Cantor Confusion
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From: mueckenh on 18 Nov 2006 08:28 Randy Poe schrieb: > Lester Zick wrote: > > On 16 Nov 2006 11:32:41 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > >David Marcus schrieb: > > > > > >> > > Indeed. If people *object* to an axiom, that is philosophy. > > >> > > > >> > But if people choose a set of axioms, that is what? > > >> > > > >> > > Everyone is welcome to choose their own axioms. > > >> > > > >> > That's mathematics? > > >> > > >> Of course. > > > > > >And if people not decide to use an axiom, that is what? > > > > Mathematics obviously. David says so. > > Any reasonable person would say so. It's mathematics on > a system which is missing that axiom. It's still perfectly good > mathematics. > > Haven't you seen people discuss ZF vs. ZFC for instance? Both are > mathematics. ZFC has the axiom of choice, people working in ZF > choose not to use that axiom. Haven't we seen people discuss the axiom of infinity? Hrbacek and Jech, for instance, report that. But some self proclaimed crank doesn't want to call that mathematics, because then he had to admit that something he did not know is mathematics nevertheless. That is why I asked the silly questions above. Of course all that is mathematics. Regards, WM
From: mueckenh on 18 Nov 2006 08:32 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> > >> http://mathworld.wolfram.com/InitialSegment.html > >> > >> What you call "complete initial segment" is not an *initial* segment > >> but the whole set of lines. > > > > What is a name? > > You should rephrase what you mean. > > > What *counts* is this: It is asserted that the number of natural > > numbers is omega > > "It" is asserted that the *set* of natural numbers is (does exist) and > is named omega. The *cardinality* of omega is aleph_0. The cardinality of omega is omega. It is usual to denote it by aleph_0, but it is allowed to denote it by omega. Even Cantor is said to have done so ... > > > (the elements of the first column = number of lines). > > The ordinal number of lines is omega. Its cardinal number is aleph_0. > Neither omega nor aleph_0 are /natural/ numbers. Neither omega nor > aleph_0 are elements of omega. Period. And there are not enough natural numbers that we could collect omega or aleph_0 of them, although this wrong definition exists. We see the lack by increasing the number of elements and the elements by 1. > > > By adding 1 element to the first column we see that, if this assertion > > is correct, the number increases to omega + 1. > > It depends on how you in the present case define "adding 1 element to > the first column". As pointed out: > > L' := { 0, 1, 2, ..., x } is of ordinal type omega + 1 > L'' := { x, 0, 1, 2, ... } is ("still") of ordinal type omega > > Besides of this | L' | = | L'' | = aleph_0 is valid. > > What "L + 1" do you mean? L' or L''? In all cases I used L'. It should be clear from what I wrote. > > > But are there enough lines? > > There are as many lines as you have defined. > > > No. > > How did you hit on that? > > > That means: The finite natural numbers are not sufficient > > to stand in bijection with all omega elements of the column, > > Here you are: > > B := { <0, 1>, <1, 2>, <2, 3>, ... } > > B is an explicit bijection between the naturals (elements of omega) and > the numbers in the first column. B does not include omega. If omega were only the fact that this bijection does include all natural numbers, then you had no problem with B. But if omega is considered a number which even can be increased, then the idea breaks down and you must assume that there are more natural numbers d_nn than natural numbers n. Regards, WM
From: mueckenh on 18 Nov 2006 08:34 David R Tribble schrieb: > Dik T. Winter wrote: > >> If you want to find absolute truth you should not look at mathematics. > > > > david petry schrieb: > >> Perhaps we should replace "absolute truth" with "culturally neutral > >> truth", or in other words, truth without any cultural, religious, or > >> philosophical bias. [...] Thinking about this question > >> leads most of us to believe that there is a core of mathematics which > >> every such civilization will accept. > > > > mueckenh wrote: > > without axioms, yes. For instance: I + I = II (after translating "+" > > and "="). Therefore I call this an absolute truth. > > Which axioms are you using to describe the "+" and "=" operators? Axioms? For which purpose? Do you think the symbols constituting the words constituting the axioms are easier or clearer to understand than the symbols "+" and "="? Take an apple and then another apple. Show the apples first apart and then together. Repeat with oranges or fingers or mixed objects, possibly. That defines all that is needed. Regards, WM
