Galileo's Paradox
in [Math]
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From: Six on 6 Dec 2006 07:35 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. Six Letters
From: Han de Bruijn on 6 Dec 2006 08:34 Bob Kolker wrote: > David Marcus wrote: > >> Bob Kolker wrote: >> >>> You obviously have no knowledge of fractal dimension or Hausdorf >>> dimension. For example the Peano space filling curves have a >>> dimension between 1 and 2. >> >> The image of a space filling curve is a square, so has dimension 2. > > I sit corrected. The Peano curve has hausdorff dimension w. Where w = 2 . See: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension Han de Bruijn
From: Bob Kolker on 6 Dec 2006 08:57 Han de Bruijn wrote: > > Where w = 2 . See: Typo. The 'w' key is just below the '2' key. Fumble fingers. Bob Kolker
From: Mike Kelly on 6 Dec 2006 10:08 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. 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. -- mike.
From: Eckard Blumschein on 6 Dec 2006 10:30
On 12/5/2006 11:10 PM, Virgil wrote: > In article <45753CB8.3080500(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> Let's rank Tonico outside but David Marcus still inside mathematics. >> >> On 12/5/2006 8:55 AM, Tonico wrote: >> > David Marcus wrote: >> >> Eckard Blumschein wrote: >> >> >> >> > Reals, as indirectly defined with DA2, >> >> >> >> Why do you think that the diagonal argument defines the reals? >> >> You all know that DA2 shows by contradiction that real numbers are >> uncountable. > > WRONG! It is not, at least in Cantor's version, a proof by contradiction. He assumes that his list of all reals is complete and shows that this is not the case. From this contradiction he was forced to conclude that the reals are uncountable but he intentionally misinterpreted the outcome by claiming there are more real than rational numbers. Look above: Galilei. > >> I carefully read how Cantor made sure that the numbers >> under test are real numbers. He did not use Dedekind cuts, nested >> intervals or anything else. He assumed numbers with actually >> indefinitely much rather than many e.g. decimals behind the decimal >> point. Strictly speaking, he did not immediately show that the reals are >> uncountable but that these representation like never ending decimals is >> uncountable. > > > Since Cantor's first proof You perhaps refer to Cantor's first diagonal argument, alias Cauchy's proof because stolen from Cauchy, which demonstrated the countability of the rarionals. > was not in any way the same and IS valid for > Dedekind cuts, any flaws in his second are irrelevant. I do not understand what you mean in this contex regarding Dedekind cuts. I do not refer to any flaw in DA2 itself, just the interpretation is wrong. |