Galileo's Paradox
in [Math]
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From: Six on 11 Dec 2006 09:06 On Sat, 9 Dec 2006 02:09:39 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Six Letters wrote: >> > >I'm not too clear on why you don't like the standard solution to this >paradox. To wit: > >Once you specify how you wish to compare sets, then you get a specific >answer as to how E and N compare. Different ways of comparing give >different answers. Some of the ways of comparing sets are containment, >cardinality, and density. There are still things I'm learning about this. I don't know if my recent answer to MoeBlee would be of any help. >For finite sets, you generally get the same answer regardless of which >way you look at it. For infinite sets, you don't. This makes infinite >sets paradoxical (if you think they should behave like finite sets), but >also more interesting. > >So, what is wrong with this solution? I don't know that's a question of thinking that infinite sets should behave like finite ones. It's that there seems to be a choice about how one responds to the fact that, or the way that they clearly don't. If that helps. Thanks, Six Letters
From: Eckard Blumschein on 11 Dec 2006 09:21 On 11/30/2006 6:39 PM, Lester Zick wrote: > On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein > <blumschein(a)et.uni-magdeburg.de> wrote: > >>On 11/29/2006 6:37 PM, Bob Kolker wrote: >>> Tony Orlow wrote: >>>> >>>> It has the same cardinality perhaps, but where one set contains all the >>>> elements of another, plus more, it can rightfully be considered a larger >>>> set. > > Tony, you know we've been over this previously. All "infinite" means > is lack of definition for a particular predicate such as numerical > size. And when you add numerical finites to numerical infinites the > result is still infinite. > > This problem mainly arises I suspect because mathematikers insist on > portraying infinites as larger than naturals and somehow coming beyond > the range of naturals such as George Gamow's famous 1, 2, 3, . . . 00. > Then mathematkers try to establish certain numerical properties for > infinities by comparative numerical analysis and mapping with > numerically defined finites. However one cannot do comparative > numerical analysis and numerical analysis with numerically undefined > infinites anymore than one can do arithmetic. Infinites are neither > large nor small; they're just numerically undefined. Neither large nor small and also not equally large. Hopefully, this was ubiquitously understood from all. Lester is not a German name but "mathematikers" is too telltale. >>> Not necessarily so, if it is an infinite set. >>> >>> Bob Kolker >> >>This time I agree with Bob. >> >> > > ~v~~
From: Eckard Blumschein on 11 Dec 2006 09:25 On 12/6/2006 8:52 PM, Virgil wrote: > In article <4576EDC6.6090307(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 11:48 PM, Virgil wrote: >> > In article <457596BC.3040307(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 12/4/2006 10:47 PM, David Marcus wrote: >> > >> >> >> Standard mathematics may lack solid fundamentals. At least it is >> >> >> understandable to me. >> >> > >> >> > If it is understandable to you, then convince us you understand it: >> >> > Please tell us the standard definitions of "countable" and >> >> > "uncountable". >> >> >> >> I do not like such unnecessary examination. >> > >> > Then stop trying to examine others. >> >> I examined others here? > > You try, but as you are proceeding from a false assumption: > that your own understanding of mathematics is superior to that of > thousands of others who have spent much more time and effort and talent > in gaining their understanding than you have. Concerning time you may be correct. However, do not underestimate the talents by Galilei, Spinoza, Gauss, Kronecker, Poincar�, Wittgenstein and many others.
From: Eckard Blumschein on 11 Dec 2006 09:38 On 12/6/2006 8:39 PM, Virgil wrote: > As I have no idea of what it would mean to well order the irreals, I > supposed you were grimacing. Correct: Wohlordnungssatz: Jede Menge kann in die Form einer wohlgeordneten Menge gebracht werden. (Fraenkel 1923, p. 141)
From: Eckard Blumschein on 11 Dec 2006 09:51
On 12/6/2006 8:02 PM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/5/2006 7:06 PM, Tony Orlow wrote: >> > Eckard Blumschein wrote: >> >> On 12/1/2006 9:59 PM, Virgil wrote: >> >> >> >>> Depends on one's standard of "size". >> >>> >> >>> Two solids of the same surface area can have differing volumes because >> >>> different qualities of the sets of points that form them are being >> >>> measured. >> >> >> >> Both surface and volume are considered like continua in physics as long >> >> as the physical atoms do not matter. >> >> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. >> >> There is no universal rule for how fine-grained the mesh has to be. >> >> Therefore one cannot ascribe more or less points to these quantities. >> >> >> >> Look at the subject: Galileo's paradox: The relations smaller, equally >> >> large, and larger are pointless in case of infinite quantities. >> > >> > Now, just a minute, Eckard. You're contradicting yourself, if these >> > objects are infinite sets of points. They have different measure. >> >> I did not consider measures. Let's get concrete. Are there more naturals >> than odd naturals? This question could easily be answered if there was a >> measure of size. > > It can easily be answered once you say what you mean by "more". Why do > you think that common English words have unambiguous mathematical > meanings? More is just inappropriate as to describe something infinite. There are not more naturals than rationals. There are not equaly many of them, there are not less naturals than rationals. > |