Cantor Confusion
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From: William Hughes on 7 Dec 2006 08:06 mueckenh(a)rz.fh-augsburg.de wrote: <statements which make it clear that certain things which were though to be settled are not settled> Terminology: If we say that X exists then we can use X in a proof. On Dec 4 I wrote: You now agree that a potentially infinite set can have a cardinal number and that this cardinal is not a natural number. As your latest post points out, this is not (or no longer) true. Stop me when I make a statement you disagree with We can then discuss this statement before proceding. -a potentially infinite set exists (this leaves open the question of whether the elements of a potentially infinite set exist.) -if we are given x and a potentially infinite set we can determine whether x is an element of the potentially infinite set. -if we are given two potentially infinite sets A and B, we can contstuct a third potentially infinite set C, consisting of ordered pairs (a,b) where a is an element of A, and B is an element of B. -a function between two potentially infinite sets A and B is a potentially infinite set of ordered pairs (a,b) where a is an element of A, and B is an element of B. -a bijection between two potentially infinite sets A and B is a potentially infinite set, C, of ordered pairs (a,b) where a is an element of A, and B is an element of B. and if a_1 and a_2 are different (a_1,b_1) and (a_2,b_2) are elements of C then b_1 and b_2 are different if b_3 is an element of B then there exists a_3 and elment of A such that (a_3,b_3) is an element of C -a bijection can exist between two potentially infinite sets -given two potentially infinite sets A and B the question "Is there a bijection between A and B?" has an answer which exists. -a cardinal number is an equivalence class on sets with respect to the equivalence relation bijection -the equivalence relation bijection can be extended to include potentially infinite sets -given a potentially infinite set A, the set C of ordered pairs (a,a) exists, where C has the property if a is an element of A then (a,a) is an element of C Call C the identity function. C is a bijection on A. -A belongs to an equivalence class with respect to the equivalence relation bijection -A has a cardinal number -the cardinal number of A is not a natural number -given two sets of natural numbers E and F where E is a potentially infinite set, and F has a largest element. there does not exist a bijection between E and F -the diagonal is the potentially infinite set of natural numbers. -every line L has a largest number -there is no bijection between the diagonal and a line L - William Hughes
From: Franziska Neugebauer on 7 Dec 2006 08:07 Han de Bruijn wrote: > Franziska Neugebauer wrote: > >> I don't know any mathematician who "pretends" an ability of >> "knowing" (what does this mean?) every integer. Who "pretends" so? > > Yeah, sure .. Everybody on earth knows what "knowing" means in common > conversation, except mathematicians. WM: "Those who pretend to be able of knowing every integer [...]" are those who "overestimate their capabilities so grossly." The questions is still unanswered: 1. Who pretends to "know" every integer? 2. Does "to know" mean the same as in the sentence "I don't know every person"? F. N. -- xyz
From: William Hughes on 7 Dec 2006 08:17 David Marcus wrote: > William Hughes wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Please clarify a point of nomenclature. > > Do you consider a potentially infinite > > set contianing only finite elements > > (e.g. the natural numbers) to be: > > > > 1. a set ? > > 2. a finite set? > > May I trouble you to restate the definition of "potentially infinite > set" that you are using? There is no settled definition. Infutively, a potentially infinite set is a set that can always be extended. The following statements are accepted by WM - if an entity X satisfies the Peano axioms then X is a potentially infinite set - the natural numbers are a potentially infinite set. - given x and a potentially infinite set S. it makes sense to ask the question "is x an element of S?" -William Hughes
From: Franziska Neugebauer on 7 Dec 2006 08:24 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> > Bob Kolker schrieb: >> >> >> [...] >> >> >> >> Rational numbers are non-genuine. Nowhere in the physical >> >> >> >> world outside of our nervouse systems do they exist. >> >> >> > >> >> >> > The nervous system is a part of the physical world. >> >> >> >> >> >> Hence this issue is to be disussed in a physics or neuro >> >> >> sciences newsgroup. >> >> >> >> >> >> > The integers belong to the nervous system. >> >> >> >> >> >> The nervous system is generally not an issue in mathematics. >> >> > >> >> > That is why many mathematicians overestimate their capabilities >> >> > so grossly. >> >> >> >> Which mathematician overestimates her or his capabilities? >> > >> > Those who pretend to be able of knowing every integer and, >> > therefore, to imagine the whole actually infinite set. >> >> I don't know any mathematician who "pretends" an ability of >> "knowing" (what does this mean?) every integer. Who "pretends" so? >> > Virgil can imagine all natural numbers, he said. imagine: v. tr. 1. To form a mental picture or image of. 2. To think; conjecture: I imagine you're right. 3. To have a notion of or about without adequate foundation; fancy: She imagines herself to be a true artist. know: v. tr. 1. To perceive directly; grasp in the mind with clarity or certainty. (http://www.thefreedictionary.com) "to imagine" is sematically distinct from "to know". If Virgil has said that he could _imagine_ all natural numbers he did not say that he _knew_ every integer. Hence Virgil is not at all a potential candidate to support your utterance "Those who pretend to be able of knowing every integer" "overestimate their capabilities so grossly." Question remains: Who else claims "to be able of knowing every integer"? F. N. -- xyz
From: William Hughes on 7 Dec 2006 08:27
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1165492756.322548.255040(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > The set in the quantifiers you are using is not finite either. The > > > > quantifier are not over a single line, but over the set of natural > > > > numbers. > > > > > > For finite natural numbers, we have finite lines only. It is not the > > > question of a single line. Every line is finite. Therefore there is no > > > line where quantifier reversal could not be applied. > > > > But the quantifier reversal is *not* applied to individual lines, it > > is applied to the set of natural numbers. > > No. > The reversal given was For every natural number n there exists a line L(n), such that every natural number m <= n is an element of L(n) There exists a line L, such that for every natural number n, every natural number m<=n, is contained in L. Note the movement of the phrase "every natural number". Please provide an alternate formulation that does not involve the set of natural numbers. - William Hughes > Regards, WM |