suspect that David means by IN - OUT the set difference defined as > {x in IN: not x in OUT} > > In this NG, this set difference is sometimes indicated with the > backslash, as "IN \ OUT". I sometimes use that notation, too! -- David Marcus
From: Tony Orlow on 27 Oct 2006 20:01 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: > > <snip> > >>> A poet would say that "A rose is still a rose by any other name"; a >>> mathematician would say that "By 'a rose' we mean a repesentative of an >>> equivalence class of those herbacious plants having the following >>> properties: thorns, leaves found on alternating sides of the stem; >>> flowers having a a sweet smell, vaselike growth pattern, ... From this, >>> we can deduce that the assertion of the heavy metal ballad, 'Every rose >>> has its thorn', logically follows." >>> >>> Sometimes these different modes of thinking overlap; but more often, >>> they lead to different conclusions about what is or isn't the state of >>> affairs. >> Very true, but like the Zen archer, we have to train our intuitions, and >> when they are in harmony with the universe, the arrow hits its mark.:) >> > > In physics those intuitions are the ones which accord with "the > universe" of real world measurements. In mathematics, those intuitions > are the ones which are instead in accord with "the universe" of logical > conclusions from agreed upon premises. > > The archers are aiming at different targets; so they develop different > intuitions. > > Cheers - Chas > That's very poetic, and as such, I don't find it mathematically compelling. The universe is the product of numbers. :) TOEknee
From: David Marcus on 27 Oct 2006 20:04 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > > >> Eat me. Do you maintain that the two theories are compatible with each > >> other? Is there, and also not, a smallest infinity. > > > > They're not in conflict, becuase 'smallest infinite' means something > > DIFFERENT in the different contexts. How many times will I say that > > while you STILL refuse to hear it? > > So, either smallest has two meanings, or infinite has tow meanings, or > both. Would you like to elucidate the matter by enumerating the various > definitions of "small" and "infinite"? A table might be nice... As many have said, "infinite" has many meanings. I'm afraid it isn't practical to produce a table. -- David Marcus
From: David Marcus on 27 Oct 2006 20:06 Tony Orlow wrote: > Lester Zick wrote: > > On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > > > >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Your question "Is there a smallest infinite number?" lacks context. You > >>>>> need to state what "numbers" you are considering. Lots of things can be > >>>>> constructed/defined that people refer to as "numbers". However, these > >>>>> "numbers" differ in many details. If you assume that all subjects that > >>>>> use the word "number" are talking about the same thing, then it is > >>>>> hardly surprising that you would become confused. > >>>> I don't consider transfinite "numbers" to be real numbers at all. I'm > >>>> not interested in that nonsense, to be honest. I see it as a dead end. > >>>> > >>>> If there is a definition for "number" in general, and for "infinite", > >>>> then there cannot both be a smallest infinite number and not be. > >>> A moot point, since there is no definition for "'number' in general", as > >>> I just said. > >>> -- > >>> David Marcus > >> A very simple example is that there exists a smallest positive > >> non-zero integer, but there does not exist a smallest positive > >> non-zero real. > > > > > > So non zero integers are not real? Or is this another zenmath > > conundrum? Just curious. > > > >> If someone were to ask "does there exist a smallest > >> positive non-zero number?", the answer depends on what sort > >> of "numbers" you are talking about. > >> > >> Stephen > > > > ~v~~ > > The positive reals>0 include elements less than any positive natural>0, > an infinite number in (0,1). Is this comment relevant in some way to what Stephen or I said or are you replying to Lester? -- David Marcus
From: Tony Orlow on 27 Oct 2006 20:08
Lester Zick wrote: > On Thu, 26 Oct 2006 23:28:04 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: > > [. . .] > >>> We JUST agreed that 'smallest infinity' means two different things when >>> referring to ordinals and when referring to certain kinds of other >>> orderings! It is AMAZING to me that even though I took special care to >>> make sure this was clear, and then you agreeed, you NOW come back to >>> conflate the two ANYWAY! >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. They appear to be unrelated. However, they cme to two >> very different conclusions regarding a basic question: is there a >> smallest infinite number? It seems clear to me there is not, for the >> very same reason that Robinson uses: if there is an infinite number, you >> can subtract 1 and get a different, smaller infinite number. It's the >> same logic y'all use to argue that there's no largest finite. It's >> correct. The Twilight Zone between finite and infinite CANNOT really be >> pinpointed that way. > > Hey, Tony. You know this is an interesting problem but I think you're > wasting your time here arguing the issue with the Holy Order of Self > Righteous Mathematikers. Let me outline my own thinking for you. Alright, but keep in mind that changing the way the world works involves changing people's minds. :) > > I think there is a smallest infinity but that subtracting finites from > infinites isn't the way to get at the problem because as far as I can > tell arithmetic operations cannot be defined between infinites and > finites any more than finite division by zero can be. Why do you think there is a smallest infinity, when removing a finite portion thereof leaves a, smaller, infinite portion? > > Instead you need a different approach altogether and I suspect the way > to get at the problem is to assess the kind of infinity according to > the number of infinitesimals in various intervals. And in this manner > I suspect you'll find the infinity associated with straight line > segments is the smallest and various kinds of curves larger. > > ~v~~ You are intuiting in the derivative sense. As segments get smaller, on whatever curve, the angle between them decreases. The trick here is to declare, as Ross aludes, to a "universe", a complete range of values for the set or sets, and measure according to that "range". I have a feeling maybe you'll get it soon. It fits, I'm sure, with some of your ideas. You do have ideas, don't you? ;) 01oo |