From: mueckenh on 18 Nov 2006 08:42 William Hughes schrieb: > > I claim case iia: There is a potentially infinite sequence N = > > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > > treat all of its elements. In particular we can never complete this > > set. We can never put it into a list > > OK. Knock yourself out. I read this several times from you. What does it mean? > Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. But it does not say that it has an ordinal number and a cardinal number. My case iia is in complete agreement with the axiom of infinity. I am in a hurry. Perhaps I will give some more comments later. Although the principle positions have been cleared. Regards, WM
From: Franziska Neugebauer on 18 Nov 2006 08:54
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> http://mathworld.wolfram.com/InitialSegment.html >> >> >> >> What you call "complete initial segment" is not an *initial* >> >> segment but the whole set of lines. >> > >> > What is a name? >> >> You should rephrase what you mean. >> >> > What *counts* is this: It is asserted that the number of natural >> > numbers is omega >> >> "It" is asserted that the *set* of natural numbers is (does exist) >> and is named omega. The *cardinality* of omega is aleph_0. > > The cardinality of omega is omega. The cardinality of omega is |omega| not omega. > It is usual to denote it by aleph_0, but it is allowed to denote it by > omega. Being imprecise does not help to clarify things. > Even Cantor is said to have done so ... Irrelevant. >> > (the elements of the first column = number of lines). >> >> The ordinal number of lines is omega. Its cardinal number is aleph_0. >> Neither omega nor aleph_0 are /natural/ numbers. Neither omega nor >> aleph_0 are elements of omega. Period. > > And there are not enough natural numbers that we could collect omega > or aleph_0 of them, In set theory there is no "task" to "collect" natural numbers. Set theory is not about "processes" of collecting numbers. > although this wrong definition exists. You have already been informed about this misconception of yours. > We see the lack by increasing the number of elements and the elements > by 1. Anything new? >> > By adding 1 element to the first column we see that, if this >> > assertion is correct, the number increases to omega + 1. >> >> It depends on how you in the present case define "adding 1 element to >> the first column". As pointed out: >> >> L' := { 0, 1, 2, ..., x } is of ordinal type omega + 1 >> L'' := { x, 0, 1, 2, ... } is ("still") of ordinal type omega >> >> Besides of this | L' | = | L'' | = aleph_0 is valid. >> >> What "L + 1" do you mean? L' or L''? > > In all cases I used L'. It should be clear from what I wrote. As W. Hughes (?) has already pointed out: There is no list (in the original sense, i. e. function with domain omega) L' (i. e. having co-domain L'). >> > But are there enough lines? >> >> There are as many lines as you have defined. >> >> > No. >> >> How did you hit on that? >> >> > That means: The finite natural numbers are not sufficient >> > to stand in bijection with all omega elements of the column, >> >> Here you are: >> >> B := { <0, 1>, <1, 2>, <2, 3>, ... } >> >> B is an explicit bijection between the naturals (elements of omega) >> and the numbers in the first column. > > B does not include omega. You tend to repeat yourself. > If omega were only the fact that this bijection does include all > natural numbers, then you had no problem with B. I have no problem with B and this bijection does include all natural numbers. > But if omega is considered a number which even can be increased, At _your_ own risk you may "consider" omega to "be" anything you want. Mathematically a definition of "increase" would be preferred. > then the idea breaks down and you must assume that there > are more natural numbers d_nn than natural numbers n. Set theory stands like a rock undaunted by WM's delusions. F. N. -- xyz |