Cantor Confusion
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From: Virgil on 20 Nov 2006 15:12 In article <1164031883.528592.97420(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > If not all digits of the diagonal number exist, then the diagonal > > > number is not an example for a number which exists but is not in the > > > list. > > > > But there is nothing that prevents any digit from existing, > > Existing are such ideas which are accessible. No others are existing, > because existence of ideas means existence in the mind of someone. That something does not exist in WM's mind does not prevent it from existing in others' minds. > > > > > > > > For example, it is definitely the case that for any given position n, > > > > the nth digit of sqrt(2) is, at least in principle, determinable, > > > > however impractical it might be to carry out such a determination. > > > > > > No, it is clear for everyone who is not a fanatic that in principle and > > > in praxis not every digit of sqrt(2) is determinable. > > > > I did not say "every", I said "any". Does WM claim that there is ANY > > digit in the decimal expansion of sqrt(2) that is not, at least > > theoretically, determinable? > > I know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In praxis there are less digits possible. Therefore the > decimal representation of this number can never be treated in a list. Engineering is about practice. mathematics is about principles. While perhaps one cannot engineer a square root of two, one can imagine it, which is all that mathematics needs.
From: Virgil on 20 Nov 2006 15:23 In article <1164032471.877294.141050(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163939685.919928.156930(a)h54g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > > > Virgil wrote: > > > > > > > > > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > >> Virgil schrieb: > > > > [...] > > > > >> And there are lines as long as the diagonal is. > > > > > > > > > > Name one. > > > > > > The elements of the diagonal are a subset of the line ends. > > > > The SET of elements of the diagonal equals the set of all line ends, but > > that does not name any line as being as long as the diagonal. > > > It is easy to see that for every line in WM's list the diagaonal must > > contain at least one more character than that line. > > Yes. And it is as easy to see that for any character of the diagonal > there exists a line which contains this one and the next one. > > Why do you see only the one side of the medal and dispel that the other > side exists too ? Is it because of the joke with the at least one cow > which is black at least at one side? WM can't even get the joke right. It was a sheep. When WM claims that there exists some line having a property, and I show that every line, without exception, fails to have that property, I have disproved WM's claim. WM claims existence of a line containing every element of the diagonal. I show that for any and every line, there is at least one element of the diagonal not in that line. I have disproved WM's claim. At least in any system governed by standard logic. What sort of logic, if any, governs WM's arguments is no longer clear.
From: Virgil on 20 Nov 2006 15:42 In article <1164034388.582046.269420(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Please learn what "definable well-order" means technically. > > > > We can easily define well-order: any ordering of a set in which every > > non-empty subset has a first element. > > Please learn what "definable well-order" means technically. This is > different from defining the expression "well-order". Why not just give you own definition, as I do not find any in the literature. > > > > > > > > > > > We have no reason to suppose the we could not recognize one if it were > > > > presented for inspection. > > > > > > We cannot recognize it, because it cannot exist. > > > > In ZFC, it must exist. In ZF, one cannot prove it impossible. its > > non-existence in ZF can be no more than an added assumption. > > > > > 1) It cannot be defined by a formula. > > > > Perhaps. Do you have any proof that it cannot be defined by a formula to > > present here? > > That has been proven in the sixties, I think by Cohen. > > > > > 2) It cannot be listed because of cardinality reasons. > > > > I am not quite sure of what you mean by "listed". Would the listing of a > > countable set of general rules constitute a listing? > > A well-order of the real numbers cannot be accomplished by a countable > set of general rules. How do you know? > > > > > 3) It cannot be determined in any other way. (Or do you have an idea?) > > > Hence, we cannot recognize it. > > > > > > > > > > > It is true that no one has actually constructed any explicit well > > > > ordering. > > > > > > > > > > > > It is true that no one will ever have constructed any explicit well > > > ordering. > > > > That is a statement of faith only. > > No. A well-order of the real numbers is not definable. If you mean that there is no way to construct an explicit well-ordering of an uncountable set, that is not what is standardly meant by "undefinable". Your non-standard use of standard terms makes your claims incomprehensible. > A well-order of > the real numbers cannot be given by a list. As Cantor proved. > There is no other means > which could