An uncountable countable set
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From: mueckenh on 27 Jun 2006 16:58 Virgil schrieb: > While that numbers may not have a known or even knowable decimal > expansion, the fact that you have named that one validates its existence. So the number pipifax which I just named does exist too? What is its approximate size? Regards, WM
From: mueckenh on 27 Jun 2006 17:01 Virgil schrieb: > Which digits in 0.111... = sum_{n in N} 10 ^(-n) are not indexed by > members of N? Best you can see it here: 1 0.1 2 0.11 3 0.111 .... ... n 0.111...n .... ... omega 0.111...omega Obviously 0.111... is not in the list enumerated by natural numbers. Obviously only digits of numbers in the list are completely indexed by natural numbers. > > > > > > You are making TO's vulgar false assumption that in order for a set of > > > naturals to be infinite, some of its members must be infinite. That is > > > no more true of sets of naturals than it is for sets of rationals or > > > sets of reals. > > > > Of course it is true, but we need not dicuss it here. > > What is false is true? that needs immediate discussion if we are to > proceed. It is true that the set of natural numbers is not an actually infinite set but a potentially infinite i.e. always finite but not limited by a finite threshold. > 0.0 0.1 0.11 0.111 .... The diagonal number 0.111... can *only then* be different from any number of the list, if it has more digits 1 than any list number. > Regards, WM
From: mueckenh on 27 Jun 2006 17:04 Virgil schrieb: > > We cannot write or read an unending string. The assumption, such > > strings existed, is wrong. > > The assumption that for every rational number we can describe precisely > the unending string representing it is right. > So some endless strings exist, even if only in the imagination. > But that is where all mathematics exists! Yes. And because it exists only there, nothing does exist, which does not precisely exist there. Such a nothing is an unending string. You may believe or even be convinced that it does exist, but you are wrong. Numbers like 5 or 1/3 do, but numbers like pi do not. > > > > > > > All numbers you will ever produce can be listed. That is enough. > > In what list? Give me any listing which you claim lists every real, and > I will give you a rule producing one nonmember of that list for every > subset of N. And that number can also be inserted in the list "g". Regards, WM
From: mueckenh on 27 Jun 2006 17:06 Daryl McCullough schrieb: > >f: 1 --> {1} and a --> { } with K_f = { }, > >g: 1 --> { } and a --> {1} with K_g = {1}. > >Here we have certainly no problem with lacking elements. > > The claim made by Cantor is that for any set A, there is > no surjection from A to P(A). Of course, if you construct > a new set B by adding extra elements to A, then there might > be a surjection from B to P(A). But why is that relevant? To show you that the condition K is impossible, independent of any surjectivity. Map |R (including |N) on P(|N) with the only condition that a natural number has to be mapped on that set K e P(|N) which contains all natural numbers which are not mapped on sets containing them. You see: It is impossible a condition. Regards, WM
From: mueckenh on 27 Jun 2006 17:09
Daryl McCullough schrieb: > mueckenh(a)rz.fh-augsburg.de says... > > >Daryl McCullough schrieb: > > >> Forget about Cantor for now. I'm asking you to demonstrate a function > >> f from N to R such that every "individualized" real number is in the > >> image of f. Can you demonstrate such a function? > > > >I cannot demonstrate a function f enumerating all reals > > So the answer is no? There is no function f enumerating > all the real numbers? So the reals are uncountable. There is a mathematical form containing all real numbers of [0, 1]. In binary representation it is given here. Why not call it a function in an extended sense? 0. / \ 0 1 / \ / \ 0 1 0 1 ...... The nodes are countable. There are less paths than nodes, down to *every* level. In order to separate into two paths, each path requires two nodes. Hence the path, i.e., the reals are countable. Regards, WM |