Galileo's Paradox
in [Math]
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From: Six on 7 Dec 2006 08:22 On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: > >Six wrote: >> On Mon, 4 Dec 2006 12:12:39 -0600, mstemper(a)siemens-emis.com (Michael >> Stemper) wrote: >> >> >In article <3stjm255vbgdfdrh9jdvrmbecu99perr0i(a)4ax.com>, Six Letters writes: >> >>On 24 Nov 2006 15:08:20 GMT, richard(a)cogsci.ed.ac.uk (Richard Tobin) wrote: >> >>>In article <f41em2h1s4dv1qm70ntnv2jped6qekfd7s(a)4ax.com>, <Six Letters> wrote: >> > >> >>>>> (a) sets in 1-1 correspondence are the same size and >> >>>>> (b) proper subsets are smaller than their supersets >> >>>> >> >>>>Exactly. It's throwing out (a) that I am trying to explore. >> >>> >> >>>This will result in bizarre consequences. For example, there will be >> >>>more decimal strings representing integers than binary strings, even >> >>>though they represent the same integers. >> >> >> >> On second thoughts I am not sure that I understand that. I rather >> >>suspect there are bizarre consequences, but I'd be really happy if you'd >> >>elaborate a little bit. >> > >> >It's pretty simple. The first eleven positive integers are represented in >> >decimal as: 1 2 3 4 5 6 7 8 9 10 11. The first three positive integers >> >are represented in binary as: 1 10 11. >> > >> >How should a map between decimal and binary representations of positive >> >integers be set up? Like this: >> > 1 1 >> > 2 10 >> > 3 11 >> > 4 100 >> > 5 101 >> > 6 110 >> > 7 111 >> > 8 1000 >> > 9 1001 >> >10 1010 >> >11 1011 >> >.. >> > >> >or like this: >> > >> > 1 1 >> > 2 >> > 3 >> > 4 >> > 5 >> > 6 >> > 7 >> > 8 >> > 9 >> >10 10 >> >11 11 >> >.. >> >> I am very grateful to you for expanding on this. While I'm almost >> certain I'm missing something, I'm afraid I still don't get it. >> >> How exactly does claiming that a 1:1 C is not necessarily >> indicative of equality of size with infinite sets presuppose an inability >> to map (eg) the binary and decimal representations of integers? >> >> There is still a 1:1 C between the two sets. It is still true that >> for any finite sets a 1:1C implies equality of size. Moreover it's still >> reasonable to suppose that a 1:1C implies equality of size in the infinite >> case unless there are other, 'functional' reasons to the contrary. (Vague, >> I know. Roughly, 1:1 C is a necessary but not sufficient condition for >> equality of size.) >> >> The idea is that the naturals (in any base) form a paradigm or >> norm, a standard against which other sets can be measured. > >The set of finite binary strings is a subset of the set of finite >decimal strings. I confess I hadn't fully appreciated this simple point, that together with the fact that the strings just are, so to speak, the natural numbers (in a given base). >Then b) precludes them being the same size. > >They are also both the same size as the set of natural numbers. > >Thus they are the same size as each other. > >Contradiction. One is driven to the conclusion that there is no base-independent size for the natural numbers. This does not make the discussion of the relative size of, for example, natural numbers and squares meaningless. It's just that a given base would have to be understood, and that whatever is said about the relative size of the two sets is understood to apply mutatis mutandis to any other base. But for me at least, it has certainly opened my eyes to the implications of the original argument. Much appreciated, Six Letters
From: Eckard Blumschein on 7 Dec 2006 09:18 On 12/7/2006 1:30 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/6/2006 5:35 AM, David Marcus wrote: >> > Eckard Blumschein wrote: >> >> >> You did not understand that I am using Fourier transform as an example. >> > >> > Example of what? >> >> a typical mistake when using the immediate value. > > Be specific: what is the mistake? Ask Hendrik van Hees who could not explain a result differing from a printed one just by the factor 2. He uttered this in sci.physics.research > >> >> I do not criticize FT but the integral tables, and I did not have a >> >> problem myself but I recall several reported cases of unexplained error >> >> by just the trifle of two. The integral tables suggest using the >> >> intermediate value. >> > >> > You are criticizing integral tables? >> >> Some of them are rather eclectic and difficult to overlook. Others are a >> bit slim. Sometimes the intermediate values are given, sometimes they >> are omitted which i consider the better decision. > > I haven't a clue why you think integral tables have any relevance to set > theory. The intermediate values, given in integral tables "for mathematical reasons", are misleading and perhaps unnecessary. The "mathematical reasons" relate to the somewhat inappropriate arbitrarily distorted notion of real numbers adapted to the illusion by Dedekind and Cantor, real numbers and infinity are numbers with full civil rights. > >> >> Experienced mathematicians should indeed know that >> >> they must avoid this use. Some tables give the intermedite value for the >> >> sake of putative mathematical correctness. >> > >> > Please give an example. A gave http://iesl.et.uni-magdeburg.de/~blumsche/M283.html >> >> I just have my old Bronstein-Semendjajew Teubner 1962 at hand. >> On p. 351, number 13 does not give intermediate values, number 15 does. > > Please quote the entire example. Not everyone is next to a library. No. 15: Integral from 0 to oo dx over sin x cos x / x = = pi/2 for |a|<1 = pi/4 for |a|=1 (intermediate value) = 0 for |a|>1 No. 13: just over tan(ax)/x > >> >> Others omit it. >> >> As long as one knows the result in advance, there is almost not risk. FT >> >> and subsequent IFT may perform an ideal check of set theory. >> > >> > What do you mean "ideal check of set theory"? >> >> Set theory leads to intermediate values. > > What do you mean? Do try to be specific. Set theory considers real numbers to be existing numbers, not just fictions. From this point of view, one cannot admit that a number may be void even if it turns out to be useless and maybe even misleading.