well-order the real numbers. There are well-ordered sets which are not ordered as lists, so that claim requires proof. >To believe that it exists > (where should it exist?) is a certificate of distinguished madness at > an advanced level. To believe that it doesn't to exist in ZFC is a certificate of distinguished incompetence at an advanced level. To believe that it doesn't to exist in ZF is optional. > > > > > > > > > The axiom of infinity does not say anything about ordinal or > > > > > > cardinal numbers. However, given that the set N exists and > > > > > > the defnition of ordinal and cardinal numbers, it is easy to > > > > > > see that if N exists it must have both an ordinal and a cardinal > > > > > > number. > > > > > > > > > > > No. You assume the possibility of a bijection of the set with itself. > > > > > > > > The identity function on any set bijects it with itself. And such > > > > functions are guaranteed, via the axiom of replacement. > > Not without a wrong interpretation of the axiom of infinity. The axiom of infinity has nothing to do with it. > > > > > > > > > > > > > That is not proven from the mere existence of the set if we cannot > > > > > recognize or treat all of its elements. > > > > > > > > It is proven in ZF, even without C. > > > > > > The recognizability of all elements is not proven. > > > > And is not needed if a rule for all cases can be provided, such as > > x |--> x. > > I do not see that any infinite set is covered. One could easily earn doctorates on what WM does not see. > Your x is always finite Maybe yours is, but mine can be omega, among other things. > and the number of numbers x treated is always finite too. So we have > potential infinity but never actual infinity. To speak of "all cases" > is a gross overstatement, unless you agree that "all cases" constitute > a set which is not actually infinite. When in ZF, one has actual infiniteness. When in WM, one has unbridled egotism. > > > > > In ZF, at least, any model of such a "triangle" has infinite diagonal. > > > > >(The diagonal is a subset of the line ends.) > > > > The diagonal contains ALL the infinitely many line ends, so its set of > > positions a SUPERset of the set of line ends. > > The diagonal is a subset and a superset of the line ends. This is just > the proof you denied above. Where? The set of members of the diagonal = the set of all line ends. I have never denied this. What I HAVE said is that for each line, the diagonal contains an element not in that line. WM sometimes sees what he wants to see, regardless of what is actually there to be seen.
From: David Marcus on 20 Nov 2006 15:55 Virgil wrote: > In article <1164034388.582046.269420(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > > > Please learn what "definable well-order" means technically. > > > > > > We can easily define well-order: any ordering of a set in which every > > > non-empty subset has a first element. > > > > Please learn what "definable well-order" means technically. This is > > different from defining the expression "well-order". > > Why not just give you own definition, as I do not find any in the > literature. That would be too simple. > > > > > It is true that no one has actually constructed any explicit well > > > > > ordering. > > > > > > > > It is true that no one will ever have constructed any explicit well > > > > ordering. > > > > > > That is a statement of faith only. > > > > No. A well-order of the real numbers is not definable. > > If you mean that there is no way to construct an explicit well-ordering > of an uncountable set, that is not what is standardly meant by > "undefinable". > > Your non-standard use of standard terms makes your claims > incomprehensible. http://sundials.org/about/humpty.htm -- David Marcus
From: Virgil on 20 Nov 2006 16:22
In article <1164036863.910044.319210(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The existence of a set does not guarantee the existence or definability > > > of a bijection or an identity mapping. > > > > No. > > > > To say that a set N exists is to say that all elements n of N exist. > > Thus the mapping n ->n for all elements n of N exists. > > That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. In what model of ZF does no identity mapping from N to itself exist? > Why should it fail if you were > right? What proof does WM have that there is ANY model of ZF for which no identity mapping from N to N is possible? > > > > Check the definitions of ordinal and cardinal below. Look for > > > > the term trichotomy. Fail to find the term trichotomy. Draw > > > > the obvious conclusion. > > > > > > My conclusion is: You don't know this term. > > > > No, this is not the obvious conclusion (it is also wrong). > > The obvious conclusion is that the > > fact that the term tricotomy is not used when defining > > either the ordinals or the cardinals, means that we > > do not need to show trichotomy to show that > > something is an ordinal or a cardinal. > > It may be called by another name but trichotomy is implied. In fact all > ordinals are asserted to stand in trichotomy with each other. Trichotomy for ordinals and cardinals is a consequence of other properties, and is not an ur-property. It is provable for cardinals in ZFC and for ordinals in ZF, but is a mere consequence of other properties of cardinality/ordinality. |