Why can no one in sci.math understand my simple point?
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From: Sylvia Else on 20 Jun 2010 02:07 On 20/06/2010 3:48 PM, |-|ercules wrote: > "Sylvia Else" <sylvia(a)not.here.invalid> wrote ... >> On 20/06/2010 6:10 AM, |-|ercules wrote: >> >>> If all digits of a single infinite expansion can be contained with >>> increasing finite prefixes, >>> and the computable set of reals has EVERY finite prefix, then all digits >>> of EVERY infinite >>> expansion are contained. >> >> It's far from clear what that actually means, but in any case you >> certainly haven't proved it. >> >> Sylvia. > > You agreed that all digits of PI (in order) are contained in this list, > right? > > 3 > 31 > 314 > ... > > So you should get the first part, > >>> If all digits of a single infinite expansion can be contained with >>> increasing finite prefixes, > > Herc I've told you which step I have reservations about. Sylvia.
From: K_h on 20 Jun 2010 03:20 "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote in message news:4c1d7dce$0$316$afc38c87(a)news.optusnet.com.au... > >>> My whole argument is that they cannot be listed in their >>> entirety, or we could use a Cantor construction to >>> produce a computable Real not on the list. >> >> You are wrongly assuming that only computable sets exist. > > No. Good. Then why wouldn't the "Cantor construction" you talk about above produce a non-computable real? _
From: WM on 20 Jun 2010 04:27 On 20 Jun., 01:58, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-19, Virgil <Vir...(a)home.esc> wrote: > > > There is, however, some question in my mind about the existence of a > > list of all and ONLY computable reals. > > Why? The computable reals can be proven countable, as you already > know, so there is a bijection between N and the set of countable > reals. That is not true. An exclusive list need not exist. The reals in a certain Cantor-list are countable. And if you form the anti-diagonal of that list and add it (for instance at first position) to the list, this new list is also countable. Again consttruct the anti-diagonal and add it to the list. Continue. The situation remains the same for all anti-diagonals you might want to construct. Therefore all reals constructed in that way are countable. Nevertheless it is impossible to put all of them in one list, because there would be another resulting anti-diagonal. Conclusion: It is impossible to obtain a bijection of all these reals with N although all "these" reals are countable with no doubt. Regards, WM
From: WM on 20 Jun 2010 04:31 On 20 Jun., 02:04, "Mike Terry" > > No, you're misunderstanding the meaning of computable. > > Hopefully you will be OK with the following definition: > > A real number r is computable if there is a TM (Turing machine) > T which given n as input, will produce as output > the n'th digit of r. Whatever might be the true meaning: The Turing machine need a finite definition. Therefore the computable number has a finite definition. There are only countable many finite definitions. And every diagonal of a defined Cantor list has also a finite definition. Therefore Cantor shows that the countable set of all real numbers with finite definitions is uncountable (or that the defined diagonal number is undefined). Regards, WM
From: Ross A. Finlayson on 20 Jun 2010 04:57
On Jun 17, 10:30 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-18, Ross A. Finlayson <ross.finlay...(a)gmail.com> wrote: > > > The rationals are well known to be countable, and things aren't both > > countable and uncountable, so to have a reason to think that > > arguments about the real numbers that are used to establish that > > they are uncountable apply also to the rationals, the integer > > fractions, has for an example in Cantor's first argument, about the > > nested intervals, that the rationals are dense in the reals, so even > > though they aren't gapless or complete, they are no- where > > non-dense, they are everywhere dense on the real number line. > > As your sentence is less than coherent, I will merely point out that > it is generally poor form to use 9 commas in a single sentence except > when listing items. I will grant that parody often benefits from > abuses of ordinary sentence structure, such as, for example, and not > in any way showing that these are the only possible forms, sentences, > like this one, which are convoluted to exhibit, by way of meandering, > that they imply that mental processes, of the original writer, that > is, which may be, perhaps, less than clear, and so in some way, to > some readers, humourous. > > - Tim Some I go back and add later. Too bad no one can read it but me. No seriously still it's clear in lots of ways, I can rewrite those paragraphs as much longer. Sorry that was a bad joke. Collected, I'm very happy with the output. Each one, in its own way, has some content. You don't agree with the rationals are dense on the line? The rationals are well known to be countable, and things aren't both countable and uncountable, so to have a reason to think that arguments about the real numbers that are used to establish that they are uncountable apply also to the rationals, the integer fractions, has for an example in Cantor's first argument, about the nested intervals, that the rationals are dense in the reals, so even though they aren't gapless or complete, they are no- where non-dense, they are everywhere dense on the real number line. The rationals are well known to be countable, and things aren't both countable and uncountable, so to have a reason to think that arguments about the real numbers that are used to establish that they are uncountable apply also to the rationals, the integer fractions, has for an example in Cantor's first argument, about the nested intervals, that the rationals are dense in the reals, so even though they aren't gapless or complete, they are no- where non-dense, they are everywhere dense on the real number line. (The rationals are well known to be countable, and things aren't both countable and uncountable, so ) to have a reason to think that arguments about the real numbers that are used to establish that they are uncountable apply also to the rationals, the integer fractions, has for an example in Cantor's first argument, about the nested intervals, that the rationals are dense in the reals, so even though they aren't gapless or complete, they are no- where non-dense, they are everywhere dense on the real number line. (...) You left out the part before and after. Arguments about the uncountability of the real numbers include those derived from the numeric property of their density. For example one of them is called "Cantor's first argument for the uncountability of the reals." The constructive (computable) sets are mappable to the countable ordinals, where, the countable ordinals is the same thing as the enumerative ordinals, because, they're each countable and that's all of them. The constructive universe is complete, each in it countable, but then the results of results are results so they are infinite and their own powersets. (Sound familiar?) The existence of the constructive universe is a result. Ha what's funny is you can still read them. Basically from having an idea to write a sentence, as it's written then the parts of it are added automatically for, as you describe, contemplative pause, as well as any emplacement of comment to provide context generally. This is in the case where the writing is for a particular medium, when there's a lot of writing back in forth (in complete sentences, thank you) then to edit for readability is deferred for real intent. But, there's not really a lot of writing going on that way, rather, I much prefer the writing with the theme and the content (on the mathematics). So, I read. Warm regards, Ross Finlayson |