From: David Marcus on 7 Dec 2006 09:57 Eckard Blumschein wrote: > On 12/7/2006 1:30 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 12/6/2006 5:35 AM, David Marcus wrote: > >> > Eckard Blumschein wrote: > >> > >> >> You did not understand that I am using Fourier transform as an example. > >> > > >> > Example of what? > >> > >> a typical mistake when using the immediate value. > > > > Be specific: what is the mistake? > > Ask Hendrik van Hees who could not explain a result differing from a > printed one just by the factor 2. He uttered this in sci.physics.research Don't be silly (or sillier than you usually are). Either you know of a mistake or you don't. If you can't tell us the mistake, then stop saying there is one. The fact that someone else is having problems with math isn't relevant. Lots of people have trouble with math. That doesn't mean there is something wrong with mathematics. > >> >> I do not criticize FT but the integral tables, and I did not have a > >> >> problem myself but I recall several reported cases of unexplained error > >> >> by just the trifle of two. The integral tables suggest using the > >> >> intermediate value. > >> > > >> > You are criticizing integral tables? > >> > >> Some of them are rather eclectic and difficult to overlook. Others are a > >> bit slim. Sometimes the intermediate values are given, sometimes they > >> are omitted which i consider the better decision. > > > > I haven't a clue why you think integral tables have any relevance to set > > theory. > > The intermediate values, given in integral tables "for mathematical > reasons", are misleading and perhaps unnecessary. No idea what "mathematical reasons" these are. Please give a full quote where the book says what the "mathematical reasons" are. > The "mathematical > reasons" relate to the somewhat inappropriate arbitrarily distorted > notion of real numbers adapted to the illusion by Dedekind and Cantor, > real numbers and infinity are numbers with full civil rights. You really should learn to speak concretely. Telling us what they "relate to" tells us nothing, especially since the rest of your sentence just contains your usual prejudiced, uninformed rant. > >> >> Experienced mathematicians should indeed know that > >> >> they must avoid this use. Some tables give the intermedite value for the > >> >> sake of putative mathematical correctness. > >> > > >> > Please give an example. > > A gave > http://iesl.et.uni-magdeburg.de/~blumsche/M283.html That link produces a page saying the search didn't produce any results. Kind of ironic since all of your postings contain no results, too. > >> I just have my old Bronstein-Semendjajew Teubner 1962 at hand. > >> On p. 351, number 13 does not give intermediate values, number 15 does. > > > > Please quote the entire example. Not everyone is next to a library. > > No. 15: Integral from 0 to oo dx over sin x cos x / x = > = pi/2 for |a|<1 > = pi/4 for |a|=1 (intermediate value) > = 0 for |a|>1 There is no "a" in int_0^oo sin x cos x / x. > No. 13: just over tan(ax)/x Huh? > >> >> Others omit it. > >> >> As long as one knows the result in advance, there is almost not risk. FT > >> >> and subsequent IFT may perform an ideal check of set theory. > >> > > >> > What do you mean "ideal check of set theory"? > >> > >> Set theory leads to intermediate values. > > > > What do you mean? Do try to be specific. > > Set theory considers real numbers to be existing numbers, not just > fictions. Nonsense. The words "existing" and "fictions" are your own creation. Set theory says nothing of the sort. If you discuss something, you should have at least some knowledge of it. > From this point of view, one cannot admit that a number may be > void even if it turns out to be useless and maybe even misleading. More made-up words: "void", "useless", "misleading". -- David Marcus
From: Tonico on 7 Dec 2006 11:46 Eckard Blumschein ha escrito: [....unbelievable stupid ranting snipped....] *************************************************** I wonder whether somebody still wonders why I got excited when Eckie boy sent a blank message and I wrote that finally we got a message from him that makes sense...*sigh*... Tonio
From: stephen on 7 Dec 2006 12:25
Six wrote: > On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: >> >>Six wrote: >>> >>> I am very grateful to you for expanding on this. While I'm almost >>> certain I'm missing something, I'm afraid I still don't get it. >>> >>> How exactly does claiming that a 1:1 C is not necessarily >>> indicative of equality of size with infinite sets presuppose an inability >>> to map (eg) the binary and decimal representations of integers? >>> >>> There is still a 1:1 C between the two sets. It is still true that >>> for any finite sets a 1:1C implies equality of size. Moreover it's still >>> reasonable to suppose that a 1:1C implies equality of size in the infinite >>> case unless there are other, 'functional' reasons to the contrary. (Vague, >>> I know. Roughly, 1:1 C is a necessary but not sufficient condition for >>> equality of size.) >>> >>> The idea is that the naturals (in any base) form a paradigm or >>> norm, a standard against which other sets can be measured. >> >>The set of finite binary strings is a subset of the set of finite >>decimal strings. > I confess I hadn't fully appreciated this simple point, that > together with the fact that the strings just are, so to speak, the natural > numbers (in a given base). >>Then b) precludes them being the same size. >> >>They are also both the same size as the set of natural numbers. >> >>Thus they are the same size as each other. >> >>Contradiction. > One is driven to the conclusion that there is no base-independent > size for the natural numbers. How can the size be base dependent? The natural numbers are not base dependent. Any natural number can be expressed in any base. There is no natural number expressible in base 16 that is not expressible in base 10, or base 9, or base 2. I suppose you could claim that there is a set of decimal numbers, and a set of base 2 numbers, and a set of hexadecimal numbers, and that they are all different, and all have different sizes. But it is a strange notion of "different size" given that all the sets represent the same thing. Stephen |