Cantor Confusion
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From: Franziska Neugebauer on 8 Feb 2007 09:00 mueckenh(a)rz.fh-augsburg.de wrote: > On 8 Feb., 11:05, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: >> mueck...(a)rz.fh-augsburg.de wrote: [...] >> >> Who has told you that? Whatever, M�ckenheim axiom 2: >> >> >> "Numbers cannot exist without name" >> >> > If you take all names (in the widest sense, including symbols and >> > notations and defining equations) from a number and attach it to >> > another number, what remains with the first? >> >> >> Let's call that numbers "named numbers". If the length of names >> >> "must" be finite one can easily prove, that any set of named >> >> numbers has cardinality <= card(omega). >> >> > Of course. That is not new to me. >> >> It seems new to you that in real mathematics numbers do not >> necessarily have a name to exist. > > What do those nameless numbers need to exist? Wrong question. You need M�ckenheim axiom 2 to erase those innocent inhabitants of Cantor's paradise. > How can we distinguish them from nonexisting numbers? Good grief! First you terminate their existence and afterwards you complain about their non-existence. F. N. -- xyz
From: MoeBlee on 8 Feb 2007 14:29 On Feb 8, 1:25 am, mueck...(a)rz.fh-augsburg.de wrote: > On 7 Feb., 21:24, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > > On Feb 7, 4:46 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > There is an > > > > > axiom which requires the existence of the non existing and seems to > > > > > make some people happy > > > > > If there is such an axiom anywhere, it is only an axiom in WM's system, > > > > not in anyone else's. > > > > > > (like the axiom which requires the finity of > > > > > the infinite). > > > > > Any such axiom exists only in WM's system, and not in anyone else's. > > > > Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: > > > "Foundations of Set Theory", 2nd edn., North Holland, Amsterdam > > > (1984): "Intuitionists reject the very notion of an arbitrary > > > sequence of integers, as denoting something finished and definite as > > > illegitimate. Such a sequence is considered to be a growing object > > > only and not a finished one." > > > > Who considers it a finished one? > > > Which particular formal intuitionistic axiomatization do you have in > > mind? (There are intuitionistic formal systems, but I don't know in > > particular which one you have in mind so that we can evaluate whatever > > it is you think holds in or about them.) > > The question here is not at all what these or those intuitionists > think, but what "others" think, according to the opinion of Frenkel et > al. > > Intuitionists reject N as something finished. Who considers N as > something finished? "Finished" is informal. Meanwhile, ExAy(yex <-> y is a natural number) (where 'is a natural number' is here a rendering of a certain defined predicate symbol) is a theorem of many recursive axiomatizations. Meanwhile, you still have not stated from what system of logic and what mathematical axioms your own mathematical conclusions are supposed to be derived from. Or perhaps (?) you said that you reject formal axiomatization? Okay, then I don't see what objective basis you offer for determining which mathematical propositions have been established and which have not. > Above you could find the outset: > > > > > > (like the axiom which requires the finity of > > > > > the infinite). "The finity of the infinite" is whose expression? If 'finity' (as opposed to 'is finite') and 'the infinite' (as opposed to 'is infinite') are supposed to be about formal set theory, then what are the set theoretic definitions and what specfic axiom do you have in mind and how do you think it states such a requirement? The axiom of infinity? It states that there exists a set that has 0 as a member and is closed under successorship. Nothing about 'the finity of the infinite'. > > > > Any such axiom exists only in WM's system, and not in anyone else's. MoeBlee
From: Virgil on 8 Feb 2007 18:29 In article <1170926117.851856.193760(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 7 Feb., 20:22, Virgil <vir...(a)comcast.net> wrote: > > In article <1170852533.563009.176...(a)j27g2000cwj.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 6 Feb., 21:15, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1170756166.580698.67...(a)p10g2000cwp.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > On 4 Feb., 20:28, Virgil <vir...(a)comcast.net> wrote: > > > > > > > > > > Not so. Induction can only prove every set of naturals which is > > > > > > > > bounded > > > > > > > > above by a natural is finite, but that does not, according to > > > > > > > > the > > > > > > > > axiom > > > > > > > > of infinity, exhaust the possible sets of naturals. > > > > > > > > > There is no natural number in this set which is not covered by > > > > > > > induction. So, which number is missing to exhaust N? > > > > > > It is not numbers but sets of naturals of which I spoke. > > > > > The empty set of natural numbers? I am only interested in sets of > > > natural numbers which contain at least one natural number and contain > > > nothing but natural numbers. They are covered by induction. > > > > One can show by induction that any set of naturals is a subset of "the > > set of all naturals", but only if one allows that "the set of all > > naturals" exists in the first place. > > > > And one cannot, by induction or any other method, show that every set of > > natural numbers is finite