Cantor Confusion
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From: William Hughes on 17 Nov 2006 12:51 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > Make it easier for you to understand it by considering the initial > > > > > segments of the first column. The initial segment > > > > > > > > > > 1 > > > > > 2 > > > > > 3 > > > > > ... > > > > > n > > > > > > > > > > stand in bijection with the line 1,2,3,...,n by the diagonal element > > > > > d_nn. > > > > There is no bijection between the line indexes and > > the initial segments of the first column. There is > > almost a bijection, but not quite. > > Congratulations! After all you got it. This "not quite" is the > observation which is necessary to recognize that there are not actually > infinitely many numbers > i.e., that it is impossible to count the > number of natural numbers by a number omega, which can be increased by > 1. No. This observation would only lead to the conclusion that no bijection exists between the lines and columns if we assume that the initial segment {1,2,3,...} must correspond to a line. > > > Each line index is associated with > > an initial segment of the form {1,2,3,...,n}, that is an > > initial segment that ends. Each initial > > segment that ends is associated to a line index. > > However, there is one initial segment, {1,2,3,...} that does not end. > > This initial segment is not associated with any > > line index. So the fact that there is no bijection > > between the initial segments of the first column > > and the lines, does not mean that there is > > no bijection between the line indexes and > > the column indexes. > > Of course not. It only mans that there is not a number omega of lines. > Equivalently, there is no actually infinite set of finite natural > numbers. > No. It means that the initial sequence {1,2,3,...} does not correspond to any line index. You are arguing there is no bijection therefore there is no actually infinite set of finite natural numbers. which you prove by there is no actually infinite set of finite numbers therefore there is no bijection. A tad circular. You will not show that making the assumption "there is a set of infinitely many finite numbers" leads to a contradiction by using circular arguments. - William Hughes
From: Lester Zick on 17 Nov 2006 13:11 On 16 Nov 2006 13:09:32 -0800, "david petry" <david_lawrence_petry(a)yahoo.com> wrote: > >Dik T. Winter wrote: > >> If you want to find absolute truth you should not look at mathematics. Correction. One should not look at mathematikers who proclaim the irrelevance of truth to mathematics. >Perhaps we should replace "absolute truth" with "culturally neutral >truth", or in other words, truth without any cultural, religious, or >philosophical bias. Sure that'll work. If truth is not absolute, mathematical, physical, cultural, religious, or philosophical it's not really clear what the content of "truth" might be or what could be true. > We can reason about this concept by asking the >question: what will the mathematics of advanced alien civilizations >(i.e. from other planets) look like? Thinking about this question >leads most of us to believe that there is a core of mathematics which >every such civilization will accept. Obviously there can only be such a core if it's true. ~v~~
From: Lester Zick on 17 Nov 2006 13:13 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. >But if people decide not to use an axiom, that is philosophy? Of course. Philosophy(x)=disagree(David). >> > Would like to do. Please le me know which words are available in your >> > universe of discourse. >> >> I told you several times that the terminology in any modern textbook is >> fine. For some reason you do not like this answer. > >I told you the terminology used in a modern textbook to show that >finished infinity is used there. For some reason you do not like to >understand it. > >"Some mathematicians object to the Axiom of Infinity on the grounds >that a collection of objects produced by an infinite process (such as >N) should not be treated as a finished entity." > >Regards, WM > > >> What would be something that is "actually infinite"? > >Read Cantor, he can explain it better than me. > >> > > e. An "infinite number" is a number other than the natural numbers. > >> > An "infinite number" would be a number other than a natural number. > >> Are you agreeing or disagreeing? > >I am astonished that you cannot understand simplest sentences. You seem >to have difficulties with conditional constructs. Should you ever >intend to study mathematics be prepared that such constructs will >appear quite frequently. > >> > If an actually infinite set of numbers existed, and if neighbouring >> > elements had a fixed distance from each other, then the set must >> > contain an infinite number. > >> Is that a "no" or a "yes"? > >Read again, simplified: If neighbouring elements have a fixed distance, >the answer is yes. >If neighbouring elements have not a fixed distance like the rational >numbers: the answer is no > >Regards, WM ~v~~
From: Lester Zick on 17 Nov 2006 13:28 On 16 Nov 2006 11:23:29 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > >Lester Zick schrieb: > > >> Pi is certainly accessible through circular arcs and their diameters. >> Are you suggesting circles and their diameters don't exist? It looks >> to me like you're confusing the accessibility of calculations for >> fractional expansions with the accessibility of the number itself. If >> one ignores geometry and makes arithmetic the sole mathematical >> paradigm of course your conclusion follows. That's the price to be >> paid for exclusive reliance on infinite set analytical techniques as >> exhaustive criteria for the properties of numbers. > >Sqrt(2) does exist as the diagonal of the square. But I call that an >idea in order to distinguish it from numbers which can be written in >lists and can be subject to a diagonal proof. I call only those >entities numbers which can be put in trichotomy with each other. I don't know what a diagonal proof and trichotomy may be. Certainly there is a diagonal proof for the square root of 2 and it would seem to be in a trichotomy with unit sides at right angles. In any event to me this distinction appears a trifle artificial since the square root of 2 certainly bears numerical relations to other numbers. > If you >use the first 10^100 digits of pi as a natural number and exchange the >last digit by 5, then you cannot determine which one is larger. Well that's my whole point. Why would you do that? You're just drawing completely arbitrary distinctions regarding transcendentals based on inexact arithmetic concepts and calculations and then complaining the universe is too small to accommodate all the precision needed. I don't see that reclassifying numbers solves any problem. That's the typical modern math approach, merely a verbal regression. Why not just see the process as one of infinitesimal subdivision of unity instead of the infinite addition of unities and let it go at that? ~v~~
From: Randy Poe on 17 Nov 2006 13:33
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. - Randy |