Cantor Confusion
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From: Virgil on 12 Oct 2006 15:57 In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Dik T. Winter wrote: > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > It is not > > > contradictory to say that in a finite set of numbers there need not be > > > a largest. > > > > It contradicts the definition of "finite set". But I know that you are > > not interested in definitions. > > Set Theory is simply not very useful. The main problem being that finite > sets in your axiom system are STATIC. They can not grow. One can quite easily construct set valued functions which do grow. WE just do not call them sets. > Which is quite > contrary to common sense. Does "Mueckenh" insist that everything must grow? THAT is quite contrary to common sense. > (I wouldn't imagine the situation that a table > in a database would have to be redefined, every time when a new row has > to be inserted, updated or deleted ...) And if one chose to save a copy of the original at each update, would those archived files have to change at each update?
From: Virgil on 12 Oct 2006 16:00 In article <452e5a9d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> > >>> "Albrecht" <albstorz(a)gmx.de> writes: > >>> > David Marcus schrieb: > >>> ... > >>> > > I don't follow. How do you know that the procedure that you gave > >>> > > actually "defines/constructs" a natural number d? It seems that you > >>> > > keep > >>> > > adding more and more digits to the number that you are constructing. > >>> > > >>> > What is the difference to the diagonal argument by Cantor? > >>> > >>> That a (to the right after a decimal point) infinite string of decimal > >>> digits defines a real number, but that a (to the left) infinite string > >>> of decimal digits does not define a natural number. > >> It defines something. > > > > But not necessarily a number. > > > >> What do you call that? If the value up to and > >> including every digit is finite, how can the string represetn anything > >> but a finite value? > > > > Because representations of finite values end, and the string doesn't > > end, so it breaks the rules of "strings that represent finite values". > > > > - Randy > > > > Can you rightly call it an infinite value? I can't. It's unbounded like > the finites themselves, but not infinite, as long as all digit positions > are finite. Then is TO claiming that the infinite binary string ...111 represents a finite natural? Note that all digit positions in ...111 are finite.
From: Virgil on 12 Oct 2006 16:02 In article <452e5acb$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > Because there are two types or strings. Strings that end and strings > > that don't end. Only strings that end represent finite values. > > > > -William Hughes > > > > And what about countably infinite strings which cannot achieve actually > infinite values? They "end" in the sense that they only have finitely many non-zero digits, so are "equivalent" to finite strings.
From: Virgil on 12 Oct 2006 16:07 In article <452e5bbb$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452d14fe$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Dik T. Winter wrote: > >>> In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> > >>> "Albrecht" <albstorz(a)gmx.de> writes: > >>> > David Marcus schrieb: > >>> ... > >>> > > I don't follow. How do you know that the procedure that you gave > >>> > > actually "defines/constructs" a natural number d? It seems that you > >>> > > keep > >>> > > adding more and more digits to the number that you are constructing. > >>> > > >>> > What is the difference to the diagonal argument by Cantor? > >>> > >>> That a (to the right after a decimal point) infinite string of decimal > >>> digits defines a real number, but that a (to the left) infinite string > >>> of decimal digits does not define a natural number. > >> It defines something. > > > > An infinite string of digits. but every standard natural number is > > defined by a finite string of digits, given a base, so those infinite > > string define nothing at all. Besides themselves. > > > > > > > > What do you call that? > > > > An infinite string. > > > >> If the value up to and > >> including every digit is finite, how can the string represetn anything > >> but a finite value? > > > > If a binary string s:N --> {01} is such that s(n) = 1 for all n in N, > > then its "value" is sum_{n in N} 2^n, which diverges. > > Of course it diverges, which means it attains an infinite value for > infinite n. But, for all finite n, sum(x=0->n: 2^x) is finite. You have > no infinite n in N. But the "value" of that string, if it is to exist at all, cannot be finite even thought all its digit positions are finite. Thus contradicting TO's claims. > > > > > But all the partial sums, sum_{n =1..k}2^n are all finite. > > Right, and that's all there is in N. There is nothing in N that is > infinite in value or in index. > > > > > So the value up to and including every digit is finite and the string > > itself cannot represent any finite value. > > It cannot represent any infinite value for that very reason. So is TO claiming to have created/discovered values between finite and infinite?
From: Virgil on 12 Oct 2006 16:11
In article <1160675140.906009.253460(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > this implies > > > > for every digit position N, > > there exists a single unary number, M, > > such that M covers 0.111... to position N > > > > > > this does not imply > > > > there exists a single unary number M such that for every digit > > position N, M covers 0.111... to position N > > Why shouldn't it? In general "for all x there is a y such that f(x,y)" does not imply "there is a y such that for all x f(x,y)". To establish the latter requires proof over and above the former. >If every digit position of 0.111... is a finite > position then exactly this is implied. Your reluctance to accept it > shows only that you do not understand how an infinite set can consist > of finite numbers. In fact, nobody can understand it, because it is > impossible. Not in ZF or NBG. |