Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Randy Poe on 21 Nov 2006 11:48 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > > > > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > > > > principle. > > > > > > > > In principle, if a is the sqrt(2) to 10^100 digits, then > > > > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > > > > > > > What "in principle" prevents me from calculating 2/a, > > > > or adding it to 1, or taking 0.5*(a + 2/a)? > > > > > > Lack of bits to represent 2/a if a already requires all bits available. > > > > That means it's hard in practice. > > > > But what prevents 2/a from existing in principle? > > It is not "hard in practice", but it is impossible. There is no chance. > That means "in principle". No, that is not the meaning of "in principle". - Randy
From: William Hughes on 21 Nov 2006 12:12 Eckard Blumschein wrote: > For the first time I do not feel unnecessarily lectured and deliberately > misunderstood. > > On 11/21/2006 3:07 PM, William Hughes wrote: > > Eckard Blumschein wrote: > > > >> While I do not question your last sentence being a correct conclusion if > >> the premise holds, I cannot not see any possibility to reasonably and > >> concisely quantify a largest possible number. > >> > >> Do we really have to define numbers? I tend to be satisfied when I have > >> an executable rule which may create as many natural numbers as I like, > >> as does Archimede's axiom. Executable means effectively about the same > >> as communicable. It excludes postulated a priori existence. > >> > >> Do not get me wrong. I consider the properties executable and > >> communicable like belonging to the abstract mathematical idea, not to > >> any physical limitation to it. The process of counting indefinitely is > >> never completely executable. Accordingly all created numbers are finite, > >> and there is no principial reason for a largest possible number to be > >> found. > > > > This is of course the standard position. It is not WM's position. > > I just described my position and wonder if it is the standard one. > > > > On the larger question of whether the set of > > all natural numbers exists. > > > > Your "executable rule" defines what is commonly referred to > > as "potential infinity". > > Exactly. > > > The problem is the difference between saying > > > > The set of all natural numbers exists > > This is what I call Dedekind/Cantor Utopia. Elsewhere i tried to explain > why it deals with sets instead of numbers. > > > and > > We can generate an arbitrarily large set > > of natural numbers > > Arbitrarily large is according to Cantor not really infinite. I share > this view of him. > > > > (i.e. the difference between actual and potential infinity) is > > mostly one of terminology. > > Yes, but I got aware that actual and potential infinity are two > different views both directed towards the same object of natural numbers > from different levels of abstraction. The potential infinite view > belongs to counting and is able to separate from each other all single > numbers. The actual infinite one provides the fiction of all natural > numbers at a higher level of abstraction. You cannot have both views at > a time. > Why not. There is nothing about assuming the set of all natural numbers exists that precludes counting or the ability to sepatate from each other all single numbers. The elements of the potentially infinite set are exactly the same elements with exactly the same properties as the elements of the actually infinite set. > > There is no element of the set > > of natural numbers that does not exist as an element of some > > arbitrarily large set. So there is absolutely no difference between > > the > > elements of "the infinite set of natural numbers" and the elements > > of "the potentially infinite set of natural numbers". > > This is what I meant when writing above "the same object". > > > About the only thing you can say about the > > "potentially infinite set of natural numbers" is that it is > > not a set. So what? You still have this potential, > > this thing that allows you to get new natural numbers > > whenever you want. This potential has > > all the properties of a set but the name. > > And the elusive belief hidden within and conveyed by this name. > Aren't I correct? > No. You can develop "potential set theory". Just define an element of a potential set to be an element of any one of the arbitrary sets that can be produced. Now go through set theory and add the word "potential" in front of each occurence of the word set. There is no difference between saying "a potentially infinite set exists" and saying "an actually infinite set exists". - William Hughes
From: Lester Zick on 21 Nov 2006 13:42 On Tue, 21 Nov 2006 03:04:42 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <8m8pl2pj1icbeven2hq7rp4hq1rufqh1u2(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: >... > > Why are square circles unimaginable? > >They are not. With the Manhattan measure of the plane, each circle is >a square. Then what is a square? ~v~~
From: Lester Zick on 21 Nov 2006 13:47 On Tue, 21 Nov 2006 03:45:54 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <1163672167.744087.89590(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: [. . .] >Never ever do you give a proof. Only assertions. Weeell one might make the same observation about mathematikers. ~v~~
From: Lester Zick on 21 Nov 2006 13:48
On 20 Nov 2006 20:56:44 -0800, imaginatorium(a)despammed.com wrote: > >mueckenh(a)rz.fh-augsburg.de wrote: >> Randy Poe schrieb: >> >> >> > > I know that at most 10^100 digits of sqrt(2) can be determined, in >> > > principle. >> > >> > In principle, if a is the sqrt(2) to 10^100 digits, then >> > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. >> > >> > What "in principle" prevents me from calculating 2/a, >> > or adding it to 1, or taking 0.5*(a + 2/a)? >> >> Lack of bits to represent 2/a if a already requires all bits available. >> >> If you don't believe me, simply try it. By about 330 calculations you >> should be able to produce the first 10^100 digits. If you have a slow >> computer you will need one second per calculation. So let it run for 5 >> minutes and you will know what prevents you from calculating 2/a. > >I think the bottom line here is that Mueckenheim doesn't have a clue >what "in principle" means. (Amongst other things) Whereas you do. ~v~~ |