without assuming it. > > Wrong. If a set is subject to complete induction, i.e., if n implies > the existence of n+1, then the set of these numbers n is infinite. What is WM's definition of finite or infinite that precedes induction? In any normal system, some form of induction must exist /before/ there is any need, or ability, to differentiate between finite sets and infinite sets. > That is the definition of infinity. Not in any standard system. Any standard system uses one of two possible definitions of finiteness versus infiniteness of sets. I prefer Dedekind's which says that a set is finite if and only every injection into itself is a bijection. > If you deny this then you should > give an upper bound. You cannot. The cardinality of the set of naturals is an upper bound on the "number" of naturals, and it is the same as the cardinality defined by bijectability with the first limit ordinal. > But you will not respond to this > paragraph. Won't I? > No, the problem is that set modern theory is not satisfied with this > simple form of infinity (because it would turn out self- > contradictive). Except that neither WM nor anyone else has ever been able to display any such self-contradiction in ZF or ZFC or NBG. Any alleged contradictions require additional assumptions not present in ZF or ZFC or NBG themselves. > Therefore you need some other form, a more infinite > infinity, even for the "smallest" infinity. Not "more infinite" but more set-like that WM's nonsense non-sets. > > > So that assuming what he wants to be able to prove has been WM's > > technique from the start. > > > > Intuitionalists are at least open about their assumptions. WM is not > > My assumptions are > that all natural numbers are finite > and that for all even positive numbers |{2,4,6,...,2n}| < n > and that all subsets of natural numbers contain natural numbers only > so that every set of natural numbers can be exhausted by induction. > > There are no further assumptions. Which should be dropped? if there are no others, how does WM justify all those non-theorems he keeps claiming, as none of them follow from what WM claims are his only axioms. Or does WM claim proof by fiat?
From: Virgil on 8 Feb 2007 18:32 In article <1170926173.228089.177950(a)v33g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 7 Feb., 20:24, Virgil <vir...(a)comcast.net> wrote: > > In article <1170852711.217577.234...(a)a75g2000cwd.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 6 Feb., 23:07, Virgil <vir...(a)comcast.net> wrote: > > > > > > > The basic way to establish IV c V is to use the numbers in their basic > > > > > form IIII c IIIII. (Numbers *are* their representations.) > > > > > > Numerals are no more numbers than names are people. > > > > > Wrong! People can exist and do exist without names. Numbers cannot. > > > > It is, as usual WM who is Wrong! Most numbers do not have names. > > Which number exists without any name? All those without names, which, for obvious reasons, cannot be named. It is a simple argument that there are at most countably many nameable ones, and even if WM denies it, that means uncountably many must remain forever unnamed.
From: Virgil on 8 Feb 2007 18:50
In article <1170934078.553109.258430(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 8 Feb., 11:02, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > > > This is not true. A tree *then* is the structure plus the set of paths > > and/or nodes. You persistently try to vaporize the essential > > constituents of a tree until it fits your preconception. > > What is wrong with my use of trees, except that it is inconvenient if > one is a bit clumsy in adapting new ideas? Among other things, it hides essential properties within naming rules for the nodes. > > This structure is given once and for all. Hence we do not need to > mention it further. Except then one is prohibited from dealing with the essential properties of trees and being tied to artificial naming rules for the nodes. > > > > Froget this complicated picture. I does not lead ahead. > > > > It leads again leads to the insight, that you posit what does not hold. > > If you mention the structure in any step, the description gets > complicated. It would seem that WM can't deal with the very slight complication which arises from considering all the tree properties, so wants to hide them from view. > The result does not differ from the case where the > structure is given once and for all. That is an unproven claim, and can only be established by allowing a similar analysis when everything is forced to be in view. > > > > > A path is an ordered set of nodes. > > > > In this picture a path is an ordered pair (S, <) of a set of nodes S and > > an order relation <. > > The path also has some nodes. And we need only this aspect here. Wrong! We need to establish that the pattern of links between nodes in that set of nodes fits the pattern required by a path in that tree. WM would gloss over all this because he is incompetent to deal with it. > > > > > But we need not use this order. > > > > To show that you are wrong even in this picture we *need* this order: > > But we don't need it for our conclusion. Therefore we do not use this > picture. Show what is wrong with the conclusion that the nodes of a > path are a subset of the set of nodes of the tree. Is every ordered subset of nodes of a tree a path in that tree? If not, WM's analysis is flawed. Given an ordered subset which is a path, is every other ordering of that set also a path? If not, WM's analysis is flawed. |