Cantor Confusion
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From: Virgil on 7 Feb 2007 14:22 In article <1170852533.563009.176100(a)j27g2000cwj.googlegroups.com>, mueckenh(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. 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.
From: Virgil on 7 Feb 2007 14:24 In article <1170852711.217577.234000(a)a75g2000cwd.googlegroups.com>, mueckenh(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.
From: Federico Ferreres Solana on 7 Feb 2007 05:09 > Then I take it you think the above is bijection from the real numbers > to the naturals? Which natural number does the real number 1/3 map > to? It is 33333...(periodic) If you do not allow infinetly large numbers in naturals, you must come to the conclusion that fractionals are more, because you have an expanded definition, allowing you to write infinetly large numbers such as 0.33...(periodic) or PI, but that you can NEVER see in practice, just imagine them. 1/3 is an abreviation for 1(all, base stuff, world, object to be parted into potentially infinite slides)/3(parts), which means in naturals, infinity/3, or 3333...(periodic). Natural number 1 is the representation of the smallest slides of pie called infinity. But 1 is also the largest fractional number imaginable, something that when we use naturals start to call infinity. A fractional can be seen as defining an integer. 1/2 defines, or names 2 slides of a pie, so it may be seen as a fraction only, but then it defines 2, as you have 1 slide, and another one. When you define 0.11...(periodic) you are basically thinking there is an infinetly large number of slides in the pie, but you can never know what it is, yet, you think you know what 0,50000...(periodic) is, because you relate it to the whole pie. 500000...(periodic) is exactly the same in relation to something we named "infinity", that is exactly similar to what we denote as "1" in fractionnals. Infinity is exatly the same in the naturals as the "1" is to fractionals. When we don't allow that, we think of the problems as separate sets. In one (fractionals) we part from infinity towards smaller and smaller parts, and allow infinetly large number of slices through notation (1/3), allowing us to use infinity as a number ("1"), and then in the naturals, we claim to not know how to divide infinity. The problem arises when only inadvertly allows infinetly large numbers of "fractions" in the fractionals (which is OK for me) but forbids from thinking about infinetly large natural number. If you defined fractionals, as parts of something (1), you had to reach by induction, you would see you could only aproximate PI, but that PI will not be in the set. And if you do prove it is in the set, then you have proven that the natural 1415...(rest of numbres of PI), also exists, as the fractional part of PI implicitly defines in the fractionals and infinitely large number of parts or slices of a pie.
From: MoeBlee on 7 Feb 2007 15:24 On Feb 7, 4:46 am, mueck...(a)rz.fh-augsburg.de wrote: > On 6 Feb., 20:58, Virgil <vir...(a)comcast.net> wrote: > > > > > > > In article <1170755634.350929.67...(a)s48g2000cws.googlegroups.com>, > > > mueck...(a)rz.fh-augsburg.de wrote: > > > The existence of the empty set is not at all guaranteed. > > > It is in ZFC. > > > ZFC requires at least one set to exist, and also requires existence of a > > subset of that set which does not contain any of the members of that set. > > > > There is an > > > axiom which requires the existsence 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 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.) MoeBlee
From: MoeBlee on 7 Feb 2007 15:30
On Feb 7, 7:12 am, Han de Bruijn <Han.deBru...(a)DTO.TUDelft.NL> wrote: > William Hughes wrote: > > And since the limit is not a finite number the fact that there > > is a contradiction for all finite numbers does not mean that > > there is a contradiction for the limit. > > In my calculus class, the fact that the "limit is not a finite number" > just always meant: the limit _does not exist_, i.e. there IS NO limit. > > Why does contemporary mathematics employ two measures for the infinite? > Why doesn't the left hand know what the right hand (calculus) is doing? There is terminological informality and amphiboly among various treatments of mathematics and even within certain treatments. But, so far, I have not found anything that can't be rectified by settling upon a formal theory with rigorous and uniform terminology. MoeBlee |