Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: stephen on 10 Dec 2006 12:43 Han.deBruijn(a)dto.tudelft.nl wrote: > step...(a)nomail.com schreef: >> Han.deBruijn(a)dto.tudelft.nl wrote: >> >> > Let us say that set theory is half the truth. Within set theory, any >> > function is a special relation between commodities and products, i.e. >> > domain and range. But the production process itself involves _labour_, >> > hence time, and this aspect is not covered by the static set theoretic >> > framework, where functions are reduced to "mappings" between sets. >> >> You can include it. You are just talking about a cost being >> associated with a function. That is trivial to do. > Exactly! In the eye of the capitalist beholder labour is identical with > "cost". And that cost is an easy thing to incorporate. But a fact is > that labour is not cost and not static and it involves time. And time > is not a set. Constructive mathematics is different from mainstream > mathematics, precisely for that sole reason: time. Time excludes actual > infinities as well, because they cannot be "done". > Han de Bruijn You did not provide an example of your mathematics that incorporates time. You seem to be avoiding every question that I ask you. Once again, how do you mathematically describe a production process that involves labour. It should be an easy question. Are you just going to ignore and snip it again? And why are you excluding things? I thought you wanted everyone to be free of restriction. Or is that something only you are entitled to? Stephen
From: mueckenh on 10 Dec 2006 13:26 William Hughes schrieb: > 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. That already depends on what you understand by "a set exists". I suspect that you understand that all its elements exist. > > > 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. I never agreed. oo is not a number, so it is not a cardinal number, it is at most a "cardinal number". > > 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. Correct. But we cannot determine every element of the set. > > > -a bijection can exist between two potentially > infinite sets That again depends on what you understand by "to exist". I suspect that you understand that all its elements exist. Even the bijection between positive and negative integers: f(n) = -n is not complete, because domain and range are not complete. > > -given two potentially infinite sets A and B > the question "Is there a bijection between > A and B?" has an answer which exists. This answer again depends on what you understand by "to exist". > > -a cardinal number is an equivalence class on > sets with respect to the equivalence relation > bijection Yes, that is correct. But it does not satisfy the order-relation "greater than" with natural cardinals. > > -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. That again depends on what you understand by "to exist". The identity function f(n) = n does not provide the existence of all natural numbers. We see it by the fact that otherwise finite (every number including the limit) = infinite (the limit of initial segments of natural numbers). > > -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 neither it is larger than every (or even any) 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 There is no complete diagonal. Regards, WM
From: mueckenh on 10 Dec 2006 13:30 William Hughes schrieb: > 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. The set N can be involved and the quatifier can be changed as noted above as long as it is asured that: 1) every line has a finite number of elements 2) there is no element of the diagonal outside of every line. If you disagree please provide a counter example (with a finite line). Regards, WM
From: mueckenh on 10 Dec 2006 13:33 Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > You cannot imagine the integer [pi*10^10^100]. > > That is not an integer, dummkopf. It is an irrational real number. In any case "Dummkopf" starts with a capital letter, and is applied by uneducated people for uneducated people. i.e., such with little knowledge or experience. By the way, have you never seen expressions like [1.8] = 1, involving Gauss brackets? Do you know Gauss? Regards, WM
From: mueckenh on 10 Dec 2006 13:34
stephen(a)nomail.com schrieb: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > stephen(a)nomail.com wrote: > >> Nobody but you has talked about "growing" sets. Sets, like numbers, do not > >> grow. You, like many other people who do not understand set theory, > >> think of sets as mutable objects, that change as we perform operations > >> on them. This is akin to thinking that numbers change when we perform > >> addition. If I add 3 to 7, neither 3 or 7 changes. > > > It is such an odd belief. Why use a set for something that a function is > > naturally for? I don't really understand why cranks insist on using sets > > for everything, while at the same time insisting that sets are useless > > or illogical or whatever. > > I do not think it is that odd. In everyday usage, the word "set" is used > to denote something that changes. But then again, so is the word "number". > The number of people in a room may change, but that does not imply that > a specific number, such as 5, changes. Only cranks may not know that a number is a set. Only cranks may not know that functions may be defined on sets. > Most people seem to have an abstract > enough concept of number that the common usage does not confuse them. However > they do not apply this abstraction to sets, so if someone says the set of people > in the room changes, they think a specific set changes. A number, according to set theory, is a set. So what is appropriate for numbers is also appropriate for sets. It is appropriate to speak of the number n, not in common use but in mathematical text books. This number is not fixed. Regards, WM |