An uncountable countable set
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From: mueckenh on 28 Aug 2006 07:42 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > Albrecht wrote: > > > > We must conclude that we can only say: we can find infinite many points > > > > on a line. This is a potential infinity. > > > > > > Do you have axioms for your mathematical theory of potential but not > > > actual infinity? > > > > Five Peano axioms. > > In what logic? There is only one logic. That which can be traced back to Aristoteles. > I have to think that it is at least second order logic, > since five Peano axioms in first order logic isn't even as strong as > the full seven Peano axioms in first order logic. 7? Originally there were more than 5, but also more than 7. > > So what is your logic (and I don't suppose it's classical?), and which > version of the Peano axioms do you have in mind? > > (I'm guessing it's second order: (1) N0, (2) Nx -> NSx, (3) Sx = Sy -> > x=y, (4) ~Ex 0=Sx, (5) Induction schema.) Nearly correct. I only start with 1 e N, because 0 is not very natural a number. It was invented 1000 years ago. The natural numbers have been known for at least 5000 years. > > Then, in this system, what are your definitions of 'potential infinity' > and 'actual infinity'? Learn it from Cantor who knew and stated it very clear and simple: Potential infinity: A quantity which is always finite but unbounded, like the variable n in n --> oo of analysis. (I, personally, compare it to a direction, in particular because it cannot be increased or completed. If n is a natural, then n+1 is a natural too. But there is no largest natural and there are not all naturals.) Actual infinity: A fixed quantity larger than any natural number. (In my opinion this is self-contradictory, because "fixed" is finished is finite.) Unfortunately today both types are intermingled because otherwise contradictions in set theory would become too obvious. For instance it is argued that all digit positions of 1/3 are countable and all levels of the binary tree are countable but not all edges of the binary tree are countable but only those which belong to finite levels (as if there were others). Regards, WM
From: mueckenh on 28 Aug 2006 07:43 Virgil schrieb: > In article <1156500736.055279.173430(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > Albrecht wrote: > > > > We must conclude that we can only say: we can find infinite many points > > > > on a line. This is a potential infinity. > > > > > > Do you have axioms for your mathematical theory of potential but not > > > actual infinity? > > > > Five Peano axioms. > > Which version? In some versions, it is explicitly required that the set > of all "naturals" exist. But as it does not and cannot exist in any sense, I prefer the original version (slimmed to 5 axioms). Regards, WM
From: mueckenh on 28 Aug 2006 07:44 Virgil schrieb: > In article <1156500782.651084.171830(a)i3g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1156148773.420187.122140(a)h48g2000cwc.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > You are dreaming. It does follow from the form of the numbers of my > > > > list. > > > > Try to come back to reality. Simply show how a digit position n can be > > > > indexed the complete number of which, i.e., the complete sequence of > > > > digit positions 1 to n is not in the list. > > > > > > The Hilbert Hotel method! > > > > Why does the Hilbert Hotel method not apply to my list, but only to > > Dik's number? > > Because it DOES apply to your list! It applies to any list. Why then is 0.111... not in my list, but Hilbert helps only Dik to defend his number being indexable? Regards, WM
From: mueckenh on 28 Aug 2006 07:45 Virgil schrieb: > In article <1156500895.295710.85000(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > The set of edges of an infinite binary tree are certainly countable, and > > > it is not difficult to construct an explicit bijection between them and > > > the set of binary representations of the naturals. > > > > > > The set of paths, however is not countable, as may is shown by > > > constructing an explicit bijection between the set of all such paths and > > > the set of all subsets on the set of naturals. > > > > > > Both the bijections referred to above have been presented here with no > > > one able to show either to be anything other than as advertised. > > > > Ok. Let's accept that. But by rational relation we find the set of > > paths being *not* larger > > than the set of edges. How is that possible? > > Because the "rational relation" in question requires that both be finite. That would contradict the axiom of infinity. But I explicitly accept that axiom for my tree. If the infinite set of natural numbers does exist, then there exists the infinite set of levels of my tree and the infinite set of digit positions of 1/3 and of pi. They all are in my tree (if they are anywhere). Regards, WM
From: mueckenh on 28 Aug 2006 07:47
Virgil schrieb: > > The problem is not indexing but "indexing without covering". > > That is easy to prove impossible for any finite natural number in unary > > representation. And, alas, there are only finite natural numbers. > > I have no idea what "indexing without covering" means, Never mind, it is impossible. > and until it has > a clear definition, I will continue to state that indexing all of them > in any way indexes all of them. > > > > > > Division was possible and was practised in fact long before rings and > > > > fields were known. > > > > > > Not outside of what later became known as rings. > > > > > > > Division of rationals by rationals or reals by reals is quite different. > > > So we have no reason to suppose that divisions involving other than > > > naturals need behave like division of naturals, or even be possible > > > without specific definitions of what it is and how it works.. > > > > > The old Greek already developed the method of geometric division, using > > the so-called Gnomon. This method makes no difference between division > > of naturals, rationals, reals or whatever deserves the name "number". > > Every well educated mathematician knows it. (Geometry is a certain > > language of mathematics.) > > But they HAD those specific definitions that are required. It is > division in the absence of such definitions that I object to. They had the theory of proportion of geometric lines and areas. That's all needed. In particular the division was (and is) not different for naturals and fractions and reals. > > > > > > > > One must have a very restricted mind to believe that without fields and > > > > rings division was impossible. > > > > > > The set of positive naturals is not even a ring, but it allows at least > > > two forms of division, and this has been recognized by most of us from > > > well before "Mueckenh". > > > > Your frequence of self contradicitions increases. You just said "Not > > outside of what later became known as rings." > > The naturals, or at least an ismorphic image of them , are contained in > lots of rings, so that arithmetic is "within" a ring. The number 7 is contained in lots of rings. Nevertheless it does not form a ring. Regards, WM |